Quasi M-convex and L-convex Functions - Semantic Scholar

RIMS Preprint No. 1306, Kyoto University

Quasi M-convex and L-convex Functions | Quasiconvexity in Discrete Optimization |3 Kazuo MUROTA

and

Akiyoshi SHIOURA

December, 2000

Keywords:

quasiconvex function, discrete optimization, matroid, base polyhedron Abstract

We introduce two classes of discrete quasiconvex functions, called quasi M-convex and L-convex functions, by generalizing the concepts of M-convexity and L-convexity due to Murota (1996, 1998). We investigate the structure of quasi M-convex and L-convex functions with respect to level sets, and show that various greedy algorithms work for the minimization of quasi M-convex and L-convex functions.

Kazuo MUROTA: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan, [email protected]. Akiyoshi SHIOURA: Department of Mechanical Engineering, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan, [email protected].

3 This work is

supported by Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.

1

2 Contents 1

Introduction

3

2

Review of Fundamental Results on M-convexity/L-convexity

6

3

Quasi M-convex Functions

4

Minimization of Quasi M-convex Functions

22

5

Quasi L-convex and Submodular Functions

28

6

Minimization of Quasi L-convex Functions

35

2.1 De nitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 M-convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 L-convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 3.2 3.3 3.4

De nitions of Quasi M-convex Functions . . . . . . . . . . . . . . . . Level Sets of Quasi M-convex Functions . . . . . . . . . . . . . . . . Operations for Quasi M-convex Functions . . . . . . . . . . . . . . . Characterization of Quasi M-convexity by Local Exchange Properties

. . . .

. . . .

. . . .

... ... ... ...

10

10 14 16 18

4.1 Properties of Minimizers of Quasi M-convex Functions . . . . . . . . . . . . . . 23 4.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 De nition of Quasi L-convex and Submodular Functions . . . . . . . . . . . . . 28 5.2 Level Sets of Quasi L-convex and Submodular Functions . . . . . . . . . . . . . 31 5.3 Operations for Quasi L-convex and Submodular Functions . . . . . . . . . . . . 33

6.1 Properties of Minimizers of Quasi L-convex Functions . . . . . . . . . . . . . . . 35 6.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A Proofs

41

A.1 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1

1

Introduction

3

Introduction

The concept of convexity for sets and functions plays a central role in continuous optimization (or nonlinear programming with continuous variable), and has various applications in the areas of mathematical economics, engineering, operations research, etc. [2, 19, 22]. The importance of convexity relies on the fact that a local minimum of a convex function is also a global minimum. Due to this property, we can nd a global minimum of a convex function by iteratively moving in descent directions, i.e., so-called descent algorithms work for the convex function minimization. Therefore, convexity for a function is a sucient condition for the success of descent methods. Most of descent methods, however, work for a fairly larger class of functions called quasiconvex functions. A function f : Rn ! R [ f+1g is said to be quasiconvex if it satis es f (x + (1 0 )y )  maxff (x); f (y )g (8x; y 2 dom f; 0 < 8 < 1); and semistrictly quasiconvex if it satis es f (x + (1 0 )y ) < maxff (x); f (y )g (8x; y 2 dom f with f (x) 6= f (y); 0 < 8 < 1); where dom f = fx 2 Rn j f (x) < +1g. It is easy to see that convexity implies semistrict quasiconvexity, and semistrict quasiconvexity implies quasiconvexity under the assumption of lower semicontinuity. Although (semistrict) quasiconvexity is a weaker property than convexity, it still has nice properties as follows:  A strict local minimum of a quasiconvex function is also a strict global minimum.  A local minimum of a semistrictly quasiconvex function is also a global minimum.  Level sets of quasiconvex functions are convex sets. Due to these properties, quasiconvexity also plays an important role in continuous optimization. See [1] for more accounts on quasiconvexity. Remark 1.1. In the literature, semistrictly quasiconvex functions above are sometimes called \strictly quasiconvex functions," \explicitly quasiconvex functions," etc. In this paper, we follow the terminology in Avriel et al. [1]. Remark 1.2. A function f : Rn ! R [ f+1g is said to be strictly quasiconvex if it satis es f (x + (1 0 )y ) < maxff (x); f (y)g (8x; y 2 dom f with x 6= y; 0 < 8 < 1): The concept of strictly quasiconvex functions is a generalization of that of strictly convex functions. A strictly quasiconvex function attains its minimum at only one point if the minimum exists. It is clear that any strictly quasiconvex function is semistrictly quasiconvex.

4

Quasi M/L-convex Functions

In the area of discrete optimization, on the other hand, discrete analogues of convexity, or \discrete convexity" for short, have been considered, with a view to identifying the discrete structure that guarantees the success of descent methods, i.e., the so-called \greedy algorithms." Examples of discrete convexity are \discretely-convex functions" by Miller [11], \integrallyconvex functions" by Favati{Tardella [5], and \M-convex and L-convex functions" by Murota [12, 13, 14, 15, 16] as well as their variants called \M\-convex functions" by Murota{Shioura [17] and \L\-convex functions" by Fujishige{Murota [7]. A function f : ZV ! R [ f+1g is called M-convex if dom f = fx 2 ZV j f (x) < +1g 6= ; and f satis es the following property: (M-EXC)

8x; y 2 dom f , 8u 2 supp (x 0 y), 9v 2 supp0(x 0 y) such that +

f (x) + f (y )  f (x 0 u + v ) + f (y + u 0 v );

(1.1)

where supp (x 0 y) = fw 2 V j x(w) > y(w)g, supp0(x 0 y) = fw 2 V j x(w) < y(w)g, and w 2 f0; 1gV is the characteristic vector of w 2 V . A function g : ZV ! R [ f+1g is called L-convex if dom g 6= ; and g satis es the following properties: (SBM) g is submodular, i.e., for all p; q 2 ZV we have +

g (p) + g (q)  g (p ^ q ) + g (p _ q ); (TRF)

9r 2 R such that g(p + 1) = g(p) + r (8p 2 ZV ; 8 2 Z),

where p ^ q; p _ q 2 ZV are de ned by (p ^ q)(w) = minfp(w); q(w)g;

(p _ q)(w) = maxfp(w); q(w)g

(w 2 V ):

M-convex and L-convex functions have various properties as discrete convex functions: (i) A local minimum of an M-convex/L-convex function is also a global minimum. (ii) M-convex/L-convex functions can be extended to ordinary convex functions. (iii) Various duality theorems hold. (iv) M-convex and L-convex functions are conjugate to each other. In particular, the property (i) shows that greedy algorithms work for the M-convex/L-convex function minimization. However, we see from results in continuous optimization that strong properties such as M-convexity/L-convexity are not required for the success of greedy algorithms, and that some properties like \quasi M-convexity/L-convexity" will suce. The main aim of this paper is to introduce the concepts of quasi M-convex and L-convex functions by generalizing those of M-convexity and L-convexity.

1

5

Introduction

Table 1: Possible sign patterns of 1f (x; v; u) and 1f (y; u; v) in (M-EXC) 1f (x; v; u) n 1f (y; u; v) 0 0 0

0

+

2

1 1 1 possible, 2 1 1 1 impossible

+

2 2

To extend the concept of M-convexity to quasi M-convexity, we relax the condition (1.1) while keeping the possible sign patterns of values 1f (x; v; u) = f (x 0 u + v ) 0 f (x) and 1f (y; u; v) = f (y + u 0 v ) 0 f (y) in mind. Table 1 shows the possible sign patterns of those values. Let f : ZV ! R [ f+1g be a function. Then, we call f quasi M-convex if dom f 6= ; and it satis es (QM): (QM) 8x; y 2 dom f , 8u 2 supp (x 0 y ), 9v 2 supp0 (x 0 y ) such that 1f (x; v; u)  0 or 1f (y; u; v)  0. Similarly, we call f semistrictly quasi M-convex if dom f 6= ; and it satis es (SQM): (SSQM) 8x; y 2 dom f , 8u 2 supp (x 0 y ), 9v 2 supp0 (x 0 y ) such that (i) 1f (x; v; u)  0 =) 1f (y; u; v)  0, and (ii) 1f (y; u; v)  0 =) 1f (x; v; u)  0. We introduce the concept of quasi L-convex functions by generalizing the submodularity of functions to quasi-submodularity. We consider two dierent generalizations of the submodularity (SBM): (QSB) For all p; q 2 ZV we have g (p ^ q )  g (p) or g (p _ q)  g (q ). (SSQSB) For all p; q 2 ZV we have both (i) and (ii): (i) g(p _ q)  g(q) =) g(p ^ q)  g(p), and (ii) g(p ^ q)  g(p) =) g(p _ q)  g(q). We call a function g : ZV ! R [f+1g quasi-submodular (resp. semistrictly quasi-submodular) if it satis es (QSB) (resp. (SSQSB)). We de ne a quasi L-convex (resp. semistrictly quasi L-convex) function as a function g : ZV ! R [ f+1g with dom g 6= ; satisfying (QSB) (resp. (SSQSB)) and (TRF). Remark 1.3. The condition (SSQSB) was introduced by Milgrom{Shannon [10], in which a function g : ZV ! R [f+1g is called quasi-supermodular if the function 0g satis es (SSQSB) above. We adopt the terminology \semistrict quasi-submodularity" for the property (SSQSB) in view of our results shown in Section 5. +

+

6

Quasi M/L-convex Functions

The organization of this paper is as follows. We rst review some fundamental results on M-convex and L-convex functions in Section 2. In Sections 3 and 5, we show some properties of level sets of quasi M-convex/L-convex functions and prove that the classes of quasi Mconvex/L-convex functions are closed under various fundamental operations. These results justify the de nitions of quasi M-convex/L-convex functions. Finally, we show that various greedy algorithms work for the minimization of (semistrictly) quasi M-convex/L-convex functions in Sections 4 and 6. We also show proximity theorems on (semistrictly) quasi M-convex/L-convex functions, which guarantee the applicability of the so-called \scaling technique" to the quasi M-convex/L-convex function minimization. The concepts of M\-convex functions by Murota{Shioura [17] and L\-convex functions by Fujishige{Murota [7] can be also extended to quasi M\-convex and L\-convex functions, and the results in this paper can be restated in obvious ways in terms of quasi M\-convex and L\-convex functions. 2

Review of Fundamental Results on M-convexity/Lconvexity

2.1 De nitions and Notation

We denote by R the set of reals, and by Z the set of integers. Also, we denote by R the set of positive reals. For any nite set X , its cardinality is denoted by jX j. Throughout this paper, we assume that V is a nonempty nite set with jV j = n(> 0). The characteristic vector of a subset X  V is denoted by X (2 f0; 1gV ), i.e., ++

X (w) =

(

1 (w 2 X ); 0 (w 2 V n X ):

In particular, we use the notation 0 = ;, 1 = V . Let x = (x(w) j w 2 V ) 2 RV . We de ne supp (x) = fv 2 V j X x(v) > 0g; kxk = jx(v)j; v 2V X hp; xi = p(v)x(v) (p 2 RV ); +

1

v 2V

supp0 (x) = fv 2 V j x(v) < 0g; kxk1 = max jx(v)j; v 2V X x(v) (X  V ): x(X ) = v 2X

For any p; q 2 RV , p ^ q and p _ q denote the vectors in RV such that (p ^ q)(w) = minfp(w); q(w)g; (p _ q)(w) = maxfp(w); q(w)g

(w 2 V ):

2

7

Review on M/L-convexity

For a : V ! Z [ f01g and b : V ! Z [ f+1g with a(v)  b(v) (v 2 V ), we de ne the interval [a; b] ( ZV ) by [a; b] = fx 2 ZV j a(v)  x(v)  b(v) (v 2 V )g: Let f : ZV ! R [ f+1g. The eective domain dom f of f is de ned by dom f = fx 2 ZV j f (x) < +1g: We denote by arg min f the set of the minimizers of f , i.e., arg min f = fx 2 ZV j f (x)  f (y) (8y 2 ZV )g: For any  2 R [ f+1g, the level set L(f; ) is de ned by L(f; ) = fx 2 ZV j f (x)  g: Note that arg min f = L(f; inf f ) and dom f = L(f; +1) are special cases of level sets. For any vector p 2 RV , the function f [p] : ZV ! R [ f+1g is given by f [p](x) = f (x) +

X

v 2V

p(v )x(v)

(x 2 ZV ):

(2.1)

For a set S  ZV , the indicator function S : ZV ! f0; +1g of S is given by S (x) =

(

0 (x 2 S ); +1 (x 26 S ):

We de ne (semistrict) quasiconvexity for functions ' : Z ! R [ f+1g in the following way: we call a function ' quasiconvex if it satis es '( )  maxf'(1 ); '(2 )g

(8 ;  ; 2 Z with  1

2

1

< < 2 );

(2.2)

and semistrictly quasiconvex if it is a quasiconvex function and satis es '( ) < maxf'(1 ); '(2 )g

(8 ;  ; 2 Z with  1

2

1

< < 2 ; '(1) 6= '(2 )):

(2.3)

For a function f : Rn ! R [ f+1g, semistrict quasiconvexity implies quasi convexity under the assumption of lower semicontinuity [1, 2]. For a function ' : Z ! R [ f+1g, on the other hand, the property (2.3) alone does not imply the quasiconvexity in general. It is convenient for our subsequent development to assume quasiconvexity in the de nition of semistrict quasiconvexity for '.

Remark 2.1.

8

Quasi M/L-convex Functions

Let ' : Z ! R [ f+1g. (i) ' is quasiconvex () for any 1; 2 2 dom ' with 1 < 2 we have

Theorem 2.2.

minf'( + 1); '( 0 1)g  maxf'( ); '( )g: 1

2

1

2

(ii) Under quasiconvexity (2.2), the condition (2.3) for ' is equivalent to the following condition: minf'( + 1); '( 0 1)g < maxf'( ); '( )g (8 ;  2 dom ' with  <  ; '( ) 6= '( )): (iii) ' is semistrictly quasiconvex () for any  ;  2 dom ' with  <  we have both '( + 1)  '( ) =) '( 0 1)  '( ), and '( 0 1)  '( ) =) '( + 1)  '( ). Proof. See Appendix. A function ' : R ! R [ f+1g is said to be nondecreasing if '()  '( ) holds for all ; 2 R with  < , and strictly increasing if for all ; 2 R with  < we have either '() < '( ) or '() = '( ) = +1. For a set T , a total order on T , denoted by , is a binary relation satisfying the conditions (i) 8a 2 T , a  a, (ii)a  b, b  a =) a = b, (iii) a  b, b  c =) a  c, and (iv) 8a; b 2 T , a  b or b  a. We call such a pair (T; ) a totally ordered set. For a; b 2 T , we denote a  b if b  a, and a  b if a  b and a 6= b. For the set of real vectors Rn , the lexicographic order is the total order  de ned as follows: for a; b 2 Rn, a  b if either a = b or there exists some k 2 f1; 2; 1 1 1 ; ng such that ai = bi for i = 1; 1 1 1 ; k 0 1 and ak < bk . 1

2

1

1

2

2

1

1

1

1

2

2

2

2

1

1

2

2

2

2

1

1

lex

lex

2.2 M-convex Functions A function f : ZV ! R [ f+1g is called M-convex if dom f 6= ; and f satis es the following

property:

(M-EXC)

8x; y 2 dom f , 8u 2 supp (x 0 y), 9v 2 supp0(x 0 y) such that f (x) + f (y )  f (x 0 u + v ) + f (y + u 0 v ): (2.4) +

For any x 2 dom f and u; v 2 V , the directional dierence of f at x w.r.t. u and v by 1f (x; u; v) = f (x + u 0 v ) 0 f (x): Then, the inequality (2.4) can be rewritten as follows in terms of directional dierences: 1f (x; v; u) + 1f (y; u; v)  0: M-convex functions can be characterized by the following (seemingly) weaker property:

2

9

Review on M/L-convexity

8x; y 2 dom f with x 6= y, 9u 2 supp (x 0 y), 9v 2 supp0 (x 0 y) satisfying (2.4).

(M-EXCw )

+

Theorem 2.3 ([13, Th. 3.1]).

For f

: ZV ! R [ f+1g, we have (M-EXC) () (M-

EXC ). We also de ne the set version of M-convexity as follows. A set B  ZV is called M-convex if B 6= ; and it satis es (B-EXC) 8x; y 2 B , 8u 2 supp (x0y ), 9v 2 supp0 (x0y) such that x0u +v 2 B and y + u 0 v 2 B . An M-convex set is nothing but (the set of integral vectors in) an integral base polyhedron [6]. It is easy to see that  B  ZV satis es (B-EXC) () B satis es (M-EXC),  f : ZV ! R [ f+1g satis es (M-EXC) =) dom f satis es (B-EXC). For x 2 B and u; v 2 V , the exchange capacity associated with x, v and u is de ned as ~cB (x; v; u) = maxf 2 R j x + (v 0 u) 2 Bg: M-convex sets can be characterized also by the following (seemingly) weaker property: (B-EXCw ) 8x; y 2 B with x 6= y , 9u 2 supp (x 0 y ), 9v 2 supp0 (x 0 y ) such that x 0 u + v 2 B and y + u 0 v 2 B . Theorem 2.4 ([23]). For B  ZV , we have (B-EXC) () (B-EXC ). A variant of M-convex functions, called M\-convex function, was introduced in [17]. A function f : ZV ! R [ f+1g is called M\-convex if the function f~ : ZV ! R [ f+1g de ned by ( f (x) (x = 0x(V )); ((x ; x) 2 ZV ) f~(x ; x) = +1 (x 6= 0x(V )); is an M-convex function, where V~ = fv g [ V . M\-convex functions are essentially equivalent to M-convex functions, whereas the class of M\-convex functions properly contains that of M-convex functions; i.e., M\n ' Mn ; Mn  M\n ; where Mn (resp. M\n ) denotes the class of M-convex (resp. M\-convex) functions de ned over Zn . w

+

+

w

~

~

0

0

0

0

0

+1

10

Quasi M/L-convex Functions

2.3 L-convex Functions Let g : ZV ! R [ f+1g. We call g submodular if it satis es the property (SBM):

(p; q 2 ZV ). A function g is called L-convex if dom g 6= ; and it satis es (SBM) and (TRF): (TRF) 9r 2 R such that g (p + 1) = g (p) + r (8p 2 dom g; 8 2 Z). We also de ne the set version of L-convexity as follows. A set D  ZV is called L-convex if D 6= ; and it satis es (DL) and (TRS): (DL) p; q 2 D =) p ^ q; p _ q 2 D, (TRS) p 2 D;  2 Z =) p + 1 2 D . It is easy to see that  D  ZV satis es (DL) (resp. (TRS)) () D satis es (SBM) (resp. (TRF)),  g : ZV ! R [ f+1g satis es (SBM) (resp. (TRF)) =) dom g satis es (DL) (resp. (TRS)). A variant of L-convex functions, called L\-convex function, was introduced in [7]. A function g : ZV ! R [ f+1g is called L\-convex if the function g~ : ZV ! R [ f+1g de ned by (SBM)

g (p) + g (q )  g (p ^ q ) + g (p _ q )

~

g~(p0; p) = g (p 0 p0 1)

((p ; p) 2 ZV ) ~

0

is L-convex, where V~ = fv g [ V . We see that L\-convex functions are essentially the same as L-convex functions, while the class of L\-convex functions properly contains that of L-convex functions; i.e., 0

L\n ' Ln ; +1

Ln  L\n ;

where Ln (resp. L\n) denotes the class of L-convex (resp. L\-convex) functions de ned over Zn . 3

Quasi M-convex Functions

3.1 De nitions of Quasi M-convex Functions Let f : ZV ! R [ f+1g. Then, we call f quasi M-convex if dom f 6= ; and it satis es (QM): (QM)

8x; y 2 dom f , 8u 2 supp (x 0 y), 9v 2 supp0 (x 0 y) such that 1f (x; v; u)  0 or 1f (y; u; v)  0. +

3

11

Quasi M-convex Functions

Similarly, we call f semistrictly quasi M-convex if dom f 6= ; and it satis es (SSQM): (SSQM)

8x; y 2 dom f , 8u 2 supp (x 0 y), 9v 2 supp0(x 0 y) such that (i) 1f (x; v; u)  0 =) 1f (y; u; v)  0, and (ii) 1f (y; u; v)  0 =) 1f (x; v; u)  0. +

Note that (SSQM) can be rewritten as follows: 8x; y 2 dom f , 8u 2 supp (x 0 y), 9v 2 supp0(x 0 y) satisfying at least one of the following: (i) 1f (x; v; u) < 0, (ii) 1f (y; u; v) < 0, (iii) 1f (x; v; u) = 1f (y; u; v) = 0. +

We also consider weaker properties than (QM) and (SSQM): (QMw )

8x; y 2 dom f with x 6= y, 9u 2 supp (x 0 y), 9v 2 supp0 (x 0 y) such that 1f (x; v; u)  0 or 1f (y; u; v)  0. +

8x; y 2 dom f with x 6= y, 9u 2 supp (x 0 y), 9v 2 supp0 (x 0 y) such

(SSQMw )

+

that

(i) 1f (x; v; u)  0 =) 1f (y; u; v)  0, (ii) 1f (y; u; v)  0 =) 1f (x; v; u)  0.

and

The set version of quasi M-convexity can be obtained by translating the properties (QM) and (QM ) for the indicator function B : ZV ! f0; +1g of a set B  ZV in terms of B. w

(Q-EXC)

8x; y 2 B, 8u 2 supp (x 0 y), 9v 2 supp0(x 0 y) such that x 0 u + v 2 B or y + u 0 v 2 B .

(Q-EXCw )

+

8x; y 2 B with x 6= y, 9u 2 supp (x 0 y), 9v 2 supp0(x 0 y) such that x 0 u + v 2 B or y + u 0 v 2 B . +

It may be noted that the properties (Q-EXC) and (Q-EXCw ) are labeled (EXC) and (EXCw ) in [21], respectively. The following properties for B  ZV can be shown easily from the fact that 1B (x; v; u) 2 f0; +1g for x 2 B and u; v 2 V .  (Q-EXC ) for B () (QM ) for B ,  (Q-EXC) for B () (QM) for B ,  (B-EXC) for B () (SSQM) for B () (SSQM ) for B . w

w

w

We show some examples of quasi M-convex functions below.

12

Quasi M/L-convex Functions

Let ' : Z ! R [ f+1g. We de ne f : Z ! R [ f+1g by ( '(x ) (x + x = 0); (3.1) f (x ; x ) = +1 (x + x 6= 0): By Theorem 2.2, f satis es (QM) (or (QMw )) if and only if ' is quasiconvex, and f satis es (SSQM) (or (SSQMw )) if and only if ' is semistrictly quasiconvex. Example 3.2. Let f : ZV ! R [ f+1g be an M-convex function, and ' : R ! R [ f+1g be a nondecreasing function. We de ne the function f~ : ZV ! R [ f+1g by ( '(f (x)) (x 2 dom f ); (3.2) f~(x) = +1 (x 62 dom f ): Then, f~ satis es (QM). Furthermore, if ' is strictly increasing, then f~ satis es (SSQM). Example 3.3. Let B  ZV be an M-convex set, p 2 RV , and  2 R. Then, the set S = fx 2 B j hp; xi  g satis es (Q-EXC). Moreover, the function f : ZV ! R [f+1g with dom f = S de ned by f (x) = hp; xi (x 2 S ) satis es (SSQM). Remark 3.4. The concept of (semistrict) quasi M-convexity can be naturally extended to functions f : S ! T with S  ZV and a totally ordered set T with total order . For example, the property (SSQM) is rewritten for such functions as follows: 8x; y 2 S , 8u 2 supp (x 0 y), 9v 2 supp0 (x 0 y) such that (i) if either x 0 u + v 62 S , or x 0 u + v 2 S and f (x 0 u + v )  f (x), then y + u 0 v 2 S and f (y + u 0 v )  f (y), and (ii) if either y + u 0 v 62 S , or y + u 0 v 2 S and f (y + u 0 v )  f (y), then x 0 u + v 2 S and f (x 0 u + v )  f (x). It is easy to see that the properties of (semistrictly) quasi M-convex functions shown in this paper still hold true. For simplicity and convenience, however, we assume in this paper that the codomain of a function is R [ f+1g. Example 3.5. Suppose that V = f1; 2; 1 1 1 ; ng (n  1). Let a : V ! Z [ f01g, b : V ! P P Z [f+1g, and  2 Z satisfy a(i)  b(i) (i 2 V ) and i2V a(i)    i2V b(i). For i 2 V , let fi : [a(i); b(i)] ! R be a semistrictly quasiconvex function. We de ne B  ZV and f : B ! RV by B = fx 2 [a; b] j x(V ) = g; f (x) = (fi (x(i)) j i 2 V ) (x 2 B ); where the total order  on the codomain RV of f is de ned by the lexicographic order. Then, f satis es (SSQM) in the extended sense (see Remark 3.4). 2

Example 3.1.

1

+

2

1

1

2

1

2

3

13

Quasi M-convex Functions

Let x; y 2 B be distinct vectors. Also, let u 2 supp (x 0 y), v 2 supp0 (x 0 y) be any elements, and w.l.o.g. assume that u < v. Then, we have x 0 u + v 2 B and y + u 0 v 2 B. If fu (x(u) 0 1) < fu (x(u)) or fu (y (u)+1) < fu (y(u)) holds, then we have f (x 0 u + v )  f (x) or f (y + u 0 v )  f (y ). Otherwise, we have fu(x(u) 0 1) = fu (x(u)) and fu(y (u)+1) = fu (y (u)) by Theorem 2.2. If fv (x(v) + 1) < fv (x(v)) or fv (y(v) 0 1) < fv (y(v)) holds, then we have f (x 0 u + v )  f (x) or f (y + u 0 v )  f (y ). Otherwise, we have fv (x(v)+1) = fv (x(v )) and fv (y(v) 0 1) = fv (y(v )), from which follows f (x 0 u + v ) = f (x) and f (y + u 0 v ) = f (y ). The relationship among various quasi M-convexity for sets and functions is summarized as follows. Note that the claim (i) of Theorem 3.6 is already shown in [21, Remark 11]. Theorem 3.6. Let S  ZV and f : ZV ! R [ f+1g. Then, we have (i) (B-EXC) =) (Q-EXC) m + (B-EXC ) =) (Q-EXC ). (ii) (M-EXC) =) (SSQM) =) (QM) m + + (M-EXC ) =) (SSQM ) =) (QM ). Remark 3.7. The converses of the statements \(B-EXC ) =) (Q-EXC)" and \(Q-EXC) =) (Q-EXC )" do not hold in general (see [21, Remark 11]). This fact shows that neither of the implications \(QM) =) (SSQM )" and \(QM ) =) (QM)" hold. In the following, we present several examples to show that implications not mentioned in Theorem 3.6 (ii) do not hold in general. [(SSQM ) 6) (QM)] The function f : Z ! R [ f+1g given by dom f = f(1; 1; 1; 0; 0; 0); (0; 1; 1; 1; 0; 0); (0; 0; 1; 1; 1; 0); (0; 0; 0; 1; 1; 1)g; f (x) = x + x + x (x 2 dom f ) satis es (SSQM ) and not (QM) since dom f does not satisfy (Q-EXC) (see Theorem 3.11 below). [(SSQM) 6) (M-EXC)] The function f : Z ! R [ f+1g given by dom f = f(0; 0); (1; 01); (2; 02)g; f (0; 0) = 0; f (1; 01) = 2; f (2; 02) = 3 satis es (SSQM) and not (M-EXC). The property (QM ) is equivalent to each of the following (seemingly) weaker conditions. maxff (x); f (y)g  u2 min ff (x 0 u + v ); f (y + u 0 v )g + x0y 0 x0y (8x; y 2 dom f with x 6= y); (3.3) v2 f (x 0 u + v ) (8x; y 2 dom f with f (x)  f (y)): (3.4) f (x)  min + x0y u2 Proof.

+

w

w

w

w

w

w

w

w

w

w

6

1

1

1

1

2

3

1

w

1

2

2

2

2

w

supp ( supp (

supp (

)

v2supp0 (x0y )

) )

2

2

14

Quasi M/L-convex Functions

: ZV ! R [ f+1g, we have (QM ) () (3.3) () (3.4). Proof. It is easy to see that (3.4) implies both (QM ) and (3.3). Hence, we prove \(QM ) =) (3.4)" and \(3.3) =) (3.4)" below. Suppose that f : ZV ! R [ f+1g satis es (QM ) or (3.3). Let x; y 2 dom f be vectors such that f (x)  f (y). We show by induction on the value jjx 0 yjj that there exist some u 2 supp (x 0 y) and v 2 supp0 (x 0 y ) such that 1f (x; v; u)  0. We may assume jjx 0 yjj > 2, since otherwise the claim holds obviously. Suppose rst that f satis es (QM ). Then, there exist some u 2 supp (x 0 y) and v 2 supp0(x 0 y) such that 1f (x; v; u)  0 or 1f (y; u; v)  0. If the latter holds, then we have f (x)  f (y0 ) for y0 = y + u 0 v and jjx 0 y 0jj < jjx 0 yjj . Hence, the inductive hypothesis yields 1f (x; v0; u0)  0 for some u0 2 supp (x 0 y0)  supp (x 0 y) and v0 2 supp0(x 0 y0)  supp0(x 0 y). We next suppose that f satis es (3.3). Then, there exist some u 2 supp (x 0 y) and v 2 supp0 (x 0 y ) such that 1f (x; v; u)  0 or f (y + u 0 v )  f (x). If the latter holds, then we have f (x)  f (y0) for y0 = y + u 0 v and jjx 0 y0jj < jjx 0 yjj . Hence, the inductive hypothesis yields 1f (x; v0; u0)  0 for some u0 2 supp (x 0 y0)  supp (x 0 y) and v 0 2 supp0(x 0 y0 )  supp0(x 0 y). Theorem 3.8.

For f

w

w

w

w

1

+

1

+

w

1

1

+

+

+

1

1

+

+

3.2 Level Sets of Quasi M-convex Functions

We show various properties of level sets of quasi M-convex functions. The following two theorems claim that level sets of quasi M-convex functions have quasi M-convexity. Furthermore, the weaker version of quasi M-convexity (QM ) for functions can be characterized by quasi M-convexity (Q-EXC ) of level sets. Lemma 3.9 ([21]). Let B  ZV . (i) If B satis es (Q-EXC ), then x(V ) = y(V ) for all x; y 2 dom f . (ii) (Q-EXC ) is equivalent to the following property: (Q-EXCw+ ) 8x; y 2 B , x 6= y , 9u 2 supp (x 0 y ); 9v 2 supp0 (x 0 y ) such that x 0 u + v 2 B . Theorem 3.10. A function f : ZV ! R [ f+1g satis es (QM ) if and only if the level set L(f; ) satis es (Q-EXC ) for all  2 R [ f+1g. In particular, if f satis es (QM ), then dom f and arg min f satisfy (Q-EXC ). Proof. [\only if" part] Let  2 R [f+1g, and x; y 2 L(f; ) be vectors with x 6= y. Applying (QM ) to x and y, we have 1f (x; v; u)  0 or 1f (y; u; v)  0 for some u 2 supp (x 0 y) and v 2 supp0 (x 0 y ). Therefore, we have x 0 u + v 2 L(f; ) or y + u 0 v 2 L(f; ). w

w

w

w

+

w

w

w

w

w

+

3

15

Quasi M-convex Functions

[\if" part] Let x; y 2 dom f be distinct vectors, and assume f (x)  f (y). By Lemma 3.9 (ii), the level set L(f; f (x)) satis es (Q-EXC ), from which follows x 0 u + v 2 L(f; f (x)) for some u 2 supp (x 0 y) and v 2 supp0(x 0 y), i.e., f (x 0 u + v )  f (x) holds. Theorem 3.11. Let f : ZV ! R [ f+1g satisfy (QM). Then, the level set L(f; ) satis es (Q-EXC) for all  2 R [ f+1g. In particular, dom f and arg min f satisfy (Q-EXC). Proof. The proof is similar to that for the \only if" part of Theorem 3.10. Theorem 3.12. Let f : ZV ! R [f+1g. Suppose that the level set L(f; ) satis es (B-EXC) for all  2 R [ f+1g. Then, f satis es (QM). Remark 3.13. The converse of the statement of Theorem 3.11 does not hold in general, as shown in the following example. Let V = fa; b; c; dg, and f : ZV ! R [ f+1g be a function given by dom f = fx 2 f0; 1gV j xa + xb + xc + xd = 2g; f (1; 1; 0; 0) = 1; f (1; 0; 1; 0) = f (1; 0; 0; 1) = 2; f (0; 0; 1; 1) = 3; f (0; 1; 0; 1) = f (0; 1; 1; 0) = 4: We can easily check that for any  2 R [ f+1g the level set L(f; ) satis es (Q-EXC). Let x = (1; 1; 0; 0), y = (0; 0; 1; 1), and u = b 2 supp (x 0 y ). Then, for any v 2 supp0 (x 0 y ) = fc; dg we have 2 = f (x 0 u + v ) > f (x) = 1; 4 = f (y + u 0 v ) > f (y) = 3: Hence, (QM) does not hold for f . Note that the level set L(f; 3) does not satisfy (B-EXC). Remark 3.14. A function does not necessarily satisfy (SSQM ) even if every level set satis es (B-EXC), as shown in the following example. Let f : Z ! R [ f+1g be a function given by dom f = f(0; 0); (1; 01); (2; 02)g; f (0; 0) = 0; f (1; 01) = f (2; 02) = 1: Every level set of f satis es (B-EXC), but (SSQM ) does not holds for f . Theorem 3.15. If f : ZV ! R [ f+1g satis es (SSQM ), then arg min f satis es (B-EXC). An M-convex function can be characterized by quasi M-convexity of level sets of functions perturbed by linear functions. Recall the de nition of a function f [p] in (2.1). Theorem 3.16 ([21, Th. 1]). Let f : ZV ! R [ f+1g. Then, f satis es (M-EXC) () 8p 2 RV , 8 2 R [ f+1g, L(f [p]; ) satis es (Q-EXC) () 8p 2 RV , 8 2 R [ f+1g, L(f [p]; ) satis es (Q-EXC ). Combining Theorems 3.10 and 3.16, we see the following: Corollary 3.17. Let f : ZV ! R [ f+1g. Then, f satis es (M-EXC) () 8p 2 RV , f [p] satis es (QM) () 8p 2 RV , f [p] satis es (QM ). w+

+

+

w

2

w

w

w

w

16

Quasi M/L-convex Functions

3.3 Operations for Quasi M-convex Functions

The class of (semistrictly) quasi M-convex functions is closed under several fundamental operations. Let f : ZV ! R [ f+1g. For any subset U  V , de ne fU : ZU ! R [ f+1g by fU (y ) = f (y; 0V nU ) (y 2 ZU ); where 0V nU 2 ZV nU denotes the zero vector. For any functions a : V ! Z [ f01g and b : V ! Z [ f+1g, de ne fab : ZV ! R [ f+1g by ( f (x) (x 2 [a; b]); (3.5) fab (x) = +1 (otherwise): Theorem 3.18. Let (3QM3 ) denote one of (QM), (QM ), (SSQM), and (SSQM ), and f : ZV ! R [ f+1g be a function with the property (3QM3 ). (i) For any a 2 ZV and  > 0, the functions  1 f (a 0 x) and  1 f (a + x) satisfy (3QM3) as a w

w

function in x. (ii) For any U  V , the function fU : ZU ! R [ f+1g satis es (3QM3). (iii) For any a : V ! Z [ f01g and b : V ! Z [ f+1g with a  b, the function fab : ZV ! R [ f+1g satis es (3QM3 ). (iv) Let fi : ZVi ! R++ [ f+1g (i = 1; 2) be functions with (3QM3). Then, the function f : ZV1 2 ZV2 ! R++ [ f+1g de ned by

f (x1 ; x2 ) = f1(x1 )f2 (x2 )

((x ; x ) 2 ZV1 2 ZV2 ) 1

2

satis es (3QM3 ). Proof. We prove (iv) only. We consider the case when (3QM3) = (SSQM). Let x = (x1 ; x2 ); y = (y1; y2) 2 dom f1 2 dom f2, and let u 2 supp+ (x 0 y), where u 2 supp+ (x1 0 y1) w.l.o.g. Then, there exists v 2 supp0 (x1 0 y1) such that

1f (x ; v; u)  0 =) 1f (y ; u; v)  0; and 1f (y ; u; v)  0 =) 1f (x ; v; u)  0: This implies that 1f (x; v; u)  0 =) 1f (y; u; v)  0; and 1f (y; u; v)  0 =) 1f (x; v; u)  0: Hence, (SSQM) holds for f . Remark 3.19. The class of (semistrictly) quasi M-convex functions is not closed under addition; in particular, it is not closed under the addition of a linear function, as shown in the following example. 1

1

1

1

1

1

1

1

3

17

Quasi M-convex Functions

Let ' : Z ! Z be a function such that '() =

(

3 ( < 0);  (  0);

and de ne f : Z ! R [ f+1g by (3.1). It is easy to see that f satis es (SSQM) (and not (M-EXC)). We also de ne a linear function g : Z ! R [ f+1g by 2

2

g (x1 ; x2 ) =

(

02x (x + x = 0); +1 (x + x 6= 0): 1

1

2

1

2

Then, we have 8 >
: +1 (x + x 6= 0); which satis es neither (SSQM) nor (QM ). Theorem 3.20. For f : ZV ! R [f+1g and ' : R ! R [f+1g, de ne f~ : ZV ! R [f+1g 1

2

1

2

1

1

2

1

1

1

2

1

1

2

w

by (3.2). (i) If f satis es (QM) (resp. (QMw )) and ' is nondecreasing, then f~ satis es (QM) (resp. (QMw )). (ii) If f satis es (SSQM) (resp. (SSQMw )) and ' is strictly increasing, then f~ satis es (SSQM) (resp. (SSQMw )).

A quasi M-convex function f~ : ZV ! R [ f+1g is not necessarily represented in the form (3.2) with an M-convex function f : ZV ! R [f+1g and a nondecreasing function ' : R ! R [ f+1g. As an example, let us consider a function f~ : Z ! R [ f+1g given by dom f~ = f(0; 0; 0); (1; 0; 01); (2; 0; 02); (2; 1; 03)g; f~(x ; x ; x ) = 02x + x (x 2 dom f~); which satis es (SSQM). Suppose that f~(x) = '(f (x)) (x 2 ZV ). Since ' is nondecreasing, we must have

Remark 3.21.

3

1

2

3

1

2

f (2; 0; 02) < f (2; 1; 03) < f (1; 0; 01) < f (0; 0; 0) < f (1; 1; 02);

which implies f (0; 0; 0) + f (2; 1; 03) < f (1; 0; 01) + f (1; 1; 02):

Hence, (M-EXC) for f does not hold for x = (0; 0; 0) and y = (2; 1; 03).

18

Quasi M/L-convex Functions

: ZV ! R [ f+1g and g : ZV ! R [ f01g be functions such that g(x) > 0 for all x 2 dom f . If the function f (1) 0 g(1) satis es (QM ) for all  2 R [ f+1g, then the function r : ZV ! R [ f+1g given by ( f (x)=g(x) (x 2 dom f ); r(x) = +1 (x 62 dom f ); also satis es (QM ). Proof. The proof is easy from Theorem 3.10. Remark 3.23. The following example shows that the statement of Theorem 3.22 cannot be strengthened by replacing (QM ) with (QM), even if f and g are linear functions. Let V = fa; b; c; dg. De ne a function r : ZV ! R [ f+1g as dom r = fx 2 f0; 1gV j xa + xb + xc + xd = 2g; r(x) = xa +2xxb++x3x+c +x 3xd (x 2 dom r): a c d Let x = (1; 1; 0; 0), y = (0; 0; 1; 1), and u = b 2 supp (x 0 y). Then, for v 2 supp0(x 0 y) = fc; dg we have Theorem 3.22.

Let f

w

w

w

+

4=3 = f (x 0 u + v ) > f (x) = 1; 4 = f (y + u 0 v ) > f (y) = 3: Therefore, (QM) does not hold for r. 3.4 Characterization of Quasi M-convexity by Local Exchange Properties

An M-convex function is known to be characterized by a localized version of the property (M-EXC): (M-EXC-loc) 8x; y 2 dom f with jjx0y jj = 4, 8u 2 supp (x0y ), 9v 2 supp0 (x 0 y ) such that 1

+

f (x) + f (y )  f (x 0 u + v ) + f (y + u 0 v ): Let f : ZV ! R [ f+1g be a function such that dom f is a nonempty set with (Q-EXCw ). Then, (M-EXC) () (M-EXC-loc). Theorem 3.24 ([13, Th. 3.1], [21, Th. 2]).

As a corollary, we also have a characterization of an M-convex set by a local exchange property: (B-EXC-loc) 8x; y 2 dom f with jjx0y jj = 4, 8u 2 supp (x0y ), 9v 2 supp0 (x 0 y ) such that x 0 u + v 2 B and y + u 0 v 2 B . 1

+

3

19

Quasi M-convex Functions

(Q-EXC ). Then, (B-EXC) () (B-EXC-loc). We show that semistrict quasi M-convexities can be characterized also by the localized version of (SSQM) and (SSQM ). (SSQM-loc) 8x; y 2 dom f with jjx0y jj = 4, 8u 2 supp (x0y ), 9v 2 supp0 (x 0 y ) such that (i) 1f (x; v; u)  0 =) 1f (y; u; v)  0, and (ii) 1f (y; u; v)  0 =) 1f (x; v; u)  0. (SSQMw -loc) 8x; y 2 dom f with jjx0y jj = 4, 9u 2 supp (x0y ), 9v 2 supp0 (x 0 y ) such that (i) 1f (x; v; u)  0 =) 1f (y; u; v)  0, and (ii) 1f (y; u; v)  0 =) 1f (x; v; u)  0. Theorem 3.26. Let f : ZV ! R [ f+1g be a function such that dom f satis es (Q-EXC ).

Theorem 3.25.

Let B  ZV be a set with

w

w

+

1

+

1

w

Then,

(i) (SSQM) () (SSQM-loc). (ii) (SSQM ) () (SSQM -loc). Proof. For both (i) and (ii), the \=)" parts are obvious. [\(=" part of (i)] Assume, to the contrary, that (SSQM) does not hold for some x; y 2 dom f and u3 2 supp (x 0 y). We also assume that (x; y) minimizes the value jjx 0 yjj of all such pairs. Note that jjx 0 yjj  6 and x(V ) = y(V ) by Lemma 3.9 (i). Claim 1: There exists u 2 supp (x 0 y ) such that y + u0 0 v 2 dom f for some v 2 supp0(x 0 y). Moreover, if x(u3) 0 y(u3) = 1 then we can assume u 6= u3. [Proof of Claim 1] By Lemma 3.9 (ii), dom f satis es (Q-EXC ). Applying (Q-EXC ) to y and x, there exist u 2 supp (x 0 y ) and v 2 supp0 (x 0 y ) with y = y + u1 0 v1 2 dom f . Hence, the former part of Claim 1 holds. In the following, we assume x(u3) 0 y(u3) = 1 and show the latter part of Claim 1. If u 6= u3 then we are done. Thus, we assume u = u3. Since jjx 0 y jj  4, we can again apply (Q-EXC ) to y and x to obtain u 2 supp (x 0 y ) = supp (x 0 y) n fu3g and v 2 supp0(x 0 y )  supp0(x 0 y) with y = y + u2 0 v2 2 dom f . Then, we apply (SSQM-loc) to y , y, and u3 2 supp (y 0y) to obtain some v 2 supp0 (y 0y) = fv ; v g such that if 1f (y; u3; v)  0 then 1f (y ; v; u3 )  0. By the choice of x and y we have 1f (y; u3; v)  0, from which follow 1f (y ; v; u3)  0. Hence, y + v 0 u3 = y + u2 0 v0 2 dom f for some v0 2 fv ; v g. [End of Claim 1] 0 , S 0 , and S 0 , where We can divide the set supp0(x 0 y) into three sets S>> > > 0 = fv 2 supp0(x 0 y ) j 1f (x; v; u3 ) > 0; 1f (y ; u3 ; v ) > 0g; S>> S>0 = fv 2 supp0(x 0 y ) j 1f (x; v; u3 ) > 0; 1f (y ; u3 ; v ) = 0g; S 0> = fv 2 supp0(x 0 y ) j 1f (x; v; u3 ) = 0; 1f (y ; u3 ; v ) > 0g: w

w

+

1

1

+

0

0

w+

+

1

1

w+

1

1

1

1

1

w+

+

2

1

1

2

+

2

1

2

2

1

2

2

2

2

=

=

=

1

1

2

2

+

=

20

Quasi M/L-convex Functions

Then, we de ne v 2 supp0(x 0 y) as follows: if 0 [ S 0 g; minff (y + u0 0 v ) j v 2 S>0 g < minff (y + u0 0 v ) j v 2 S>> > then let v be any element in arg minff (y + u0 0 v ) j v 2 S>0 g, and otherwise let v be any 0 [ S 0 g. Put y0 = y + u 0 v . The, y 0 2 dom f element in arg minff (y + u0 0 v ) j v 2 S>> 0 0 > by Claim 1. 0

=

=

0

0

=

=

Claim 2:

1f (y0; u3; v)  0 (8v 2 supp0(x 0 y0)); (3.6) 1f (y0; u3; v) > 0 (8v 2 supp0 (x 0 y0) \ S 0>): (3.7) [Proof of Claim 2] For v 2 supp0(x 0 y0), put y00 = y 0 + u3 0 v = y + u0 + u3 0 v0 0 v : We may assume y00 2 dom f , since otherwise the claim holds obviously. Applying (SSQM-loc) to y00; y, and u3 2 supp (y00 0 y), we have 1f (y 00; v0; u3)  0 =) 1f (y; u3; v0)  0; (3.8) 0 00 0 1f (y; u3; v )  0 =) 1f (y ; v ; u3)  0 (3.9) for some v0 2 fv ; vg. Since 1f (y; u3; v0)  0, (3.9) implies that f (y00 )  f (y 00 + v0 0 u3 ) = f (y + u0 0 v0 0 v + v0 )  f (y + u0 0 v0 ) = f (y0 ): (3.10) This proves (3.6). Next, we assume v 2 S 0>. If 1f (y; u3; v0) > 0, then (3.8) implies that the rst inequality in (3.10) holds with strict inequality, i.e., (3.7) holds. Hence, we assume 1f (y; u3; v0) = 0; (3.11) which implies v 0 = v 2 S>0 since v 2 S 0>. Due to the choice of v , we have f (y 0) < f (y + u0 0 v ): (3.12) By (3.11) and (3.9), we have f (y00)  f (y 00 + v0 0 u3 ) = f (y + u0 0 v ): (3.13) From (3.12) and (3.13) follows (3.7). [End of Claim 2] Since u3 2 supp (x 0 y0) and jjx 0 y 0jj < jjx 0 yjj , Claim 2 contradicts the choice of x and y. This concludes the proof. =

+

0

=

0

+

=

0

=

1

1

3

21

Quasi M-convex Functions

[\(=" part of (ii)] We show (SSQM ) for x; y 2 dom f by induction on the value jjx 0 yjj . We may assume that jjx 0 yjj > 4 and 1f (x; v; u)  0; 1f (y; u; v)  0 (8u 2 supp (x 0 y); 8v 2 supp0(x 0 y)); (3.14) since otherwise the claim holds immediately. We are to show 1f (x; v; u) = 1f (y; u; v) = 0 for some u 2 supp (x 0 y) and v 2 supp0 (x 0 y). Claim 1: Let y 0 be any vector in [x ^ y; x _ y ] \ dom f . Put k = jjx 0 y 0 jj and x = x. For i = 1; 2; 1 1 1 ; k , we de ne xi 2 ZV by xi 2 arg minff (xi0 0 u + v ) j u 2 supp (xi0 0 y0 ); v 2 supp0(xi0 0 y 0)g: (3.15) Then, we have xi 2 [x ^ y; x _ y] \ dom f and f (xi)  f (xi0 ) (i = 1; 1 1 1 ; k). In particular, we have f (y0)  f (x). [Proof of Claim 1] From (Q-EXC ) for dom f follows xi 2 [x ^ y; x _ y] \ dom f (i = 1; 1 1 1 ; k). We show the inequality f (xi )  f (xi) by induction on i. The inequality f (x )  f (x ) follows from (3.14) and the fact that supp (x 0 y0)  supp (x 0 y), supp0 (x 0 y 0)  supp0 (x 0 y). We then suppose i  1. Since jjxi0 0 xi jj = 4, we can apply (SSQM -loc) to xi0 and xi to obtain some u 2 supp (xi0 0 xi ) and v 2 supp0(xi0 0 xi ) such that if 1f (xi0 ; v; u)  0 then 1f (xi ; u; v)  0. By the inductive hypothesis and the choice of xi , we have 1f (xi0 ; v; u)  f (xi) 0 f (xi0 )  0. Hence, we have f (xi )  f (xi + u 0 v )  f (xi). [End of Claim 1] In a similar way, we can show that f (y0)  f (y) (8y0 2 [x ^ y; x _ y]). Hence, we have f (x) = f (y). We de ne a set S  ZV by S = fx0 2 [x ^ y; x _ y ] \ dom f j f (x0) = f (x) = f (y )g: Claim 2: S satis es (B-EXC). [Proof of Claim 2] From Theorem 3.25 it suces to show (Q-EXC ) and (B-EXC-loc) for S . We rst prove (Q-EXC ) for x~; y~ 2 S with jjx~0y~jj < jjx0yjj . By the inductive hypothesis, we can apply (SSQM ) to x~ and y~ to obtain (a) 1f (~x; v; u) < 0, (b) 1f (~y; u; v) < 0, or (c) 1f (~x; v; u) = 1f (~y; u; v) = 0 for some u 2 supp (~x 0 y~) and v 2 supp0 (~x 0 y~). Since x~ 0 u + v ; y~ + u 0 v 2 [x ^ y; x _ y ], we have 1f (~x; v; u)  0 and 1f (~y; u; v)  0. Therefore (c) must hold, i.e., x~ 0 u + v 2 S and y~ + u 0 v 2 S . This fact also yields (B-EXC-loc) for x~; y~ 2 S with jjx~ 0 y~jj = 4. We next prove (Q-EXC ) for x~; y~ 2 S with jjx~ 0 y~jj = jjx 0 yjj . Then, we have fx~; y~g = fx; yg. For y0 = y and i = 0; 1; 1 1 1 ; jjx 0 yjj , de ne xi 2 ZV by (3.15). By Claim 1, we have xi 2 S for each i = 0; 1; 1 1 1 ; k . Hence, (Q-EXC ) holds for x; y 2 S . [End of Claim 2] Applying (B-EXC) to x; y, we have x 0 u + v 2 S and y + u 0 v 2 S for some u 2 supp (x 0 y ) and v 2 supp0 (x 0 y). Hence follows 1f (x; v; u) = 1f (y ; u; v ) = 0. w

1

1

+

+

1

+

1

1

0

1

1

w

+1

1

+

+

+1 1

+

0

1

+1

1

+1

0

0

1

w

1

1

+1

+1

1

1

+1

+1

0

0

0

w

w

0

1

0

1

w

+

0

0

0

1

w

0

1

1

1

0

w

0

0

+

0

22

Quasi M/L-convex Functions

The localized version of (QM) does not characterize (QM) in general. Let ! R [ f+1g be a function such that

Remark 3.27.

f:Z

2

dom f = f(0; 0); (1; 01); (2; 02); (3; 03)g;

f (0; 0) = f (3; 03) = 0; f (1; 01) = f (2; 02) = 1:

Then, dom f satis es (Q-EXC ), and (QM) holds for any x; y 2 dom f with jjx 0 yjj = 4. However, (QM) does not hold for x = (0; 0) and y = (3; 03). w

4

1

Minimization of Quasi M-convex Functions

In this section, we use the following weaker properties than (SSQM) and (SSQM ): w

(SSQM6= )

such that

+

(i) 1f (x; v; u)  0 =) 1f (y; u; v)  0, (ii) 1f (y; u; v)  0 =) 1f (x; v; u)  0.

(SSQM6= w)

such that

8x; y 2 dom f with f (x) 6= f (y), 8u 2 supp (x 0 y), 9v 2 supp0 (x 0 y)

and

8x; y 2 dom f with f (x) 6= f (y), 9u 2 supp (x 0 y), 9v 2 supp0 (x 0 y) +

(i) 1f (x; v; u)  0 =) 1f (y; u; v)  0, (ii) 1f (y; u; v)  0 =) 1f (x; v; u)  0.

and

These properties imply neither (SSQM) nor (SSQM ); an example is f : Z ! R [f+1g with dom f = f(0; 0); (1; 01); (2; 02)g given by f (0; 0) = f (2; 02) = 0 and f (1; 01) = 1. Note that f does not satisfy (QM ). The property (SSQM6 ) is equivalent to each of the following two conditions: 2

w

w

= w

maxff (x); f (y)g u>2 min ff (x 0 u + v ); f (y + u 0 v )g + x0y supp (

)

v2supp0 (x0y)

f (x) >

min

u2supp+ (x0y) v2supp0 (x0y )

f (x 0 u + v )

(8x; y 2 dom f with f (x) 6= f (y)); (4.1) (8x; y 2 dom f with f (x) > f (y)): (4.2)

The condition (4.2) is also considered in in [18]. Theorem 4.1.

Proof.

For f

: ZV ! R [ f+1g, (SSQM6 ) () = w

(4.1)

()

(4.2).

The proof is quite similar to that for Theorem 3.8. See Appendix for details.

4

23

Quasi M-convex Function Minimization

4.1 Properties of Minimizers of Quasi M-convex Functions

Global minimality of quasi M-convex functions is characterized by local minimality. Theorem 4.2. Let f : ZV ! R [ f+1g and x 2 dom f . (i) Assume (QM ) for f . Then, 1f (x; v; u) > 0 (8u; v 2 V; u 6= v) () f (x) < f (y) (8y 2 ZV n fxg). (ii) Assume (SSQM6 ) for f . Then, 1f (x; v; u)  0 (8u; v 2 V ) () f (x)  f (y) (8y 2 ZV ). w

= w

We show the \=)" part of (ii) only. The \(=" part of (ii) is easy to prove, and the proof of (i) can be done in a similar way as that of (ii) by using Theorem 3.8. Assume, to the contrary, that there exists some y 2 dom f such that f (y) < f (x). By Theorem 4.1, f satis es (4.2), which implies that there exist some u0 2 supp (x 0 y) and v 0 2 supp0(x 0 y ) such that 1f (x; v0 ; u0 ) < 0, a contradiction to the assumption for x. If f satis es (SSQM6 ), then any vector in dom f can be easily separated from some minimizer of f (cf. [20, Th. 2.2, Cor. 2.3]). This property will be used as a basis of the domain reduction method in Section 4.2. Theorem 4.3. Let f : ZV ! R [ f+1g be a function with (SSQM6 ). Assume arg min f 6= ;. (i) For x 2 dom f and v 2 V , let u 2 V be such that f (x 0 u + v ) = mins2V f (x 0 s + v ). Then, there exists x3 2 arg min f with x3 (u)  x(u) 0 1 + v (u). (ii) For x 2 dom f and u 2 V , let v 2 V be such that f (x 0 u + v ) = mint2V f (x 0 u + t). Then, there exists x3 2 arg min f with x3 (v)  x(v ) 0 u(v ) + 1. (iii) For x 2 dom f n arg min f , let u; v 2 V be such that f (x 0 u + v ) = mins;t2V f (x 0 s + t). Then, there exists x3 2 arg min f with x3 (u)  x(u) 0 1 and x3 (v )  x(v ) + 1. Proof.

+

=

=

(i): Put x0 = x 0 u + v . We may assume x0 62 arg min f , since otherwise the claim holds immediately. Assume, to the contrary, that there is no x 2 arg min f with x(u)  x0(u). Let x3 2 arg min f minimize x3(u). Then, we have x3(u) > x0(u). Since f (x3) 6= f (x0), we can apply (SSQM6 ) to x3, x0, and u to obtain some w 2 supp0 (x3 0 x0) such that if 1f (x3; w; u) > 0 then 1f (x0; u; w) < 0. Due to the choice of x3, we have 1f (x3; w; u) > 0. Hence, f (x0) > f (x0 + u 0 w ) = f (x 0 w + v ) holds, a contradiction to the de nition of u2V. (ii): The proof is similar to that for (i) and therefore omitted. (iii): Put x0 = x 0 u + v (6= x). We may assume x0 62 argmin f , since otherwise the claim holds immediately. By (i), there exists some x3 2 arg min f such that x3(u)  x(u) 0 1, and we may assume that x3 maximize x3(v) among all such vectors. To the contrary assume x3 (v) < x0(v ). Since f (x3 ) 6= f (x0 ), we can apply (SSQM6 ) to x0, x3 , and v 2 supp (x0 0 x3 ) to obtain some w 2 supp0 (x0 0 x3) satisfying at least one of the following: Proof.

=

=

+

24

Quasi M/L-convex Functions

(a) 1f (x0; w; v) < 0, (b) 1f (x3; v; w) < 0, (c) 1f (x0; w; v) = 1f (x3; v; w) = 0. Due to the choice of u; v 2 V , we have 1f (x0; w; v)  0 since x0 0 v + w = x 0 u + w . We also have 1f (x3; v; w)  0 since x3 2 arg min f . Therefore, we have (c), which implies x3 + v 0 w 2 arg min f , a contradiction to the choice of x3 . Remark 4.4. The following example shows that the statements in Theorem 4.3 do not hold even if a function satis es the property (SSQM ) (and not (SSQM)). Let V = fa; b; c; d; e; gg, and f : ZV ! R [ f+1g be a function de ned as follows: dom f = f(1; 1; 1; 0; 0; 0); (0; 1; 1; 1; 0; 0); (0; 0; 1; 1; 1; 0); (0; 0; 0; 1; 1; 1); (1; 0; 1; 1; 0; 0); (1; 0; 0; 1; 0; 1)g; f (1; 1; 1; 0; 0; 0) = 0; f (0; 1; 1; 1; 0; 0) = 1; f (0; 0; 1; 1; 1; 0) = 2; f (0; 0; 0; 1; 1; 1) = 3; f (1; 0; 1; 1; 0; 0) = 4; f (1; 0; 0; 1; 0; 1) = 5: We can easily check that f satis es (SSQM ). Put x = (1; 0; 0; 1; 0; 1), v = e, u = a, and x0 = x 0 u + v = (0; 0; 0; 1; 1; 1). Then, we have f (x0) = mins2V f (x 0 s + v ). The unique minimizer x3 = (1; 1; 1; 0; 0; 0) of f , however, does not satisfy the inequality x3(u)  x0(u) since x3 (u) = 1 and x0 (u) = 0. Hence, the statement (i) does not hold for f . We can show in the similar way that the statement (ii) does not hold for this f . We apply the scaling technique to the minimization of quasi M-convex functions in Section 4.2. Let f : ZV ! R [ f+1g be a semistrictly quasi M-convex function. For x 2 dom f and  2 Z , we de ne f : ZV ! R [ f+1g by f (x) = f (x + x) (x 2 ZV ): (4.3) Remark 4.5. The following example shows that a function f de ned by (4.3) is not quasi M-convex in general, even if f is an M-convex function. De ne B  Z by P B = fx 2 Z j i xi = 4; 0  xi  2 (i = 1; 2; 3; 4)g nf(2; 0; 2; 0); (0; 2; 0; 2); (0; 2; 2; 0); (2; 0; 0; 2)g and f : Z ! R [ f+1g by f = B . Since B is an M-convex set, f is an M-convex function. For x = 0 and  = 2, de ne f : Z ! R [ f+1g by (4.3). Then, f is the indicator function of the set f(1; 1; 0; 0); (0; 0; 1; 1)g, which does not satisfy (Q-EXC ). Hence, f is not quasi M-convex. Let y 2 dom f be a local minimum of f, i.e., x = x + y satis es f (x )  f (x + (v 0 u)) (8u; v 2 V ): (4.4) The following theorem shows that a global minimum of a semistrictly M-convex function exists in the neighborhood of x. This generalizes [8, Th. 4.1]. w

w

0

++

0

4

4

4 =1

4

0

4

w

0

4

25

Quasi M-convex Function Minimization

Let f : ZV ! R [ f+1g be a function with (SSQM6= ), and  2 Z++ . Suppose that x 2 dom f satis es (4.4). Then, arg min f 6= ; and there exists some x3 2 arg min f such that Theorem 4.6.

jx(v) 0 x3 (v)j  (n 0 1)( 0 1)

(v 2 V ):

(4.5)

It suces to show that for any 2 R with > inf f , there exists some x3 2 dom f satisfying f (x3)  and (4.5). Let x3 2 dom f satisfy f (x3)  , and suppose that x3 minimizes the value jjx3 0 xjj among all such vectors. In the following, we x v 2 V and prove x(v) 0 x3(v)  (n 0 1)( 0 1). The inequality x3(v) 0 x(v)  (n 0 1)( 0 1) can be shown similarly. We may assume x(v) > x3(v). Put

Proof.

1

8
x3(v). Since jjy 0 xjj < jjx3 0 xjj , we have f (y) > f (x3). By (SSQM6 ) applied to y, x3, and v 2 supp (y 0 x3)  supp (x 0 x3), we have some w 2 supp0 (y 0 x3)  supp0(x 0 x3) such that if 1f (x3; v; w) > 0 then 1f (y; w; v) < 0. By the choice of x3, we have 1f (x3; v; w) > 0 since jj(x3 + v 0 w ) 0 xjj < jjx3 0 xjj . Therefore, f (y 0 v + w ) < f (y), which is a contradiction since y 0 v + w 2 S . [End of Claim 1] Let y~ = x 0 ~v + Pf~w w j w 2 supp0 (x 0 x3)g 2 arg minff (y0) j y0 2 S g. Claim 2: For any w 2 supp0 (x 0 x3 ) with  ~w > 0 and  2 [0; ~w 0 1], we have

Claim 1:

1

=

+

1

+

1

1

x 0 (v 0 w ) 2 dom f;

f (x 0 ( + 1)(v 0 w )) < f (x 0 (v 0 w )):

(4.6)

[Proof of Claim 2] For  2 [0; ~w 0 1], put x0 = x 0 (v 0 w ) and suppose x0 2 dom f . Note that x0 2 S and x0(v) > x3(v). Therefore, Claim 1 yields f (x0) > f (~y). Since supp0 (~y 0 x0) = fvg, (SSQM6 ) applied to y~, x0, and w 2 supp (~y 0 x0 ) implies that if 1f (~y; v; w) > 0 then 1f (x0; w; v) < 0. By Claim 1, we have 1f (~y; v; w) > 0, from which (4.6) follows. [End of Claim 2] Claim 2 and (4.4) imply ~w   0 1 for w 2 supp0(x 0 x3). Thus, =

+

x (v) 0 x3(v ) = x (v) 0 y~(v ) =

X

w2supp0 (x 0x3 )

where the second equality is by Lemma 3.9 (i).

~w  (n 0 1)( 0 1);

26

Quasi M/L-convex Functions

4.2 Algorithms Let f : ZV ! R [ f+1g be a function such that dom f is a nonempty bounded set, and put L = maxfjx(v ) 0 y (v )j j x; y 2 dom f; v 2 V g:

Assume (SSQM6 ) for f . Then, Theorem 4.2 immediately leads to the following algorithm. Algorithm Descent M Step 0: Let x be any vector in dom f . Step 1: If f (x) = s;tmin f (x 0 s + t ) then stop. [x is a minimizer of f .] 2V Step 2: Find u; v 2 V with f (x 0 u + v ) < f (x). Step 3: Set x := x 0 u + v . Go to Step 1. Algorithm Descent M terminates in at most jdom f j  (L +1)n0 iterations since it generates a distinct x in each iteration. To the end of this section we assume (SSQM6 ) for f . Based on Theorem 4.6, we apply the scaling technique to Algorithm Descent M to obtain a faster algorithm. Algorithm Scaling Descent M Step 0: Let x be any vector in dom f . Put  := 2d 2 Le, B := dom f . Step 1: Step 1-1: If f (x) = minff (x 0 (s 0 t)) j s; t 2 V; x 0 (s 0 t) 2 Bg, then go to Step 2. Step 1-2: Find u; v 2 V with x 0 (u 0 v ) 2 B satisfying f (x 0 (u 0 v )) < f (x). Step 1-3: Set x := x 0 (u 0 v ). Go to Step 1-1. Step 2: If  = 1 then stop. [x is a minimizer of f .] Step 3: Put B := B \ fy 2 ZV j jy(v) 0 x(v)j  (n 0 1)( 0 1) (v 2 V )g and  := =2. Go to Step 1. The number of scaling phases is dlog Le, and each scaling phase terminates in (4n)n0 iterations. Therefore, Algorithm Scaling Descent M runs in (4n)n0 dlog Le iterations. We then propose another elaboration of Algorithm Descent M. Note that the algorithm Steepest Descent M reduces the set B iteratively in Step 3 by exploiting Theorem 4.3 (iii). Algorithm Steepest Descent M Step 0: Let x be any vector in dom f . Set B := dom f . Step 1: If f (x) = s;tmin f (x 0 s + t ) then stop. [x is a minimizer of f .] 2V Step 2: Find u; v 2 V with x 0 u + v 2 B satisfying f (x 0 u + v ) = minff (x 0 s + t) j s; t 2 V; x 0 s + t 2 B g: (4.7) Step 3: Set x := x 0 u + v and B := B \ fy 2 ZV j y (u)  x(u) 0 1; y (v )  x(v) + 1g: (4.8) = w

1

=

log

1

2

1

2

4

27

Quasi M-convex Function Minimization

Go to Step 1. By Theorem 4.3 (iii), the set B always contains a minimizer of f . Hence, SteepestP Descent M nds a minimizer of f . To analyze the number of iterations, we consider the value w2V fuB (w)0 lB (w)g, where lB (w ) = miny2B y (w) and uB (w ) = maxy2B y(w ) (w 2 V ). This value is bounded by nL and decreases at least by two in each iteration. Therefore, Steepest Descent M terminates in O(nL) iterations. In particular, if dom f  f0; 1gV then the number of iterations is O(n ). It is shown in [20] that the minimization of an M-convex function can be done in polynomial time by the domain reduction method explained below. We show that the domain reduction method also works for the minimization of a function with (SSQM6 ) if its eective domain is a bounded M-convex set. Given a bounded M-convex set B  ZV , we de ne the set NB  B by NB = fy 2 B j lB0  y  u0B g, where        0l (w) = 1 0 1 l (w) + 1 u (w) ; u0 (w) = 1 l (w ) + 1 0 1 u (w) (w 2 V ): 2

=

B

n

B

n

Lemma 4.7 ([20, Th. 2.4]).

B

B

n

B

n

B

NB is a (nonempty) M-convex set.

The next algorithm maintains a set B ( dom f ) which is an M-convex set containing a minimizer of f . It reduces B iteratively by exploiting Theorem 4.3 (iii) and nally nds a minimizer. Algorithm Domain Reduction Step 0: Set B := dom f . Step 1: Find a vector x 2 NB . Step 2: If f (x) = s;tmin f (x 0 s + t ) then stop. [x is a minimizer of f .] 2V Step 3: Find u; v 2 V with x 0 u + v 2 B satisfying (4.7). Step 4: Set B by (4.8). Go to Step 1. We analyze the number of iterations of Domain Reduction. Denote by Bi the set B in the i-th iteration, and let li(w) = lBi (w), ui(w) = uBi (w) (w 2 V ). It is clear that ui(w) 0 li(w) is nonincreasing w.r.t. i. Furthermore, we have the following property: Lemma 4.8 ([20, Lemma 3.1]). ui (w ) 0 li (w ) < (1 0 1=n)fui (w) 0 li (w)g for w 2 fu; vg, where u; v 2 V are the elements found in Step 3. This lemma implies that Algorithm Domain Reduction terminates in O(n log L) iterations. We now consider the time complexity of each step. Steps 2, 3, and 4 can be done in O(n ) time. In Step 1, we use the exchange capacity to compute the values lB (w) and uB (w) and to nd a vector in NB . For any w 2 V , the values lB (w) and uB (w) can be computed by evaluating +1

+1

2

2

28

Quasi M/L-convex Functions

Table 2: Possible sign patterns of g(p ^ q) 0 g(p) and g(p _ q) 0 g(q) in submodular inequality g(p ^ q) 0 g(p) n g (p _ q ) 0 g(q )

0 0 0

0

+

2

1 1 1 possible, 2 1 1 1 impossible

+

2 2

the exchange capacity at most n times, provided that a vector in B is given [6, Th. 3.27]. A vector in NB can be found by evaluating the exchange capacity at most n times, provided that a vector in B is given [20, Th. 2.5]. The exchange capacity can be computed in O(log L) time by binary search. Hence, Step 1 requires O(n log L) time. 2

2

Suppose that f : ZV ! R [ f+1g satis es (SSQM6=) and that dom f is a bounded M-convex set. If a vector in dom f is given, Algorithm Domain Reduction nds a minimizer of f in O(n4 (log L)2) time. Theorem 4.9.

5

Quasi L-convex and Submodular Functions

5.1 De nition of Quasi L-convex and Submodular Functions

To extend the concept of L-convexity to quasi L-convexity, we relax the submodularity condition (SBM) while keeping in mind the possible sign patterns of the values g(p ^ q) 0 g(p) and g(p _ q ) 0 g (q ). Table 2 shows the possible sign patterns of those values. Let g : ZV ! R [ f+1g be a function. We call g quasi-submodular if it satis es (QSB): (QSB)

For all p; q 2 ZV we have g(p ^ q)  g(p) or g(p _ q)  g(q),

and call g quasi L-convex if dom g 6= ; and it satis es (QSB) and (TRF). Since p and q are interchangeable, (QSB) implies g(p^q)  g(q) or g(p_q)  g(p). Similarly, we call g semistrictly quasi-submodular if it satis es the following property: For all p; q 2 ZV we have both (i) and (ii): (i) g(p _ q)  g(q) =) g(p ^ q)  g(p), and (ii) g(p ^ q)  g(p) =) g(p _ q)  g(q), (SSQSB)

and call g semistrictly quasi L-convex if dom g 6= ; and it satis es (SSQSB) and (TRF). We also consider weaker properties than (QSB) and (SSQSB) by keeping in mind the possible sign patterns of the four values g(p ^ q) 0 g(p), g(p ^ q) 0 g(q), g(p _ q) 0 g(p), and g(p _ q) 0 g(q).

5

29

Quasi L-convex Functions

For any p; q 2 dom g, we have maxfg(p); g(q)g  minfg(p ^ q); g(p _ q)g. (SSQSBw ) For any p; q 2 dom g, we have either of (i) and (ii): (i) maxfg(p); g(q)g > minfg(p ^ q); g(p _ q)g, (ii) g(p) = g(q) = g(p ^ q) = g(p _ q). The property (SSQSB ) says that either (i) at least one of the values g(p^q)0g(p), g(p^q)0g(q), g(p _ q ) 0 g (p), and g (p _ q ) 0 g (q ) is negative or (ii) all the four values are equal to zero. Similarly, (QSB ) says that at least one of the four values is nonpositive. The set version of quasi-submodularity can be obtained by translating the property (QSB) for the indicator function D : ZV ! f0; +1g of a set D  ZV in terms of D. (QDL) p; q 2 D =) p ^ q 2 D or p _ q 2 D. The following properties for D  ZV can be shown easily:  (QDL) for D () (QSB) for D () (QSB )for D ,  (DL) for D () (SSQSB) for D () (SSQSB ) for D ,  (TRS) for D () (TRF) for D . We show some examples of quasi L-convex/submodular functions below. Example 5.1. Let ' : Z ! R [f+1g. We de ne g : Z ! R [f+1g by g(p ; p ) = '(p 0 p ) ((p ; p ) 2 Z ). Then, g satis es (TRF) with r = 0. Moreover, g satis es (QSB) (or (QSB )) if and only if ' is quasiconvex, and g satis es (SSQSB) (or (SSQSB )) if and only if ' is semistrictly quasiconvex. Proof. [(QSB ) for g =) quasiconvexity for '] For any  ;  2 Z with  <  , we have maxf'( ); '( )g = maxfg( ; 0); g( 0 1; 01)g  minfg( ; 01); g( 0 1; 0)g = minf'( + 1); '( 0 1)g by (QSB ). This implies the quasiconvexity of ' by Theorem 2.2 (i). [quasiconvexity for ' =) (QSB) for g] Let p; q 2 Z , and we may assume p > q and p < q . Since q 0 q < p 0 q < p 0 p , we have maxfg(p ; p ); g(q ; q )g = maxf'(p 0 p ); '(q 0 q )g  '(p 0 q ) = g(p ; q ): We can prove the statement \g satis es(SSQSB ) () ' is semistrictly quasiconvex" in the similar way by using Theorem 2.2 (ii). Example 5.2. Let g : ZV ! R [ f+1g be a submodular function, and ' : R ! R [ f+1g be a nondecreasing function. We de ne the function g~ : ZV ! R [ f+1g by ( '(g(p)) (p 2 dom g ); (5.1) g~(p) = +1 (p 62 dom g): (QSBw )

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Then, g~ satis es (QSB). Furthermore, if ' is strictly increasing, then g~ satis es (SSQSB). Note that if g satis es (TRF) then g~ also does. Example 5.3. Let D  ZV satisfy (DL), and x 2 RV ,  2 R. Then, the set S = fp 2 D j hp; xi  g satis es (QDL). Moreover, the function g : ZV ! R [ f+1g with dom g = S de ned by g(p) = hp; xi (p 2 S ) satis es (SSQSB) and (TRF) with r = p(V ). These properties are obvious from the equation hp; xi + hq; xi = hp ^ q; xi + hp _ q; xi. Remark 5.4. The concept of (semistrict) quasi submodularity/L-convexity can be naturally extended to functions g : S ! T with S  ZV and a totally ordered set T with total order . For example, the property (SSQSB) is rewritten for such functions as follows: For any p; q 2 S , we have both (i) and (ii): (i) if p _ q 62 S , or p _ q 2 S and g(p _ q)  g(q), then p ^ q 2 S and g(p ^ q)  g(p), and (ii) if p ^ q 62 S , or p ^ q 2 S and g(p ^ q)  g(p), then p _ q 2 S and g(p _ q)  g(q). It is easy to see that the properties of (semistrictly) quasi submodular/L-convex functions shown in this paper still hold true. For simplicity and convenience, however, we assume in this paper that the codomain of a function is R [ f+1g. Example 5.5. Suppose that V = f1; 2; 1 1 1 ; ng (n  1) and put V 0 = f1; 1 1 1 ; n 0 1g. Let a : V 0 ! Z [ f01g, b : V 0 ! Z [ f+1g satisfy a(i)  b(i) (i 2 V 0 ). For i 2 V 0, let fi : [a(i); b(i)] ! R be a semistrictly quasiconvex function. We de ne D  ZV and g : D ! RV 0 by D = fp 2 ZV j a(i)  p(i) 0 p(n)  b(i)(i 2 V 0)g; g (p) = (gi (p(i) 0 p(n)) j i 2 V 0 ) (p 2 D); where the total order  on the codomain RV 0 of g is given by the lexicographic order. Then, g satis es (TRF) with r = 0 and (SSQSB) in the extended sense (see Remark 5.4). Proof. We show (SSQSB) for g only. Let p; q 2 [a; b]. Then, p ^ q; p _ q 2 [a; b], and at least one of (a), (b) or (c) holds for each i 2 V 0 (see Example 5.1): (a) gi (p(i) ^ q(i) 0 p(n) ^ q(n)) < gi(p(i) 0 p(n)), (b) gi(p(i) _ q(i) 0 p(n) _ q(n)) < gi(q(i) 0 q(n)), (c) gi(p(i) ^ q(i) 0 p(n) ^ q(n)) = gi(p(i) 0 p(n)) and gi(p(i) _ q(i) 0 p(n) _ q(n)) = gi (q(i) 0 q(n)). If (c) holds for all i 2 V 0, then we have g(p _ q) = g(q) and g(p ^ q) = g(p), implying (SSQSB). Otherwise, let i3 2 V 0 be the minimum element satisfying (a) or (b). Then, we have g(p _ q )  g(q ) or g (p ^ q )  g (p) since (c) holds for all i 2 f1; 2; 1 1 1 ; i3 0 1g.

5

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Quasi L-convex Functions

The relationship among various quasi-submodularity is summarized as follows. Theorem 5.6. For a function g : ZV ! R [ f+1g, we have (SBM) =) (SSQSB) =) (QSB) + + (SSQSB ) =) (QSB ). Remark 5.7. It is easy to see that (DL) =) (QDL), but the converse does not hold in general, even under the condition (TRS). For example, the set f(p + ; p + ; ) 2 Z j  2 Z; (p ; p ) is either (0; 0); (1; 0); or (0; 1)g satis es (QDL) and (TRS) and not (DL). This fact shows that the implication \(QSB) =) (SSQSB )" does not hold necessarily, even under the condition (TRF). We present some examples to show that implications not mentioned in Theorem 5.6 do not hold in general, even if functions are assumed to satisfy (TRF). For i = 1; 2, let gi : Z ! R [ f+1g be a function with dom gi = f0; 1g such that gi (0; 0) = 1; gi (1; 0) = 2; gi (1; 1) = 4 (i = 1; 2); g (0; 1) = 0; g (0; 1) = 2; and de ne g~i : Z ! R [ f+1g by g~i(p ; p ; p ) = gi(p 0 p ; p 0 p ) ((p ; p ; p ) 2 Z ). Then, g~ satis es (SSQSB ) and (TRF) and not (QSB), and g~ satis es (SSQSB) and (TRF) and not (SBM). Due to the de nitions of quasi L-convexity/submodularity, most of the properties of quasisubmodular functions can be naturally restated in terms of quasi L-convex functions, and vice versa. In the following sections, we state properties mainly in terms of quasi-submodular functions and omit those for quasi L-convex functions whenever the restatements are immediate. w

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5.2 Level Sets of Quasi L-convex and Submodular Functions

We show various properties of level sets of quasi L-convex/submodular functions. The following two theorems claim that level sets of quasi-submodular functions have nice properties such as (DL) and (QDL). Furthermore, the weaker version of quasi-submodularity (QSB ) for functions can be characterized by the property (QDL) of level sets. Theorem 5.8. A function g : ZV ! R [ f+1g satis es (QSB ) if and only if the level set L(g; ) satis es (QDL) for every  2 R [ f+1g. In particular, if g satis es (QSB ), then dom g and arg min g satisfy (QDL). Proof. We show the \if" part only. Let p; q 2 dom g , and put  = maxfg (p); g (q )g. Since the level set L(g; ) satis es (QDL), we have either p ^ q 2 L(g; ) or p _ q 2 L(g; ), implying maxfg(p); g(q)g  minfg(p ^ q); g(p _ q)g. w

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Quasi M/L-convex Functions

Let g : ZV ! R [ f+1g, and suppose that the level set L(g; ) satis es (DL) for every  2 R [ f+1g. Then, g satis es (QSB). Theorem 5.9.

Let p; q 2 dom g and assume g(p)  g(q). Since the level set L(g; g(p)) satis es (DL) and contains p and q, we have p ^ q; p _ q 2 L(g; g(p)), implying g(p ^ q)  g(p). Remark 5.10. A function does not necessarily satisfy (SSQSB ) even if every level set satis es (DL), as shown in the following example. Let g : Z ! R [ f+1g be a function given by dom g = fp 2 ZV j p 0 p is either of 0, 1, and 2g; g(p ; p ) = 0 if p 0 p = 0; g(p ; p ) = 0 if p 0 p 2 f1; 2g: Every level set of g satis es (DL), but (SSQSB ) does not hold for g. Theorem 5.11. If g : ZV ! R [ f+1g satis es (SSQSB ), then arg min g satis es (DL). A submodular function over integer lattice can be characterized by using level sets of functions perturbed by linear functions. Recall the de nition of the function g[x] : ZV ! R [f+1g in (2.1). Theorem 5.12 ([10, Th. 10]). A function g : ZV ! R [ f+1g satis es (SBM) if and only if for all x 2 RV and  2 R the level set L(g[x]; ) satis es (QDL). Proof. The \only if" part follows from Theorem 5.8 and submodularity of g[x]. We prove the \if" part. Let p; q 2 dom g. Since p; q 2 L(g; maxfg(p); g(q)g), we have either p ^ q 2 L(g; maxfg(p); g(q)g)  dom g or p _ q 2 L(g; maxfg(p); g(q)g)  dom g. Assume, w.l.o.g., that p ^ q 2 dom g and p ^ q 6= p; q . For any " > 0, we can choose some x 2 RV and  2 R such that  = g[x](p) = g [x](q ) = g [x](p ^ q ) 0 ". By (QDL) for L(g[x]; ), we have p _ q 2 L(g [x]; ). This implies that

Proof.

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g[x](p) + g[x](q ) = 2  g [x](p ^ q ) + g[x](p _ q) 0 ":

Since " can be chosen arbitrarily, we have g [x](p) + g [x](q )  g [x](p ^ q ) + g [x](p _ q );

which is equivalent to the submodular inequality g(p) + g(q)  g(p ^ q) + g(p _ q). Combining Theorems 5.8 and 5.12, we see the following: Corollary 5.13. Let g : ZV ! R [ f+1g. Then, g satis es (SBM) () 8x 2 RV , g [x] satis es (QSB) () 8x 2 RV , g[x] satis es (QSB ). w

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5.3 Operations for Quasi L-convex and Submodular Functions

The class of (semistrictly) quasi L-convex/submodular functions is closed under several fundamental operations. Let g : ZV ! R [ f+1g be a function. For any subset U  V , we de ne g U : ZU ! R [ f61g by gU (p) = inf fg (p; q ) j q 2 ZV nU g (p 2 ZU ): Theorem 5.14. Let (3QSB3 ) be one of the properties (QSB), (QSB ), (SSQSB), and (SSQSB ), and g : ZV ! R [ f+1g be a function with the property (3QSB3 ). (i) For any a 2 ZV , 2 Z, and  > 0, the function  1 g(a + p) satis es (3QM3) as a function w

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in x. (ii) For any U  V , the function gU : ZU ! R [ f61g satis es (3QSB3) if gU > 01. (iii) For any a : V ! Z [ f01g and b : V ! Z [ f+1g with a  b, the function gab : ZV ! Z [ f+1g de ned by (3.5) satis es (3QSB3 ). (iv) Let gi : ZVi ! R++ [ f+1g (i = 1; 2) be functions with (3QSB3). Then, the function g : ZV1 2 ZV2 ! R++ [ f+1g de ned by g (p1 ; p2 ) = g1 (p1 )g2 (p2) (pi 2 ZVi ; i = 1; 2) satis es (3QSB3).

Proof is similar to that for Theorem 3.18 and therefore omitted. Remark 5.15. The class of (semistrictly) quasi-submodular functions is not closed under addition; in particular, it is not closed under the addition of a linear function. For i = 1; 2, let gi : Z ! Z [ f+1g be functions such that dom g = f(0; 0); (1; 0); (0; 1)g; g (0; 0) = 0; g (1; 0) = g (0; 1) = 1; g (p ; p ) = 02(p + p ) ((p ; p ) 2 Z ): It is easy to see that g satis es (SSQSB) (and not (SBM)), and that g is linear. The sum g = g + g , however, does not even satisfy (QSB ) since g (1; 0) = g (0; 1) = 01 < 0 = minfg(0; 0); g(1; 1)g. Theorem 5.16. For g : ZV ! R [f+1g and ' : R ! R [f+1g, de ne g ~ : ZV ! R [f+1g Proof.

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by (5.1). (i) Suppose that ' is nondecreasing. If g satis es (QSB) (resp. (QSBw )), then g~ also satis es (QSB) (resp. (QSBw )). (ii) Suppose that ' is strictly increasing. If g satis es (SSQSB) (resp. (SSQSBw )), then g~ also satis es (SSQSB) (resp. (SSQSBw )).

In [10], a semistrictly quasi-submodular function g~ : ZV ! R [f+1g is called submodularizable if there exists some strictly increasing function : R ! R such that the

Remark 5.17.

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function g : ZV ! R [ f+1g with dom g = dom g~ de ned by g(p) = (~g(p)) (p 2 dom g) is submodular as a function in p 2 ZV ; in other words, a function g~ is submodularizable if and only if it is represented in the form (5.1) with a submodular function g : ZV ! R [f+1g and a strictly increasing function ' : R ! R. The following example, which is essentially equivalent to Example 1 in [10], shows that a (semistrictly) quasi-submodular function is not necessarily given as the form (5.1) with a submodular function g and a nondecreasing function '. Let us consider the function g~ : Z ! R [ f+1g given by dom g~ =8fp 2 Z j 0  p 0 p  p 0 p  p 0 p  1; 0  p 0 p  1g; > 0 if (p 0 p ; p 0 p ; p 0 p ; p 0 p ) = (1; 1; 0; 1); > > > > > if (p 0 p ; p 0 p ; p 0 p ; p 0 p ) = (1; 0; 0; 1); < 1 g~(p) = 2 if (p 0 p ; p 0 p ; p 0 p ; p 0 p ) 2 f(0; 0; 0; 1); (1; 1; 1; 1)g; > > > 3 if (p 0 p ; p 0 p ; p 0 p ; p 0 p ) 2 f(1; 0; 0; 0); (1; 1; 0; 0)g; > > > : 4 if (p 0 p ; p 0 p ; p 0 p ; p 0 p ) 2 f(0; 0; 0; 0); (1; 1; 1; 0)g: Then, g~ satis es (SSQSB) and (TRF) with r = 0. Suppose that g~ is given in the form (5.1) in terms of an L-convex function g : ZV ! R [ f+1g and a strictly increasing function ' : R ! R [ f+1g. Then, the submodularity of g implies that g (0; 0; 0; 1; 0) + g (1; 0; 0; 0; 0)  g (0; 0; 0; 0; 0) + g (1; 0; 0; 1; 0); g (1; 1; 0; 1; 0) + g (1; 1; 1; 0; 0)  g (1; 1; 0; 0; 0) + g (1; 1; 1; 1; 0); whereas we have g (0; 0; 0; 1; 0) = g (1; 1; 1; 1; 0); g (1; 0; 0; 0; 0) = g (1; 1; 0; 0; 0); g (1; 1; 0; 1; 0) < g (1; 0; 0; 1; 0); g (1; 1; 1; 0; 0) = g (0; 0; 0; 0; 0); since ' is strictly increasing. Hence, we have a contradiction. Theorem 5.18. Let f : ZV ! R [ f+1g and g : ZV ! R [ f01g be functions such that g(p) > 0 for all p 2 dom f . Suppose that the function f (1) 0 g (1) satis es (QSB ) for all  2 R [ f+1g. Then, the function r : ZV ! R [ f+1g given by 5

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Quasi L-convex Function Minimization

De ne a function r : Z ! R [ f+1g as 3

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The function r, however, does not satisfy (QSB) since g (1; 0; 0) = 2; g (0; 1; 0) = 01; g(0; 0; 0) = 0; g(1; 1; 0) = 3:

6

Minimization of Quasi L-convex Functions

In this section, we consider the minimization of quasi L-convex functions. To the end of this section we assume r = 0 in (TRF) since otherwise quasi L-convex functions have no minimizer. Under this assumption, the minimization of a function g : ZV ! R [ f+1g is equivalent to the minimization of g : ZV nfv0g ! R [ f+1g which is de ned as 0

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6.1 Properties of Minimizers of Quasi L-convex Functions

Global minimality of quasi L-convex functions is characterized by local minimality. Let g : ZV ! R [ f+1g satisfy (TRF) with r = 0. (i) Assume (QSBw ) for g. Then, for all p; q 2 ZV and  2 Z we have

Lemma 6.1.

maxfg(p); g(q)g  minfg(p _ (q 0 1)); g((p + 1) ^ q)g:

(6.2)

In particular, for all p; q 2 dom g and  2 [0; 1 0 2 ] we have

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(6.3)

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 2 [0; 1 0 2 ] we have (6.3) with strict inequality. (iii) Assume (SSQSB) for g. Then, for all p; q 2 ZV and  2 Z we have the following properties: g(p _ (q 0 1))  g (p) =) g ((p + 1) ^ q )  g (q); g ((p + 1) ^ q )  g (q ) =) g (p _ (q 0 1))  g (p): In particular, for all p; q 2 dom g and  2 [0; 1 0 2 ] we have

g(p + X )  g (p) =) g (q 0 X )  g (q); g(q 0 X )  g(q ) =) g(p + X )  g (p); (6.5) where 1 2 Z, 2 2 Z [ f01g, and X

 V are given by (6.4). Proof. We show the proof of (i) only. Proofs of (ii) and (iii) are similar to that for (i). The inequality (6.2) can be shown as follows:

LHS of (6.2) = maxfg(p); g(q 0 1)g  minfg(p _ (q 0 1)); g(p ^ (q 0 1))g = minfg(p _ (q 0 1)); g(p ^ (q 0 1) + 1)g = RHS of (6.2). The inequality (6.3) is obvious from (6.2) since p _ fq 0 ( 0 )1g = p + X ; (p + ( 0 )1) ^ q = q 0 X (8 2 [0;  0  ]): 1

Theorem 6.2.

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p 2 dom g . (i) Assume (QSBw ) for g. Then, g(p) < g(q) for all q 2 ZV such that q 0 p is not a multiple of 1 if and only if g (p) < g (p + X ) for all X  V with X 62 f;; V g. (ii) Assume (SSQSBw ) for g. Then, g(p)  g(q) (8q 2 ZV ) () g(p)  g(p + X ) (8X  V ).

Proof. We prove the \if" part of (i) by contradiction. Suppose that g (q )  g(p) holds for some q 2 dom g such that q 0 p is not a multiple of 1. We may assume q  p by (TRF) for g , and also assume that q minimizes the value maxv2V fq(v) 0 p(v)g among all such vectors. Put X = arg maxv2V fq (v ) 0 p(v )g, where X 6= V . By applying Lemma 6.1 (i) to p and q , we obtain

g (p) = maxfg(p); g (q)g  minfg (p + X ); g(q 0 X )g:

Due to the choice of q, we have g(p) < g(q 0 X ). Hence, g(p)  g(p + X ) follows, a contradiction to the strict local minimality of p. The \only if" part of (i) is obvious, and (ii) can be shown similarly by using Lemma 6.1 (ii).

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a function g : ZV ! R [ f+1g satisfying (TRF) with r = 0, de ne : ! [ f+1g by (6.1). Let p 2 dom g0 . (i) Assume (QSBw ) for g. Then, g0(p) < g0(q) (8q 2 ZV nfv0g n fpg) () g0(p) < g0(p 6 X ) (; 6= 8X  V ). (ii) Assume (SSQSBw ) for g. Then, g0(p)  g0(q) (8q 2 ZV nfv0 g) () g0(p)  g0(p 6 X ) (8X  V ). Corollary 6.3. For g0 ZV nfv0g R

We see from its proof that the statement of Theorem 6.2 (i) holds even if (QSB ) is replaced with the following weaker condition: For any distinct p; q 2 dom g, (i) or (ii) holds: (i) minfg(p + X ); g(q 0 X )g  maxfg(p); g(q)g for X = arg max fq(v) 0 p(v)g, v 2V (ii) minfg(p 0 X ); g(q + X )g  maxfg(p); g(q)g for X = arg min fq(v) 0 p(v)g. v 2V Remark 6.4.

w

This property is strictly weaker than (QSB ) under (TRF), as shown in the following example. Let g : ZV ! R [ f+1g be a function such that 8 > ((p 0 p ; p 0 p ) = (0; 1)); < 0 g (p ; p ; p ) = 1 ((p 0 p ; p 0 p ) is either (1; 0); (2; 0); or (1; 1)); > : +1 (otherwise): It is easy to see that the function g satis es the property above. For p = (2; 0; 0); q = (0; 1; 0) 2 dom g, neither p ^q nor p _q is contained in dom g, i.e., dom g does not satisfy (QDL). Therefore, g does not satisfy (QSB ) by Theorem 5.8. Remark 6.5. We see from its proof that the statement of Theorem 6.2 (ii) holds even if (SSQSB ) is replaced with the following weaker condition: For any p; q 2 dom g with g(p) 6= g(q), (i) or (ii) holds: (i) minfg(p + X ); g(q 0 X )g < maxfg(p); g(q)g for X = arg max fq(v) 0 p(v)g, v 2V (ii) minfg(p 0 X ); g(q + X )g < maxfg(p); g(q)g for X = arg min fq(v) 0 p(v)g. v 2V w

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w

w

This property is strictly weaker than (SSQSB ) under (TRF), as shown in the following example. Let D  ZV be any set which satis es (TRF) and not (QDL), and consider its indicator function D : ZV ! f0; 1g. Since D (p) = D (q) for any p; q 2 dom D , the function D satis es the property above. However, D does not satisfy (QSB ). We apply the scaling technique to the minimization of quasi L-convex functions in Section 6.2. Let g : ZV ! R [ f+1g be a semistrictly quasi L-convex function. For p 2 dom g and  2 Z , we de ne g : ZV ! R [ f+1g by g (x) = g (p + p) (p 2 ZV ). We here consider w

w

0

++

0

38

Quasi M/L-convex Functions

the relationship between a global minimum of g and a local minimum of g. Let q 2 dom g be a local minimum of g, i.e., p = p + q satis es 0

g(p )  g(p + X )

(8X  V ):

(6.6)

The following theorem shows that a global minimum of a semistrictly L-convex function exists in the neighborhood of p. This generalizes an observation in [9]. Theorem 6.6. Let g : ZV ! R [ f+1g be a function satisfying (SSQSB) and (TRF) with r = 0, and  2 Z . Suppose that p 2 dom g satis es (6.6). Then, arg min g 6= ; and there exists some q3 2 arg min g with ++

p  q3  p + (n 0 1)( 0 1)1:

(6.7)

It suces to show that for any 2 R with > inf g, there exists some q3 2 dom g satisfying g(q3)  and (6.7). Assume, w.l.o.g., p = 0. By (TRF) for g, there exists some q3 2 dom g such that g (q3 )  and q3  0. We assume that q3 is minimal (w.r.t. the partial order ) among all such vectors. This assumption implies q3(v) = 0 for some v 2 V , i.e., supp (q3) 6= V , and Proof.

+

g (q3 0 X ) > g (q3 )

(8X  supp (q3)):

(6.8)

+

Then, there exist some Xi  supp (q3) (i = 1; 1 1 1 ; k) and figki  Z (0  k  n 0 1) such that +

=1

;  X  X  1 1 1  Xk  V; 1

Claim 1:

2

q3 =

k X i=1

++

i Xi :

For any j = 1; 1 1 1 ; k and  2 [0; j 0 1], we have g(

j 01

X

i=1

j 01

X

i Xi + Xj ) > g (

i=1

i Xi + ( + 1)Xj ):

[Proof of Claim 1] Put p = Pji 0 iXi +Xj and suppose p 2 dom g. Then, arg maxv2V fq3(v)0 p(v)g = Xj . Since Xj  supp (q3 ), we have g (q3 0 Xj ) > g (q3 ) by (6.8). This fact, together with (6.5), yields g(p + Xj ) < g(p). [End of Claim 1] Claim 2 g (Xj ) > g (( + 1)Xj ) holds for any j = 1; 1 1 1 ; k and  2 [0; j 0 1]. [Proof of Claim 2] Suppose Xj 2 dom g. Put p = Pji iXi and q = Xj . Then, argmaxv2V fq(v) 0 p(v)g = V n Xj . Since g(p + V nXj ) = g(p 0 Xj ) > g(p) by Claim 1, (6.5) implies that g(q) > g(q 0 V nXj ) = g(q + Xj ). [End of Claim 2] 1 =1

+

=1

6

39

Quasi L-convex Function Minimization

From Claim 2 and (6.6) follows i <  for i = 1; 2; 1 1 1 ; k. Hence, we have 0

 q3  ( 0 1)

k X i=1

Xi  (n 0 1)( 0 1)1:

: ZV ! R [ f+1g satisfying (SSQSB) and (TRF) with r = 0, de ne g : ZV nfv0g ! R [ f+1g by (6.1). Let  2 Z . Suppose that p 2 dom g satis es g (p )  g (p 6 X ) (8X  V ). Then, there exists some q3 2 arg min g such that jq3 (v) 0 p(v)j  (n 0 1)( 0 1) (v 2 V ): Corollary 6.7.

Given a function g

0

++

0

0

0

0

6.2 Algorithms Let g : ZV ! R [ f+1g satisfy (SSQSB ) and (TRF) with r = 0, and de ne g : ZV nfv0g ! R [ f+1g by (6.1). In the following, we explain two minimization algorithms for g . w

0

0

By Corollary 6.3, we can nd a minimizer of g by a descent method. Algorithm Descent L Step 0: Let p be any vector in dom g . Step 1: If g (p) = minfg (p 6 X ) j X  V g then stop. [p is a minimizer] Step 2: Find X  V and  2 f1; 01g such that g (p + X ) < g (p). Step 3: Set p := p + X . Go to Step 1. If dom g is bounded, the algorithm Descent L terminates in at most jdom g j  K n0 iterations, where K is given by K = maxfjp(v ) 0 q (v)j j p; q 2 dom g ; v 2 V g: We further assume (SSQSB) for g. Based on Corollary 6.7, we apply the scaling technique to Algorithm Descent L to obtain a faster algorithm. Algorithm Scaling Descent L Step 0: Put  := 2d 2 Ke , D := dom g . Let p3 be any vector in dom g . Step 1: Find q 2 ZV nfv0 g such that p3 + q 2 D and g (p3 + q ) = minfg (p3 + q 0) j q0 2 ZV nfv0g ; p3 + q0 2 D g: Step 2: If  = 1 then stop. [p3 + q is a minimizer of g .] Step 3: Put p3 := p3 + q, D := D \ fp 2 ZV j jp(v) 0 p3(v)j  (n 0 1)( 0 1) (v 2 V )g, and  := =2. Go to Step 1. The number of scaling phases is dlog K e. Therefore, if we could perform Step 1 in each iteration in polynomial time, Algorithm Scaling Descent L would run in polynomial time. Unfortunately, we do not know yet such a polynomial-time algorithm for Step 1. 0

0

0

0

0

0

0

0

0

log

0

0

0

0

0

2

1

40

Quasi M/L-convex Functions

References

[1] M. Avriel, W. E. Diewert, S. Schaible and I. Zang, Generalized Concavity, Plenum Press, New York (1988). [2] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithm (Second Edition), John Wiley and Sons, New York (1993). [3] A. W. M. Dress and W. Wenzel, \Valuated matroid: A new look at the greedy algorithm," Applied Mathematics Letters 3, 33{35 (1990). [4] A. W. M. Dress and W. Wenzel, \Valuated matroids," Advances in Mathematics 93, 214{ 250 (1992). [5] P. Favati and F. Tardella, \Convexity in nonlinear integer programming," Ricerca Operativa 53, 3{44 (1990). [6] S. Fujishige, Submodular Functions and Optimization, Annals of Discrete Mathematics 47, North-Holland, Amsterdam (1991). [7] S. Fujishige and K. Murota, \Notes on L-/M-convex functions and the separation theorems," Mathematical Programming 88, 129{146 (2000). [8] D. S. Hochbaum, \Lower and upper bounds for the allocation problem and other nonlinear optimization problems," Mathematics of Operations Research 19, 390{409 (1994). [9] S. Iwata and M. Shigeno, \Conjugate scaling technique for Fenchel-type duality in discrete convex optimization," Information Processing Society of Japan, SIG Notes 98-AL-65, 33{ 40 (1998). [10] P. Milgrom and C. Shannon, \Monotone comparative statics," Econometrica 62, 157{180 (1994). [11] B. L. Miller, \On minimizing nonseparable functions de ned on the integers with an inventory application," SIAM Journal on Applied Mathematics 21, 166{185 (1971). [12] K. Murota, \Submodular ow problem with a nonseparable cost function," Combinatorica 19, 87{109 (1999). [13] K. Murota, \Convexity and Steinitz's exchange property," Advances in Mathematics 124, 272{311 (1996). [14] K. Murota, \Discrete convex analysis," Mathematical Programming 83, 313{371 (1998).

A

41

Appendix

[15] K. Murota, Discrete Convex Analysis, Kyoritsu-Shuppan, Tokyo (2001). [In Japanese] [16] K. Murota, Discrete Convex Analysis, in preparation. [17] K. Murota and A. Shioura, \M-convex function on generalized polymatroid," Mathematics of Operations Research 24, 95{105 (1999). [18] K. Murota and A. Tamura, \New characterizations of M-convex functions and connections to mathematical economics," RIMS preprint, No. 1307, Kyoto University (2000). [19] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton (1970). [20] A. Shioura, \Minimization of an M-convex function," Discrete Applied Mathematics 84, 215-220 (1998). [21] A. Shioura, \Level set characterization of M-convex functions," IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E83-A, 586{589 (2000). [22] J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimensions I, SpringerVerlag, Berlin (1970). [23] N. Tomizawa, \Theory of hyperspaces (I) { supermodular functions and generalization of concept of `bases'," Papers of the Technical Group on Circuit and System Theory, Institute of Electronics and Communication Engineers of Japan, CAS80-72, 1980. [In Japanese] A

Proofs

A.1 Proof of Theorem 2.2 Proof of (i): The \=)" part is easy to see. Hence, we show the \(=" part only. Let  ;  2 dom ' with  <  , where we assume '( )  '( ). We show the claim by induction on the value  0  . From the assumption, we have (i) '( + 1)  '( ) or (ii) '( 0 1)  '( ). If (i) holds, then the inductive assumption implies '( )  maxf'( + 1); '( )g ( + 1 < 8 <  ). If (ii) holds, then the inductive assumption implies '( )  maxf'( ); '( 0 1)g ( < 8 <  0 1). Therefore, we have (2.2). Proof of (ii): Proof is quite similar to that for (i). The \=)" part is easy to see. Hence, we show the \(=" part only. Let  ;  2 dom ' with  <  and '( ) =6 '( ), where we assume '( ) > '( ). We show the claim by induction on the value  0  . From the 1

1

2

2

1

1

2

1

1

2

1

1

2

2

1

1

1

2

2

1

1

2

2

2

1

2

1

2

2

1

42

Quasi M/L-convex Functions

assumption, we have (i) '( + 1) < '( ) or (ii) '( 0 1) < '( ). If (i) holds, then the quasiconvexity of ' implies 1

1

2

'( )  maxf'(1 + 1); '(2 )g < '(1 )

1

( + 1 < 8 <  ): 1

2

If (ii) holds, then the inductive assumption implies '( ) < maxf'(1); '(2 0 1)g = '(1)

(

1

< 8 < 2 0 1):

Therefore, we have (2.3). Proof of (iii): [\=)" part] Let  ;  2 dom ' with  <  . We may assume that  0  2. It suces to show that we have '( + 1) < '( ), '( 0 1) < '( ), or '( + 1) = '( ) and '( 0 1) = '( ). (Case 1: '( ) 6= '( )) We have 1

2

1

1

2

2

1

2

2

2

1

1

1

2

1

2

'(1 + 1) < maxf'(1); '(2 )g; '(2 0 1) < maxf'(1 ); '(2 )g;

from which we have '( + 1) < '( ) or '( 0 1) < '( ). (Case 2: '( ) = '( )) The quasiconvexity of ' implies 1

1

1

2

2

2

'(1 + 1)  maxf'(1 ); '(2 )g = '(1 ); '(2 0 1)  maxf'(1 ); '(2)g = '(2 );

from which the claim follows. [\(=" part] The assumption for ' and the property (i) immediately yield the quasiconvexity of '. It suces from (ii) to prove that minf'( + 1); '( 0 1)g < maxf'( ); '( )g (A.9) (8 ;  2 dom ' with  <  ; '( ) 6= '( )): We assume '( ) > '( )), and prove '( + 1) < '( ) by induction on the value  0  . From the assumption, we have '( + 1) < '( ) or '( 0 1)  '( ). If the latter holds, then the inductive hypothesis yields '( + 1) < '( ). Hence, we have (A.9). 1

2

1

1

1

2

2

2

1

1

2

1

2

1

1

1

1

2

2

1

2

1

A.2 Proof of Theorem 4.1

It is easy to see that (4.2) implies both (SSQM6 ) and (4.1). Hence, we prove \(SSQM6 ) =) (4.2)" and \(4.1) =) (4.2)" below. Suppose that f : ZV ! R [ f+1g satis es (SSQM6 ) or (4.1). Let x; y 2 dom f be vectors such that f (x) > f (y). We show by induction on the value jjx 0 yjj that there exist some u 2 supp (x 0 y) and v 2 supp0 (x 0 y) such that 1f (x; v; u) < 0. We may assume jjx 0 y jj > 2, since otherwise the claim holds obviously. = w

= w

= w

1

+

1

A

43

Appendix

Suppose that f satis es (SSQM6 ). Then, there exist some u 2 supp (x 0 y) and v 2 supp0(x 0 y) such that 1f (x; v; u) < 0 or 1f (y; u; v)  0. If the latter holds, then we have f (x) > f (y0 ) for y 0 = y + u 0 v and jjx 0 y0 jj < jjx 0 y jj . Hence, the inductive hypothesis yields 1f (x; v0; u0) < 0 for some u0 2 supp (x 0 y0)  supp (x 0 y) and v0 2 supp0(x 0 y0)  supp0(x 0 y). We next suppose that f satis es (4.1). Then, there exist some u 2 supp (x 0 y) and v 2 supp0 (x 0 y) such that 1f (x; v; u) < 0 or f (y + u 0 v ) < f (x). If the latter holds, then we have f (x) > f (y0) for y0 = y + u 0 v and jjx 0 y0jj < jjx 0 yjj . Hence, the inductive hypothesis yields 1f (x; v0; u0) < 0 for some u0 2 supp (x 0 y0)  supp (x 0 y) and v 0 2 supp0(x 0 y0 )  supp0(x 0 y). = w

+

1

+

1

+

+

1

+

1

+