Symmetrization of generalized natural residual ... - Semantic Scholar
to appear in Operations Research Letters, 2015
Symmetrization of generalized natural residual function for NCP Yu-Lin Chang 1 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan Jein-Shan Chen 2 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan Chin-Yu Yang 3 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan January 20, 2015 (first revised on March 19, 2015) (second revised on April 7, 2015) Abstract. In contrast to the generalized Fischer-Burmeister function that is a natural extension of the popular Fischer-Burmeister function NCP-function, the generalized natural residual NCP-function based on discrete extension, recently proposed by Chen, Ko, and Wu, does not possess symmetric graph. In this paper we symmetrize the generalized natural residual NCP-function, and construct not only new NCP-functions and merit functions for the nonlinear complementarity problem, but also provide parallel functions to the generalized Fischer-Burmeister function. Keywords. Symmetric, NCP, natural residual, complementarity. 1
E-mail: [email protected] E-mail: [email protected] 3 Corresponding author. E-mail: [email protected] 2
1
1
Motivation
The nonlinear complementarity problem (NCP for short) has attracted much attention since 1970s because of its wide applications in the fields of economics, engineering, and operations research, see [11, 12, 18] and references therein. The mathematical format for NCP is to find a point x ∈ Rn such that x ≥ 0,
F (x) ≥ 0,
hx, F (x)i = 0,
where h·, ·i is the Euclidean inner product and F = (F1 , . . . , Fn )T is a map from Rn to Rn . For solving NCP, the so-called NCP-function φ : R2 → R defined as φ(a, b) = 0
⇐⇒
a, b ≥ 0, ab = 0,
plays a crucial role. More specifically, with such NCP-functions, the NCP can be recast as nonsmooth equations [23, 24, 29] or unconstrained minimization [13, 14, 17, 20, 21, 25, 28]. During the past four decades, around thirty NCP-functions are proposed, see [16] for a survey. Among them, two popular NCP-functions, the Fischer-Burmeister (denoted by FB) function [14, 15] and the natural residual (denoted by NR) function [22, 26], are frequently employed and most of the existing NCP-functions are indeed variants of these two functions. In particular, the Fischer-Burmeister function φFB : R2 → R is defined by √ φFB (a, b) = a2 + b2 − (a + b), whereas the natural residual function φNR : R2 → R is given by φNR (a, b) = a − (a − b)+ = min{a, b}. Recently, a generalized Fischer-Burmeister function φpFB : R2 → R, which includes the Fischer-Burmeister function as a special case, was considered in [1, 2, 3, 7, 19, 27]. The function φpFB is defined as φpFB (a, b) = k(a, b)kp − (a + b),
p>1
(1)
and this natural extension is based on “continuous generalization” in such a way that the 2-norm in FB function is replaced by general p-norm. In addition, its geometric view is depicted in [27] and the effect of perturbing p for different kinds of algorithms are investigated in [4, 5, 7, 8, 9]. More recently, a generalization of natural residual function, denoted by φpNR , is proposed in [6] and defined as φpNR (a, b) = ap − (a − b)p+
with p > 1 being a positive odd integer.
Notice that when p = 1, φpNR reduces to the natural residual function φNR , i.e. φ1NR (a, b) = a − (a − b)+ = min{a, b} = φNR (a, b). 2
(2)
In contrast to φpFB , the function φpNR is obtained by “discrete generalization” and surprisingly possesses twice differentiability, see [6]. This feature enables us that many methods such as Newton method can be employed directly for solving NCP. However, unlike the graph of φpFB , the graph of φpNR is not symmetric which may causes some difficulty in further analysis in designing solution methods. To this end, we try to symmetrize the function φpNR . More specifically, we offer two ways to obtain symmetrizations of this ”generalized natural residual function”, which still satisfy NCP-conditions. In other words, we construct not only new NCP-functions and merit functions for the nonlinear complementarity problem, but also provide parallel “symmetric” functions to the generalized Fischer-Burmeister function. To close this section, we present the ideas about how we symmetrize the “generalized natural residual function”. The first step is looking into the graph of φpNR given in [27]. Because we wish to symmetrize the graph of φpNR , we need to consider subcases of a ≥ b and a ≤ b, respectively. In view of the definition of φpNR , we propose the first symmetrization of φpNR , denoted by φpS−NR : R2 → R, which is defined by
φpS−NR (a, b) =
ap − (a − b)p
if a > b,
ap = b p
if a = b,
bp − (b − a)p
if a < b,
(3)
where p > 1 being a positive odd integer. We will see that φpS−NR is an NCP-function with symmetric graph in Section 2. However, φpS−NR is not differentiable in general, it is natural to ask whether there exists a symmetrization function that has not only symmetric graph but also is differentiable. To this end, we see that the induced family of merit functions kφpS−NR k2 will fit this purpose. Nonetheless, we can construct another simpler merit function by modifying φpS−NR . In summary, we wish to construct a symmetrized function which is also differentiable. Fortunately, we figure out the second symmetrization of φpNR , p denoted by ψS−NR : R2 → R+ , which is defined by
p ψS−NR (a, b) =
ap bp − (a − b)p bp
if a > b,
ap bp = a2p
if a = b,
ap bp − (b − a)p ap
if a < b,
(4)
where p > 1 being a positive odd integer. The pictures and differentiable properties of p p ψS−NR will be depicted in Section 3. We point it out that the value of ψS−NR is always p nonnegative which indicates that ψS−NR is a merit function for NCP. Here, due to the symmetric feature, we denote these two functions as “S-NR” standing for symmetrization of NR function.
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2
The first symmetrization function φpS−NR
In this section, we show that the function φpS−NR defined in (3) is an NCP-function. It is not differentiable on the whole R2 , but it is twice continuously differentiable on Ω := {(a, b) | a 6= b}. Proposition 2.1. Let φpS−NR be defined in (3) with p > 1 being a positive odd integer. Then, φpS−NR is an NCP-function and is positive only on the first quadrant Ω = {(a, b) | a > 0, b > 0}. Proof. It is straightforward to verify that φpS−NR is positive only on the first quadrant. Next, we continue to show φpS−NR is an NCP-function. We will proceed it by discussing three cases. Suppose a > b and φpS−NR (a, b) = 0. Then, we have ap − (a − b)p = 0, which implies that a = a − b. Thus, we see that a > b = 0. Similarly, when a < b and φpS−NR (a, b) = 0, we have 0 = a < b. For the third case φpS−NR (a, b) = 0 and a = b, it is easy to see that a = b = 0. It is trivial to check the converse way. In summary, φpS−NR satisfies that φpS−NR (a, b) = 0 if and only if a, b ≥ 0, ab = 0; and hence, it is an NCP-function. 2 We elaborate more about the function φpS−NR as below: (i) For p being an even integer, φpS−NR is not an NCP-function. A counterexample is given as below: φ2S−NR (−2, −4) = (−2)2 − (−2 + 4)2 = 0. (ii) The function φpS−NR is neither convex nor concave function. To see this, taking p = 3 and using the following argument, we can verify the assertion. 1 1 0 8 1 = φ3S−NR (1, 1) < φ3S−NR (0, 0) + φ3S−NR (2, 2) = + = 4. 2 2 2 2 1 1 0 0 1 = φ3S−NR (1, 1) > φ3S−NR (2, 0) + φ3S−NR (0, 2) = + = 0. 2 2 2 2 Proposition 2.2. Let φpS−NR be defined in (3) with p > 1 being a positive odd integer. Then, the following hold. (a) An alternative expression of φpS−NR is p φNR (a, b) if a > b, φpS−NR (a, b) = ap = b p if a = b, p φNR (b, a) if a < b. 4
(b) The function φpS−NR is not differentiable. However, φpS−NR is continuously differentiable on the set Ω := {(a, b) | a 6= b} with ( p [ ap−1 − (a − b)p−1 , (a − b)p−1 ]T if a > b, ∇φpS−NR (a, b) = p [ (b − a)p−1 , bp−1 − (b − a)p−1 ]T if a < b. In a more compact form, ( ∇φpS−NR (a, b) =
p−1 p [ φNR (a, b), (a − b)p−1 ]T if a > b, p−1 p [ (b − a)p−1 , φNR (b, a) ]T if a < b.
(c) The function φpS−NR is twice continuously differentiable on the set Ω = {(a, b) | a 6= b} with " # ap−2 − (a − b)p−2 (a − b)p−2 if a > b, p(p − 1) (a − b)p−2 −(a − b)p−2 2 p # " ∇ φS−NR (a, b) = −(b − a)p−2 (b − a)p−2 if a < b. p(p − 1) (b − a)p−2 bp−2 − (b − a)p−2 In a more compact form, " p(p − 1) 2 p " ∇ φS−NR (a, b) = p(p − 1)
p−2 (a, b) φNR
(a − b)p−2
#
(a − b)p−2 −(a − b)p−2 # −(b − a)p−2 (b − a)p−2 (b − a)p−2
φp−2 (b, a) NR
Proof. The arguments are just direct computations, we omit them.
if a > b, if a < b. 2
At last, we present some other variants of φpS−NR . Indeed, analogous to those functions in [26], the variants of φpS−NR as below can be verified being NCP-functions. φe1 (a, b) = φpS−NR (a, b) + α(a)+ (b)+ , α > 0. φe2 (a, b) = φpS−NR (a, b) + α ((a)+ (b)+ )2 , α > 0. φe3 (a, b) = φp (a, b) + α ((ab)+ )4 , α > 0. S−NR
φe4 (a, b) = φpS−NR (a, b) + α ((ab)+ )2 , α > 0. 2 φe5 (a, b) = φpS−NR (a, b) + α (a)+ )2 ((b)+ , α > 0.
Proposition 2.3. All the above functions φei (a, b) for i ∈ {1, 2, 3, 4, 5} are NCP-functions. 5
Proof. We only show that φe1 (a, b) is an NCP-function and the same argument can be applied to the other cases. First, we denote Ω := {(a, b) | a > 0, b > 0} the first quadrant and suppose that φe1 (a, b) = 0. If (a, b) ∈ Ω, then φpS−NR (a, b) > 0 by Proposition 2.1; and hence, φe1 (a, b) > 0. This is a contradiction. Therefore, we must have (a, b) ∈ Ωc which says (a)+ (b)+ = 0. This further implies φpS−NR (a, b) = 0 which is equivalent to a, b ≥ 0, ab = 0 by applying Proposition 2.1 again. Thus, φe1 is an NCP-function. 2
3
p The second symmetrization function ψS−NR
p In this section, we show that the function ψS−NR defined in (4) is not only an NCPp function, but also a merit function. In particular, ψS−NR possesses symmetric graph and is twice differentiable. p be defined in (4) with p > 1 being a positive odd integer. Proposition 3.1. Let ψS−NR p Then, ψS−NR is an NCP-function and is positive on the set
Ω = {(a, b) | ab 6= 0} ∪ {(a, b) | a < b = 0} ∪ {(a, b) | 0 = a > b}. p (a, b) = a2p > 0. Similarly, when Proof. First of all, when a < b = 0, we have ψS−NR p (a, b) = b2p > 0. For 0 6= a > b 6= 0, suppose that b > 0. Then, 0 = a > b, we have ψS−NR a > (a − b) which implies ap > (a − b)p and bp > 0, and hence ap bp − (a − b)p bp > 0. On the other hand, suppose that b < 0. Then, a < (a − b) which implies ap < (a − b)p and bp < 0. Thus, we also have ap bp − (a − b)p bp > 0. For a = b 6= 0, it is clear that ap bp = a2p > 0. For the remaining case: 0 6= a < b 6= 0, the proof is similar to the case p is positive on the set Ω. of 0 6= a > b 6= 0. From all the above, we prove that ψS−NR p Next, we go on showing that ψS−NR is an NCP-function. Suppose that a > b and p p p p p p b a b − (a − b) b = [a − (a − b) ]b = 0. If b = 0, then we have a > b = 0. Otherwise, we have a = (a − b) which also yields that a > b = 0. Similarly, the condition a < b and ap bp − (b − a)p ap = 0 implies that b > a = 0. The remaining case a = b and ap bp = 0 p gives that a = b = 0. Thus, from all the above, ψS−NR is an NCP-function. 2
p p We can conclude from Proposition 3.1 that ψS−NR is a merit function, since ψS−NR is positive on Ω and is identically zero on the set {(a, b) | a ≥ b = 0} ∪ {(a, b) | 0 = a ≤ b}. p Next, we elaborate more about the function ψS−NR as below: p (i) For p being an even integer, ψS−NR is not an NCP-function. A counterexample is given as below. 2 ψS−NR (−2, −4) = (−2)2 (−4)2 − (−2 + 4)2 (−4)2 = 0.
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p (ii) The function ψS−NR is neither convex nor concave function. To see this, taking p = 3 and using the following argument verify the assertion.
1 3 1 3 0 64 3 1 = ψS−NR (1, 1) < ψS−NR (0, 0) + ψS−NR (2, 2) = + = 32. 2 2 2 2 1 3 0 0 1 3 3 (2, 0) + ψS−NR (0, 2) = + = 0. 1 = ψS−NR (1, 1) > ψS−NR 2 2 2 2 p Proposition 3.2. Let ψS−NR be defined as in (4) with p > 1 being a positive odd integer. Then, the following hold.
(a) An alternative expression of φpS−NR is p p φNR (a, b)b if a > b, p ψS−NR (a, b) = ap bp = a2p if a = b, p φNR (b, a)ap if a < b. p (b) The function ψS−NR is continuously differentiable with
p−1 p p−1 p p p−1 − (a − b)p bp−1 + (a − b)p−1 bp ]T if a > b, p [ a b − (a − b) b , a b p ∇ψS−NR (a, b) = p [ ap−1 bp , ap bp−1 ]T = pa2p−1 [1 , 1 ]T if a = b, p [ ap−1 bp − (b − a)p ap−1 + (b − a)p−1 ap , ap bp−1 − (b − a)p−1 ap ]T if a < b. In a more compact form, p−1 p p p−1 + (a − b)p−1 bp ]T if a > b, p [ φNR (a, b)b , φNR (a, b)b p ∇ψS−NR (a, b) = p [ a2p−1 , a2p−1 ]T if a = b, (b, a)ap ]T if a < b. p [ φpNR (b, a)ap−1 + (b − a)p−1 ap , φp−1 NR
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p (c) The function ψS−NR is twice continuously differentiable with
p−2 − (a − b)p−2 ]bp (p − 1)[a p (p − 1)(a − b)p−2 bp +p[ap−1 − (a − b)p−1 ]bp−1 " (p − 1)ap−2 bp pap−1 bp−1 p p ∇2 ψS−NR (a, b) = p−1 p−1 (p − 1)ap bp−2 pa b (p − 1)[bp − (b − a)p ]ap−2 +2p(b − a)p−1 ap−1 −(p − 1)(b − a)p−2 ap p (p − 1)(b − a)p−2 ap +p[bp−1 − (b − a)p−1 ]ap−1
(p − 1)(a − b)p−2 bp
+p[ap−1 − (a − b)p−1 ]bp−1 p p p−2 if a > b, (p − 1)[a − (a − b) ]b p−1 p−1 +2p(a − b) b p−2 p # −(p − 1)(a − b) b if a = b, p−2 p
(p − 1)(b − a) +p[b
p−1
a
p−1
− (b − a)
]a
p−1
(p − 1)[bp−2 − (b − a)p−2 ]ap
if a < b.
Proof. (a) It is clear to see this part. p (a, b) on the set {(a, b) | a > (b) It is easy to verify the continuous differentiability of ψS−NR b or a < b}. We only need to check the differentiability along the line a = b. Suppose that h > k, we observe that p p ψS−NR (a + h, a + k) − ψS−NR (a, a)
= (a + h)p (a + k)p − (h − k)p bp − a2p
= pa2p−1 (1, 1), (h, k) + R(a, h, k). Here the remainder R(a, h, k) is o(h, k) function of h and k, since the degree of h and k of R(a, h, k) is at least 2. Similarly, from the other two cases h = k and h < k, we can p p conclude that ∇ψS−NR (a, a) = pa2p−1 (1, 1)T . In addition, the continuity of ∇ψS−NR (a, b) along the line a = b is easy to verify. (c) The arguments for this part are similar to those for part(b). We omit them.
2
p Again, we present some other variants of ψS−NR . Indeed, analogous to those functions p in [26], the variants of ψS−NR as below can be verified being NCP-functions.
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p ψe1 (a, b) = ψS−NR (a, b) + α(a)+ (b)+ , α > 0. p ψe2 (a, b) = ψS−NR (a, b) + α ((a)+ (b)+ )2 , α > 0. ψe3 (a, b) = ψ p (a, b) + α ((ab)+ )4 , α > 0. S−NR
p ψe4 (a, b) = ψS−NR (a, b) + α ((ab)+ )2 , α > 0. 2 p ψe5 (a, b) = ψS−NR (a, b) + α (a)+ )2 ((b)+ , α > 0.
Proposition 3.3. All the above functions ψei (a, b) for i ∈ {1, 2, 3, 4, 5} are NCP-functions. Proof. We only show that ψe1 is an NCP-function and the same argument can be applied to the other cases. Let Ω := {(a, b) | ab 6= 0} and suppose that ψe1 (a, b) = 0. If (a, b) ∈ Ω, p (a, b) > 0 by Proposition 3.1; and hence, ψe1 (a, b) > 0. This is a contradiction. then ψS−NR Therefore, we must have (a, b) ∈ Ωc which says (a)+ (b)+ = 0. This further implies p (a, b) = 0 which is equivalent to a, b ≥ 0, ab = 0 by applying Proposition 3.1 again. ψS−NR Thus, ψe1 is an NCP-function. 2
4
Concluding remarks
p for the single value Due to space limitation, we illustrate the functions φpS−NR and ψS−NR p = 3, see Figure 1. Nonetheless, we make some remarks about the surfaces of φpS−NR and p ψS−NR , as well as say a few words about their algebraic properties. First of all, it is clear p p to see that φpS−NR (a, b) = φpS−NR (b, a) and ψS−NR (a, b) = ψS−NR (b, a), which mean that the p p surfaces of φS−NR and ψS−NR are both symmetric with respect to the line a = b. As for the algebraic structure, we can verify that 2 p (a, b) = min(a, b) for p = 1. ψS−NR
To see this, for example if a > b, we check that a1 b1 − (a − b)1 b1 = b2 = min(a, b)2 . p On the other hand, for large p = 3, 5, 7, · · · , the function ψS−NR does not coincide with 2p p min(a, b) . Nonetheless, when we restrict ψS−NR (a, b) on the line a = b and two axes a = 0 and b = 0, we really have that 2p p ψS−NR (a, b) = min(a, b) . p In summary, ψS−NR can be viewed as a merit function relative to the original natural p residual NCP-function φNR (a, b) = min(a, b). Besides, we have to mention that ψS−NR is
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twice continuously differentiable so that it is good enough to develop a lot of algorithms based on this property. However, it does not satisfy that p p ∇a ψS−NR (a, b) · ∇b ψS−NR (a, b) ≥ 0. (cf. Property 2.2(d) in [2]) 3 3 For example, taking p = 3 and (a, b) = (0, −1) gives ∇a ψS−NR (0, −1) = 3 and ∇b ψS−NR (0, −1) = −6. This may cause some difficulty in analyzing the convergence rate.
×10 -12
2
8
z-axis
z-axis
0 -2 -4
6 4 2
-6
0 -0.01
-8 -1
-0.005 -0.5
1 0
x-axis
0.01 0
0.005
0.5 0
0.5
x-axis
-0.5 1
-1
0
0.005 -0.005 0.01
y-axis
(a) the graph of φpS−NR
-0.01
y-axis
p (b) the graph of ψS−NR
p with p = 3. Figure 1: The surfaces of φpS−NR and ψS−NR
As it can be seen, the surface of φpS−NR looks like “two-wings” of an eagle and there is cusp along x = y. Moreover, the graph of φpS−NR is neither convex nor concave. The p is smooth and it is neither convex nor concave. surface of ψS−NR To sum up, we propose new NCP-functions and merit functions in this short paper. p Both of them possess symmetric graphs. With our discovery of φpS−NR , ψS−NR in this short paper, there are many future directions to be explored. We list some of them. • Discovering benefits for such symmetrization. p • Doing numerical comparisons among φpFB , φpNR , φpS−NR , and ψS−NR involved in various algorithms.
• Studying the effect when perturbing the parameter p, applying this new family of NCP-functions to suitable optimization problems. • Extending these functions as the complementarity function associated with the second-order cone and symmetric cones.
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• Developing some analytic properties on φpS−NR such as directional differentiability, Lipchitz continuity, semismoothness. p Finally, we would like to point out that the p-th root of φpS−NR and ψS−NR are also NCP-functions. In other words, the functions
p p1 φS−NR ,
p p1 ψS−NR
are NCP-functions, too. The proof is routine, so we omit it. Acknowledgements. This work is supported by Ministry of Science and Technology, Taiwan. The second author is a member of Mathematics Division, National Center for Theoretical Sciences.
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[22] C. Kanzow, N. Yamashita, and M. Fukushima, New NCP-functions and their properties, Journal of Optimization Theory and Applications, 94(1997), 115-135. [23] O. L. Mangasarian, Equivalence of the Complementarity Problem to a System of Nonlinear Equations, SIAM Journal on Applied Mathematics, 31(1976), 89-92. [24] J.-S. Pang, Newton’s Method for B-differentiable Equations, Mathematics of Operations Research, 15(1990), 311-341. [25] J.-S. Pang and D. Chan, Iterative methods for variational and complemantarity problems, Mathematics Programming, 27(1982), 284-313. [26] D. Sun and L. Qi, On NCP-functions, Computational Optimization and Applications, 13(1999), 201-220. [27] H.-Y. Tsai and J.-S. Chen, Geometric views of the generalized FischerBurmeister function and its induced merit function, Applied Mathematics and Computation, 237(2014), 31-59, 2014. [28] N. Yamashita and M. Fukushima, On stationary points of the implict Lagrangian for nonlinear complementarity problems, Journal of Optimization Theory and Applications, 84(1995), 653-663. [29] N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Mathematical Programming, 76(1997), 469-491.
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Symmetrization of generalized natural residual function for NCP Yu-Lin Chang 1 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan Jein-Shan Chen 2 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan Chin-Yu Yang 3 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan January 20, 2015 (first revised on March 19, 2015) (second revised on April 7, 2015) Abstract. In contrast to the generalized Fischer-Burmeister function that is a natural extension of the popular Fischer-Burmeister function NCP-function, the generalized natural residual NCP-function based on discrete extension, recently proposed by Chen, Ko, and Wu, does not possess symmetric graph. In this paper we symmetrize the generalized natural residual NCP-function, and construct not only new NCP-functions and merit functions for the nonlinear complementarity problem, but also provide parallel functions to the generalized Fischer-Burmeister function. Keywords. Symmetric, NCP, natural residual, complementarity. 1
E-mail: [email protected] E-mail: [email protected] 3 Corresponding author. E-mail: [email protected] 2
1
1
Motivation
The nonlinear complementarity problem (NCP for short) has attracted much attention since 1970s because of its wide applications in the fields of economics, engineering, and operations research, see [11, 12, 18] and references therein. The mathematical format for NCP is to find a point x ∈ Rn such that x ≥ 0,
F (x) ≥ 0,
hx, F (x)i = 0,
where h·, ·i is the Euclidean inner product and F = (F1 , . . . , Fn )T is a map from Rn to Rn . For solving NCP, the so-called NCP-function φ : R2 → R defined as φ(a, b) = 0
⇐⇒
a, b ≥ 0, ab = 0,
plays a crucial role. More specifically, with such NCP-functions, the NCP can be recast as nonsmooth equations [23, 24, 29] or unconstrained minimization [13, 14, 17, 20, 21, 25, 28]. During the past four decades, around thirty NCP-functions are proposed, see [16] for a survey. Among them, two popular NCP-functions, the Fischer-Burmeister (denoted by FB) function [14, 15] and the natural residual (denoted by NR) function [22, 26], are frequently employed and most of the existing NCP-functions are indeed variants of these two functions. In particular, the Fischer-Burmeister function φFB : R2 → R is defined by √ φFB (a, b) = a2 + b2 − (a + b), whereas the natural residual function φNR : R2 → R is given by φNR (a, b) = a − (a − b)+ = min{a, b}. Recently, a generalized Fischer-Burmeister function φpFB : R2 → R, which includes the Fischer-Burmeister function as a special case, was considered in [1, 2, 3, 7, 19, 27]. The function φpFB is defined as φpFB (a, b) = k(a, b)kp − (a + b),
p>1
(1)
and this natural extension is based on “continuous generalization” in such a way that the 2-norm in FB function is replaced by general p-norm. In addition, its geometric view is depicted in [27] and the effect of perturbing p for different kinds of algorithms are investigated in [4, 5, 7, 8, 9]. More recently, a generalization of natural residual function, denoted by φpNR , is proposed in [6] and defined as φpNR (a, b) = ap − (a − b)p+
with p > 1 being a positive odd integer.
Notice that when p = 1, φpNR reduces to the natural residual function φNR , i.e. φ1NR (a, b) = a − (a − b)+ = min{a, b} = φNR (a, b). 2
(2)
In contrast to φpFB , the function φpNR is obtained by “discrete generalization” and surprisingly possesses twice differentiability, see [6]. This feature enables us that many methods such as Newton method can be employed directly for solving NCP. However, unlike the graph of φpFB , the graph of φpNR is not symmetric which may causes some difficulty in further analysis in designing solution methods. To this end, we try to symmetrize the function φpNR . More specifically, we offer two ways to obtain symmetrizations of this ”generalized natural residual function”, which still satisfy NCP-conditions. In other words, we construct not only new NCP-functions and merit functions for the nonlinear complementarity problem, but also provide parallel “symmetric” functions to the generalized Fischer-Burmeister function. To close this section, we present the ideas about how we symmetrize the “generalized natural residual function”. The first step is looking into the graph of φpNR given in [27]. Because we wish to symmetrize the graph of φpNR , we need to consider subcases of a ≥ b and a ≤ b, respectively. In view of the definition of φpNR , we propose the first symmetrization of φpNR , denoted by φpS−NR : R2 → R, which is defined by
φpS−NR (a, b) =
ap − (a − b)p
if a > b,
ap = b p
if a = b,
bp − (b − a)p
if a < b,
(3)
where p > 1 being a positive odd integer. We will see that φpS−NR is an NCP-function with symmetric graph in Section 2. However, φpS−NR is not differentiable in general, it is natural to ask whether there exists a symmetrization function that has not only symmetric graph but also is differentiable. To this end, we see that the induced family of merit functions kφpS−NR k2 will fit this purpose. Nonetheless, we can construct another simpler merit function by modifying φpS−NR . In summary, we wish to construct a symmetrized function which is also differentiable. Fortunately, we figure out the second symmetrization of φpNR , p denoted by ψS−NR : R2 → R+ , which is defined by
p ψS−NR (a, b) =
ap bp − (a − b)p bp
if a > b,
ap bp = a2p
if a = b,
ap bp − (b − a)p ap
if a < b,
(4)
where p > 1 being a positive odd integer. The pictures and differentiable properties of p p ψS−NR will be depicted in Section 3. We point it out that the value of ψS−NR is always p nonnegative which indicates that ψS−NR is a merit function for NCP. Here, due to the symmetric feature, we denote these two functions as “S-NR” standing for symmetrization of NR function.
3
2
The first symmetrization function φpS−NR
In this section, we show that the function φpS−NR defined in (3) is an NCP-function. It is not differentiable on the whole R2 , but it is twice continuously differentiable on Ω := {(a, b) | a 6= b}. Proposition 2.1. Let φpS−NR be defined in (3) with p > 1 being a positive odd integer. Then, φpS−NR is an NCP-function and is positive only on the first quadrant Ω = {(a, b) | a > 0, b > 0}. Proof. It is straightforward to verify that φpS−NR is positive only on the first quadrant. Next, we continue to show φpS−NR is an NCP-function. We will proceed it by discussing three cases. Suppose a > b and φpS−NR (a, b) = 0. Then, we have ap − (a − b)p = 0, which implies that a = a − b. Thus, we see that a > b = 0. Similarly, when a < b and φpS−NR (a, b) = 0, we have 0 = a < b. For the third case φpS−NR (a, b) = 0 and a = b, it is easy to see that a = b = 0. It is trivial to check the converse way. In summary, φpS−NR satisfies that φpS−NR (a, b) = 0 if and only if a, b ≥ 0, ab = 0; and hence, it is an NCP-function. 2 We elaborate more about the function φpS−NR as below: (i) For p being an even integer, φpS−NR is not an NCP-function. A counterexample is given as below: φ2S−NR (−2, −4) = (−2)2 − (−2 + 4)2 = 0. (ii) The function φpS−NR is neither convex nor concave function. To see this, taking p = 3 and using the following argument, we can verify the assertion. 1 1 0 8 1 = φ3S−NR (1, 1) < φ3S−NR (0, 0) + φ3S−NR (2, 2) = + = 4. 2 2 2 2 1 1 0 0 1 = φ3S−NR (1, 1) > φ3S−NR (2, 0) + φ3S−NR (0, 2) = + = 0. 2 2 2 2 Proposition 2.2. Let φpS−NR be defined in (3) with p > 1 being a positive odd integer. Then, the following hold. (a) An alternative expression of φpS−NR is p φNR (a, b) if a > b, φpS−NR (a, b) = ap = b p if a = b, p φNR (b, a) if a < b. 4
(b) The function φpS−NR is not differentiable. However, φpS−NR is continuously differentiable on the set Ω := {(a, b) | a 6= b} with ( p [ ap−1 − (a − b)p−1 , (a − b)p−1 ]T if a > b, ∇φpS−NR (a, b) = p [ (b − a)p−1 , bp−1 − (b − a)p−1 ]T if a < b. In a more compact form, ( ∇φpS−NR (a, b) =
p−1 p [ φNR (a, b), (a − b)p−1 ]T if a > b, p−1 p [ (b − a)p−1 , φNR (b, a) ]T if a < b.
(c) The function φpS−NR is twice continuously differentiable on the set Ω = {(a, b) | a 6= b} with " # ap−2 − (a − b)p−2 (a − b)p−2 if a > b, p(p − 1) (a − b)p−2 −(a − b)p−2 2 p # " ∇ φS−NR (a, b) = −(b − a)p−2 (b − a)p−2 if a < b. p(p − 1) (b − a)p−2 bp−2 − (b − a)p−2 In a more compact form, " p(p − 1) 2 p " ∇ φS−NR (a, b) = p(p − 1)
p−2 (a, b) φNR
(a − b)p−2
#
(a − b)p−2 −(a − b)p−2 # −(b − a)p−2 (b − a)p−2 (b − a)p−2
φp−2 (b, a) NR
Proof. The arguments are just direct computations, we omit them.
if a > b, if a < b. 2
At last, we present some other variants of φpS−NR . Indeed, analogous to those functions in [26], the variants of φpS−NR as below can be verified being NCP-functions. φe1 (a, b) = φpS−NR (a, b) + α(a)+ (b)+ , α > 0. φe2 (a, b) = φpS−NR (a, b) + α ((a)+ (b)+ )2 , α > 0. φe3 (a, b) = φp (a, b) + α ((ab)+ )4 , α > 0. S−NR
φe4 (a, b) = φpS−NR (a, b) + α ((ab)+ )2 , α > 0. 2 φe5 (a, b) = φpS−NR (a, b) + α (a)+ )2 ((b)+ , α > 0.
Proposition 2.3. All the above functions φei (a, b) for i ∈ {1, 2, 3, 4, 5} are NCP-functions. 5
Proof. We only show that φe1 (a, b) is an NCP-function and the same argument can be applied to the other cases. First, we denote Ω := {(a, b) | a > 0, b > 0} the first quadrant and suppose that φe1 (a, b) = 0. If (a, b) ∈ Ω, then φpS−NR (a, b) > 0 by Proposition 2.1; and hence, φe1 (a, b) > 0. This is a contradiction. Therefore, we must have (a, b) ∈ Ωc which says (a)+ (b)+ = 0. This further implies φpS−NR (a, b) = 0 which is equivalent to a, b ≥ 0, ab = 0 by applying Proposition 2.1 again. Thus, φe1 is an NCP-function. 2
3
p The second symmetrization function ψS−NR
p In this section, we show that the function ψS−NR defined in (4) is not only an NCPp function, but also a merit function. In particular, ψS−NR possesses symmetric graph and is twice differentiable. p be defined in (4) with p > 1 being a positive odd integer. Proposition 3.1. Let ψS−NR p Then, ψS−NR is an NCP-function and is positive on the set
Ω = {(a, b) | ab 6= 0} ∪ {(a, b) | a < b = 0} ∪ {(a, b) | 0 = a > b}. p (a, b) = a2p > 0. Similarly, when Proof. First of all, when a < b = 0, we have ψS−NR p (a, b) = b2p > 0. For 0 6= a > b 6= 0, suppose that b > 0. Then, 0 = a > b, we have ψS−NR a > (a − b) which implies ap > (a − b)p and bp > 0, and hence ap bp − (a − b)p bp > 0. On the other hand, suppose that b < 0. Then, a < (a − b) which implies ap < (a − b)p and bp < 0. Thus, we also have ap bp − (a − b)p bp > 0. For a = b 6= 0, it is clear that ap bp = a2p > 0. For the remaining case: 0 6= a < b 6= 0, the proof is similar to the case p is positive on the set Ω. of 0 6= a > b 6= 0. From all the above, we prove that ψS−NR p Next, we go on showing that ψS−NR is an NCP-function. Suppose that a > b and p p p p p p b a b − (a − b) b = [a − (a − b) ]b = 0. If b = 0, then we have a > b = 0. Otherwise, we have a = (a − b) which also yields that a > b = 0. Similarly, the condition a < b and ap bp − (b − a)p ap = 0 implies that b > a = 0. The remaining case a = b and ap bp = 0 p gives that a = b = 0. Thus, from all the above, ψS−NR is an NCP-function. 2
p p We can conclude from Proposition 3.1 that ψS−NR is a merit function, since ψS−NR is positive on Ω and is identically zero on the set {(a, b) | a ≥ b = 0} ∪ {(a, b) | 0 = a ≤ b}. p Next, we elaborate more about the function ψS−NR as below: p (i) For p being an even integer, ψS−NR is not an NCP-function. A counterexample is given as below. 2 ψS−NR (−2, −4) = (−2)2 (−4)2 − (−2 + 4)2 (−4)2 = 0.
6
p (ii) The function ψS−NR is neither convex nor concave function. To see this, taking p = 3 and using the following argument verify the assertion.
1 3 1 3 0 64 3 1 = ψS−NR (1, 1) < ψS−NR (0, 0) + ψS−NR (2, 2) = + = 32. 2 2 2 2 1 3 0 0 1 3 3 (2, 0) + ψS−NR (0, 2) = + = 0. 1 = ψS−NR (1, 1) > ψS−NR 2 2 2 2 p Proposition 3.2. Let ψS−NR be defined as in (4) with p > 1 being a positive odd integer. Then, the following hold.
(a) An alternative expression of φpS−NR is p p φNR (a, b)b if a > b, p ψS−NR (a, b) = ap bp = a2p if a = b, p φNR (b, a)ap if a < b. p (b) The function ψS−NR is continuously differentiable with
p−1 p p−1 p p p−1 − (a − b)p bp−1 + (a − b)p−1 bp ]T if a > b, p [ a b − (a − b) b , a b p ∇ψS−NR (a, b) = p [ ap−1 bp , ap bp−1 ]T = pa2p−1 [1 , 1 ]T if a = b, p [ ap−1 bp − (b − a)p ap−1 + (b − a)p−1 ap , ap bp−1 − (b − a)p−1 ap ]T if a < b. In a more compact form, p−1 p p p−1 + (a − b)p−1 bp ]T if a > b, p [ φNR (a, b)b , φNR (a, b)b p ∇ψS−NR (a, b) = p [ a2p−1 , a2p−1 ]T if a = b, (b, a)ap ]T if a < b. p [ φpNR (b, a)ap−1 + (b − a)p−1 ap , φp−1 NR
7
p (c) The function ψS−NR is twice continuously differentiable with
p−2 − (a − b)p−2 ]bp (p − 1)[a p (p − 1)(a − b)p−2 bp +p[ap−1 − (a − b)p−1 ]bp−1 " (p − 1)ap−2 bp pap−1 bp−1 p p ∇2 ψS−NR (a, b) = p−1 p−1 (p − 1)ap bp−2 pa b (p − 1)[bp − (b − a)p ]ap−2 +2p(b − a)p−1 ap−1 −(p − 1)(b − a)p−2 ap p (p − 1)(b − a)p−2 ap +p[bp−1 − (b − a)p−1 ]ap−1
(p − 1)(a − b)p−2 bp
+p[ap−1 − (a − b)p−1 ]bp−1 p p p−2 if a > b, (p − 1)[a − (a − b) ]b p−1 p−1 +2p(a − b) b p−2 p # −(p − 1)(a − b) b if a = b, p−2 p
(p − 1)(b − a) +p[b
p−1
a
p−1
− (b − a)
]a
p−1
(p − 1)[bp−2 − (b − a)p−2 ]ap
if a < b.
Proof. (a) It is clear to see this part. p (a, b) on the set {(a, b) | a > (b) It is easy to verify the continuous differentiability of ψS−NR b or a < b}. We only need to check the differentiability along the line a = b. Suppose that h > k, we observe that p p ψS−NR (a + h, a + k) − ψS−NR (a, a)
= (a + h)p (a + k)p − (h − k)p bp − a2p
= pa2p−1 (1, 1), (h, k) + R(a, h, k). Here the remainder R(a, h, k) is o(h, k) function of h and k, since the degree of h and k of R(a, h, k) is at least 2. Similarly, from the other two cases h = k and h < k, we can p p conclude that ∇ψS−NR (a, a) = pa2p−1 (1, 1)T . In addition, the continuity of ∇ψS−NR (a, b) along the line a = b is easy to verify. (c) The arguments for this part are similar to those for part(b). We omit them.
2
p Again, we present some other variants of ψS−NR . Indeed, analogous to those functions p in [26], the variants of ψS−NR as below can be verified being NCP-functions.
8
p ψe1 (a, b) = ψS−NR (a, b) + α(a)+ (b)+ , α > 0. p ψe2 (a, b) = ψS−NR (a, b) + α ((a)+ (b)+ )2 , α > 0. ψe3 (a, b) = ψ p (a, b) + α ((ab)+ )4 , α > 0. S−NR
p ψe4 (a, b) = ψS−NR (a, b) + α ((ab)+ )2 , α > 0. 2 p ψe5 (a, b) = ψS−NR (a, b) + α (a)+ )2 ((b)+ , α > 0.
Proposition 3.3. All the above functions ψei (a, b) for i ∈ {1, 2, 3, 4, 5} are NCP-functions. Proof. We only show that ψe1 is an NCP-function and the same argument can be applied to the other cases. Let Ω := {(a, b) | ab 6= 0} and suppose that ψe1 (a, b) = 0. If (a, b) ∈ Ω, p (a, b) > 0 by Proposition 3.1; and hence, ψe1 (a, b) > 0. This is a contradiction. then ψS−NR Therefore, we must have (a, b) ∈ Ωc which says (a)+ (b)+ = 0. This further implies p (a, b) = 0 which is equivalent to a, b ≥ 0, ab = 0 by applying Proposition 3.1 again. ψS−NR Thus, ψe1 is an NCP-function. 2
4
Concluding remarks
p for the single value Due to space limitation, we illustrate the functions φpS−NR and ψS−NR p = 3, see Figure 1. Nonetheless, we make some remarks about the surfaces of φpS−NR and p ψS−NR , as well as say a few words about their algebraic properties. First of all, it is clear p p to see that φpS−NR (a, b) = φpS−NR (b, a) and ψS−NR (a, b) = ψS−NR (b, a), which mean that the p p surfaces of φS−NR and ψS−NR are both symmetric with respect to the line a = b. As for the algebraic structure, we can verify that 2 p (a, b) = min(a, b) for p = 1. ψS−NR
To see this, for example if a > b, we check that a1 b1 − (a − b)1 b1 = b2 = min(a, b)2 . p On the other hand, for large p = 3, 5, 7, · · · , the function ψS−NR does not coincide with 2p p min(a, b) . Nonetheless, when we restrict ψS−NR (a, b) on the line a = b and two axes a = 0 and b = 0, we really have that 2p p ψS−NR (a, b) = min(a, b) . p In summary, ψS−NR can be viewed as a merit function relative to the original natural p residual NCP-function φNR (a, b) = min(a, b). Besides, we have to mention that ψS−NR is
9
twice continuously differentiable so that it is good enough to develop a lot of algorithms based on this property. However, it does not satisfy that p p ∇a ψS−NR (a, b) · ∇b ψS−NR (a, b) ≥ 0. (cf. Property 2.2(d) in [2]) 3 3 For example, taking p = 3 and (a, b) = (0, −1) gives ∇a ψS−NR (0, −1) = 3 and ∇b ψS−NR (0, −1) = −6. This may cause some difficulty in analyzing the convergence rate.
×10 -12
2
8
z-axis
z-axis
0 -2 -4
6 4 2
-6
0 -0.01
-8 -1
-0.005 -0.5
1 0
x-axis
0.01 0
0.005
0.5 0
0.5
x-axis
-0.5 1
-1
0
0.005 -0.005 0.01
y-axis
(a) the graph of φpS−NR
-0.01
y-axis
p (b) the graph of ψS−NR
p with p = 3. Figure 1: The surfaces of φpS−NR and ψS−NR
As it can be seen, the surface of φpS−NR looks like “two-wings” of an eagle and there is cusp along x = y. Moreover, the graph of φpS−NR is neither convex nor concave. The p is smooth and it is neither convex nor concave. surface of ψS−NR To sum up, we propose new NCP-functions and merit functions in this short paper. p Both of them possess symmetric graphs. With our discovery of φpS−NR , ψS−NR in this short paper, there are many future directions to be explored. We list some of them. • Discovering benefits for such symmetrization. p • Doing numerical comparisons among φpFB , φpNR , φpS−NR , and ψS−NR involved in various algorithms.
• Studying the effect when perturbing the parameter p, applying this new family of NCP-functions to suitable optimization problems. • Extending these functions as the complementarity function associated with the second-order cone and symmetric cones.
10
• Developing some analytic properties on φpS−NR such as directional differentiability, Lipchitz continuity, semismoothness. p Finally, we would like to point out that the p-th root of φpS−NR and ψS−NR are also NCP-functions. In other words, the functions
p p1 φS−NR ,
p p1 ψS−NR
are NCP-functions, too. The proof is routine, so we omit it. Acknowledgements. This work is supported by Ministry of Science and Technology, Taiwan. The second author is a member of Mathematics Division, National Center for Theoretical Sciences.
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