A conceptual method for solving generalized semi ... - Semantic Scholar
Eur. J. Oper. Res. 157, No. 1, 3-15 (2004).
Continuous Optimization
A conceptual method for solving generalized semi-infinite programming problems via global optimization by exact discontinuous penalization Abebe Geletu, Armin Hoffmann Technical University of Ilmenau, Institute of Mathematics, PF 100565, D98684 Ilmenau, Germany, [email protected], [email protected]
Abstract We consider a generalized semi-infinite programming problem (GSIP) with one semi-infinite constraint where the index set depends on the variable to be minimized. Keeping in mind the integral global optimization method of Zheng & Chew and its modifications we would like to outline theoretical considerations for determining coarse approximations of a solution of (GSIP) via global optimization of an exact discontinuous penalty approach. We consider an auxiliary parametric semi-infinite programming problem and the behavior of its marginal functional. In so doing we extend the theory of robust analysis to study robustness of marginal functions and robustness of set valued mappings with given structures. Keywords: Global optimization, Penalty methods, Generalized semi-infinite programming, Robustness and sensitivity analysis, Piecewise lower (upper) semi-continuous. MSC 2000: 90C34, 49J52, 49J53, 90C31
1
Introduction
programming problem (PSIP) characterizing the admissible set of the (GSIP) and enabeling us to formulate the (GSIP) via some discontinuous penalty [29] equivalently as a global optimization problem. For satisfying the assumptions of (IGOM) we do not need the lower or upper semi-continuity of the index mapping with respect to the full domain X. We have to ensure the upper robustness and measurability of the penalty term which mainly consists of the marginal function of (PSIP). Hence we give in this paper a possible frame work how to use the theory of robust analysis [28, 17, 16] to warranty the upper robustness of marginal functions. Additionally, we give conditions working for some kind of (GSIP). Roughly spoken, we use the continuity of the semi-infinite constraint function over its compact domain, some piecewise continuity of the index mapping and some metric regularity of the constraint function w.r.t. the index mapping in a modified sense. The paper is organized as follows. In Section 2 we formulate the problem and the main troubles associated with (GSIP). In Section 3 we give an obvious characterization of the feasible set of (GSIP) to be used in Section 4 for the con-
We attempt to discuss the possibility of numerical solvability of a class of generalized semi-infinite programming problems (GSIP) by the integral global optimization method (IGOM) which was proposed by Zheng and Chew [3] and was further elaborated and extended by Zheng et al [3, 16, 28, 29, 17], Hoffmann and Ph´ u [10, 5, 6], Hichert [4]. The (IGOM) has been found computationally viable in case when the data of the optimization problem posses some relevant discontinuity properties, which are characterized by the notions of robustness [28]. Moreover, the last ten years have seen an avid interest on the theoretical and numerical investigation of the class of problems known as (GSIP). Among major publications on theoretical aspects we find those works by R¨ uckmann and Shapiro [14, 13] Still, Stein and Still [23, 24, 21], Weber [25] etc. Recently, we also find first attempts on the numerical aspects of (GSIP), see Still [24], Stein & Still [22], Levitin & Tichatschke [9], Pickl & Weber [11] and Weber [26, 27]. In our case, we define an auxiliary parametric semi-infinite 1
2
struction of a global optimization problem which is equivalent to (GSIP) under mild conditions. In Section 5 we give a brief review over the (IGOM) and discuss the main assumptions being necessary for its application. In the remaining two sections we develop the main results to ensure the robustness and measurability properties of the penalty function.
2
Problem and motivation
3
We consider the problem f (x) → G(x, t) x
≥ ∈
inf 0, ∀ t ∈ B(x), X,
where we make the Assumption (A1): The sets X ⊂ Rn , T ⊂ Rm are compact and non empty, the functions f : X → R and G : X × T → R are upper semi continuous (u.s.c.) on X and continuous on X×T , respectively. The set-valued mapping (SVM) B : X ⇒ T is at least compact valued but may have empty values for some x ∈ X. We assume neither the upper nor the lower semicontinuity of B on X. Naturally, in case of continuity of B, we feel that usual treatments are more preferable than the one we suggested here. The admissible set of the (GSIP) defined by M := {x ∈ X | G(x, t) ≥ 0, ∀ t ∈ B(x)} may posses strange properties [18, 19, 20, 23]. Especially, M might be not closed, not connected, not locally convex, etc. which is a large handicap for solving such problems numerically. The following example could give an impression about the situation. 2
Example 2.1 [7] Let X = [−10, 10] and put ½ √ √ [− x1 , x1 ] if x1 ≥ 0, B(x) := ∅ if x1 < 0.
Obviously, B is compact valued, and continuous1 w.r.t. the relative topology when restricted to each of the sets {x ∈ X | x1 ≥ 0} and {x ∈ X | x1 < 0}, u.s.c but not l.s.c. on the whole set X. Already with one linear semi-infinite constraint M := {x ∈ X | G(x, t) = t + x2 ≥ 0 ∀ t ∈ B(x)}, we easily find that M = {x ∈ X | x1 < 0} ∪ {x ∈ X | x1 ≥ 0,
√
Problem of feasibility
We define now some kind of distance function p from some open superset of X − X = {y − x | y, x ∈ X} into the nonnegative reals with the following additional properties.
(GSIP)
which is not a closed set.
Jongen et al [7] have shown that if B is a continuous set valued mapping, then M will be a closed set. Nevertheless, we make no such assumption over the whole set X, except on some subset of it, as could be seen later. Consequently, the (GSIP) may not posses a solution. Even then, we may assume that there is a minimizing sequence (see Section 4) for (GSIP), the existence of which may be assured with some further assumptions on the problem data.
x1 ≤ x2 },
Assumptions (A2): i) The function p : W → R+ is u.s.c. on W and continuous in some neighborhood of x = 0; and ii) p (x) = 0 if and only if x = 0. We consider, for each parameter x ∈ X, the problem of feasibility s.t.
p(x − ξ)
→
inf
G(ξ, t) ξ
≥ ∈
0, ∀ t ∈ B(x), X.
(PSIP)
Problem (PSIP) is a parametric semi-infinite programming problem, in which, if we fix x ∈ X, the resulting problem is an ordinary semi-infinite programming problem (SIP). If we let M (x) := {ξ ∈ X | G(ξ, t) ≥ 0, ∀ t ∈ B(x)},
it can be seen that the sets M (x), x ∈ X, of (PSIP) and M of (GSIP) have, in general, entirely different structures. The set valued map M : X ⇒ X has closed values by the continuity of the function G. For instance, for the example given above we have M (x) = {ξ ∈ X | G(ξ, t) = t + ξ2 ≥ 0, ∀ t ∈ B(x)},
which yields ½ √ {ξ ∈ X | x1 ≤ ξ2 } if x1 ≥ 0, M (x) = X if x1 < 0.
We now define a (generally discontinuous) marginal value function ( inf p(x − ξ) if M (x) 6= ∅, ξ∈M (x) ϕ(x) := +∞ if M (x) = ∅.
An obvious but important property of the function ϕ is given in the following proposition.
1 From the usual definitions of the semi-continuity (see e.g. [1, 2]) it follows that a set valued map is also l.s.c. at those points where it has empty image and u.s.c. at points with empty image whenever the image is empty in some neighborhood of such points.
3
Proposition 3.1 Assume that (A1) and (A2) are satisfied. Then we obtain the equivalences x ∈ M ⇐⇒ x ∈ M (x) ⇐⇒ ϕ(x) = 0. Proof. Obviously, without using the assumptions, we find by definition of M and M (x) the implications x ∈ M ⇐⇒ G(x, t) ≥ 0, ∀ t ∈ B(x) ⇐⇒ x ∈ M (x) =⇒ ϕ(x) = 0. Now let ϕ(x) = 0. Then 0 = inf ξ∈M (x) p(x − ξ) yields by (A1) a convergent sequence (ξn )n∈N with limit x ¯ ∈ M (x) and by (A2) such one that 0 = limn→∞ p(x − ξn ) = p (x − x ¯). Hence x ∈ M (x). ¤ Remark 3.2 As a special case of the function p we may take p(x − ξ) := kx − ξk, or p(x − ξ) := r (kx − ξk) , where r : R+ → R+ is continuous on R+ and zero only at zero. Consequently, we have in the first case ϕ(x) = inf ξ∈M (x) kx − ξk = dist(x, M (x)), for M (x) 6= ∅ where dist (x, M (x)) is the well known distance from the point x to the set M (x).
4
Exact penalty approach
In this section we introduce a discontinuous penalty function and verify, under some assumptions, the penalized problem and (GSIP) posses the same set of minimizing sequences. Following Proposition 3.1 we have, for fixed d > 0, the discontinuous penalty function ½ 0 if x ∈ M, ϕd (x) := ϕ(x) + d if x ∈ / M, of the admissible set M of (GSIP) and consider the associated penalty problem f (x) + λϕd (x) → x ∈
inf, X.
(PPλ d)
A sequence {xn } ⊂ X is called a minimizing sequence of (GSIP) iff 1. ∃ n0 ∈ N : ∀ n ≥ n0 : xn ∈ M , 2. limn→∞ f (xn ) = inf x∈M f (x) =: α∗ . A sequence {yn } ⊂ X is called a minimizing sequence of (PPλd ) iff limn→∞ {f (yn ) + λϕd (yn )} = inf y∈X {f (y) + λϕd (y)} =: β∗ . Theorem 4.1 Let M be non-empty, f be Lipschitzcontinuous with module L, D be the diameter of X and let d > 0. If {xn } in X is a minimizing sequence of (PPλd ) and λd > DL, then {xn } is a minimizing sequence of (GSIP).
Proof. Let {xn } be a minimizing sequence of (PPλd ) and ε > 0 be given. Then, there is some n0 (ε) such that for all n ≥ n0 (ε)
β∗ ≤ f (xn ) + λϕd (xn ) ≤ β∗ + ε
(1)
kxnε − xn k ≥ dist(xn , M ) ≥ kxnε − xn k − ε.
(2)
and corresponding xnε ∈ M such that It follows that f (xn )
+ λϕd (xn ) − ε ≤ β∗
≤ f (xnε ) + λϕd (xnε ) = f (xnε )
(3)
and |f (xnε ) − f (xn )| + ε λ L ε ≤ kxnε − xn k + λ λ ε L (dist(xn ,M ) + ε) + (4) ≤ λ λ L ε ≤ (D + ε) + , λ λ since dist(xn ,M ) ≤ D. Hence for xn ∈ / M, n ≥ n0 (ε) and ϕd (xn ) ≤
(d − L λ D) λ =: ε0 L+1 we get then the contradiction ε
0 and if there is some γ > 0 such that ϕ(x) ≥ γ dist(x, M ), ∀x ∈ X
and γ > L λ , then {xn } is a minimizing sequence of (GSIP). Furthermore, {xn } is a minimizing sequence for each other d > 0 with the above parameter λ.
4 Proof. Let {xn } be a minimizing sequence of (PPλd ) and ε > 0 be given. Then, there is some n0 (ε) such that for all n ≥ n0 (ε) and associated xnε ∈ M we have again (1) – (4). Hence for xn ∈ / M, n ≥ n0 (ε) and
Consequently, {xn } is a minimizing sequence of (PPλd ). ¤
ε
0. ¤
0
< d−(
Corollary 4.4 Let any assumption be satisfied, which ensures that whenever {xn } is a minimizing sequence for (PPλd ) thus {xn } is a minimizing sequence of (GSIP). Then each minimizing sequence of (GSIP) is also a minimizing sequence of (PPλd ). Proof. Let {xn } be a minimizing sequence of (GSIP). Then it follows by definition that ∃ n0 : ∀ n ≥ n0 : ϕ(xn ) = 0 and lim f (xn ) = inf f (x) = α∗ . n→∞
x∈M
Let ε > 0. Then for some n1 (ε) with n ≥ n1 > n0 and for a minimizing sequence {yn } of (PPλd ) we have α∗ + ε
≥ f (xn ) = f (xn ) + λϕd (xn )
≥
Coarse global optimization approach
Assuming that (PSIP) could be handled for fixed x by known algorithms of semi-infinite programming problems, we want to determine a coarse approximation to the solution of (GSIP) by solving (PPλd ) with suitable λ and d keeping in mind the integral global optimization method (IGOM). However, (IGOM) has its roots in robust analysis, measure and integration theory. Hence, we briefly mention next some important properties. We take X ⊂ Rn , with int(X) 6= ∅, as a topological space; moreover, X is Lebesgue measurable, bounded in Rn and µ the Lebesgue measure in Rn .
5.1
Essential infimum and integral global optimization
Let f : X → R be Lebesgue measurable. Then α ∈ R is an essential lower bound of f iff f (x) ≥ α almost everywhere (a.e.) in X, i.e. µ{x ∈ X | f (x) < α} = 0. The essential infimum of f over X is the supremum over all essential lower bounds of f [10], i.e. ess inf f = sup {α ∈ R | µ{x ∈ X | f (x) < α} = 0 } . The (IGOM) theoretically uses iterations of the form
αk+1 :=
R
[f ≤αk ]
f (x)dµ(x)
µ[f ≤ αk ]
,
where [f ≤ αk ] := {x ∈ X | f (x) ≤ αk } are the level sets of f at the level αk . (IGOM) determines the essential infimum of f over X. We have
inf [f (x) + λϕd (x)]
x∈X
≥ f (yn ) + λϕd (yn ) − ε ≥ f (yn ) − ε ≥ α∗ − ε.
Theorem 5.1 ([3]) If f ∈ L∞ (X) and µ(X) < ∞, then limk→∞ αk = ess inf x∈X f (x).
5
Hichert [4] has developed a software package, known as BARLO, as an implementation of the IGOM. The coded routines of BARLO include: Monte-Carlo Sampling methods and Mean-value/Rieman sum methods, which are developed by Zheng [28] for computation of integrals; and branch and bound methods for level set approximation (Hichert [4]). Furthermore, the algorithms suggested by Chew/Zheng [3] are improved and sped up using some duality and Newton techniques (Hichert et al. [5, 6]) through the volume function introduced by Ph´ u/Hoffmann [10]. To apply this package for our purpose we have to ensure that the main assumptions of this method are valid and that the infimum and the essential infimum are equal in order to find minimizing sequences of (GSIP).
5.2
When is min = inf = ess inf ?
Consider X as a topological space and let D ⊂ X. Then D is called a robust set [28] iff cl D = cl(int D), where cl D and int D denote the closure and the interior of D, resp., in the topology of X. Remark 5.2 In [28] we find that ∅, X and open sets are robust, the union of an arbitrary collection of robust sets is again robust and the intersection of two robust sets may be not robust. However, the intersection of an open and a robust set is again robust. We say [28] that a function f : X → R is upper robust (u.r.) on X iff for all c ∈ R the set Fc := {x ∈ X | f (x) < c} =: [f < c] is a robust set. Among a lots of properties of upper robust functions we find the following statements.
Proposition 5.7 Let ϕ and ϕd be as above. If ϕ is an upper robust and measurable function and M is a robust and measurable set, then ∀d > 0 : ϕd is also upper robust and measurable. Proof. Given c ∈ R {x ∈ X | ϕd (x) < c} (8) if c < 0 ∅ M if 0 < c ≤ d : = {x ∈ X | ϕ(x) < c − d} if c > d. ¤
Consequently, it remains for us to show upper robustness and measurability of the marginal function ϕ of (PSIP) and the robustness and measurability of the admissible set M of (GSIP), under suitable assumptions. This is the remaining subject of this paper.
6
Upper robustness of marginal functions
6.1
General ideas
We first cite further relevant definitions and properties of robustness. Let D ⊂ X. x ∈ cl(D) is said to be a robust point to D [28] if N (x) ∩ int(D) 6= ∅ for each neighborhood N (x) of x. If, further, x ∈ D, then x is said to be a robust point of D. Proposition 6.1 ([28])
Corollary 5.3 (Zheng [28]) Let f : X → R. If X is a robust set and f is u.s.c., then f is u.r.
1. D is a robust subset of X if and only if each point x ∈ D is a robust point of D.
Theorem 5.4 (Ph´ u/Hoffmann [10]) Let f ∈ L∞ (X). If X is robust, Lebesgue measurable and f is u.r., then ess inf x∈X f (x) = inf x∈X f (x).
2. Any accumulation point of a set of robust points to D is also a robust point to D.
Corollary 5.5 Let f ∈ L∞ (X). If X is robust, Lebesgue measurable, f is u.r. and l.s.c., then ess inf x∈X f (x) = minx∈X f (x). For example, a function f of one variable is u.r. and l.s.c. whenever f is a l.s.c. regularization. That means the concept of upper robustness and lower semi-continuity allows us to minimize special kinds of discontinuous functions using the IGOM. However the set of u.r. functions is not a linear space. The sum of two or more u.r. functions generally need not to be u.r. Proposition 5.6 (Zheng [28]) Let f and ϕd be as above. If f is u.s.c. and ϕd is upper robust, then f + λϕd is upper robust, for every λ > 0.
Similarly, as each open set is a neighborhood of all its points, the robustness of a set is connected with a weaker notion of a neighborhood. D is called a semi-neighborhood of x iff x is a robust point of the set D. Corollary 6.2 A robust set D neighborhood of each of its points.
is
a
semi-
We also have the following properties, which we would frequently make use of in our discussions. Proposition 6.3 1. If D is a semi-neighborhood of x and int(D) ⊂ A, then A is also a semi-neighborhood of x.
6 2. If D is a semi-neighborhood of x and x ∈ O, where O is an open set, then D ∩ O is also a semi-neighborhood of x ([28]).
Choosing ε > 0 such that 0 < 2ε < c − ϕ(x0 ), we have Q ⊂ Φc . Hence Φc is a semi-neighborhood of x0 . ¤
Let M : X ⇒ Y be a set valued mapping. For y ∈ Y and C ⊂ Y , we define S M −1 (y) = {x ∈ X | y ∈ −1 M (x)} and M (C) = y∈C M −1 (y). Thus, M (·) is lower robust at x ∈ X iff for each y ∈ M (x) and each neighborhood U (y) ⊂ Y of y, M −1 (U (y)) is a semi-neighborhood of x in X. M (·) is lower robust (l.r.) iff M (·) is lower robust at x, for all x ∈ X.
Corollary 6.7 Let (X, ρ) be a metric space and M : X ⇒ Y be a set valued mapping. If M (·) is l.r., r : R+ → R+ is continuous and strictly increasing on R+ , then the function ϕ(x) = r (d(x, M (x))) :=
−1
Corollary 6.4 M (·) is l.r. iff M (U ) is a robust set in X for every nonempty open set U ⊂ Y . Corollary 6.5 If M : X ⇒ Y is l.s.c., then M (·) is l.r. A function f : X → R is called upper robust (u.r.) at x ∈ X iff for each c > f (x) the level set [f < c] is a semi-neighborhood of x. It is easy to see that f is u.r. on X if and only if f is u.r. at x for all x ∈ X. The set-valued mapping [1, 4] if x > 0 {4} if x = 0 M (x) := [2, 3] if x < 0
is a simple example of a set valued mapping which is l.r., but not l.s.c. at x = 0. Let us now come back to our original intention of investigating the behavior of marginal functions w.r.t. robustness properties of the data. Hence, we consider the marginal function ϕ defined by ϕ(x) :=
inf
y∈M (x)
ψ(x, y).
(9)
Theorem 6.6 (upper robustness of infimum) Let ψ : X × Y → R be u.s.c. on {x0 } × M (x0 ), where M : X ⇒ Y is a l.r. set valued mapping, then ϕ is u.r. at x0 . Proof. Let c ∈ R and x0 ∈ Φc := {x | ϕ(x) < c}. We have to show that Φc is a semi-neighborhood of x0 . Let ε > 0 be arbitrary, then there exists y ε ∈ M (x0 ) such that ψ(x0 , y ε ) < ϕ(x0 ) + ε. Since ψ is u.s.c. on {x0 } × M (x0 ), there exist open neighborhoods N (x0 ) of x0 and N (y ε ) of y ε such that ∀x ∈ N (x0 ), ∀y ∈ N (y ε ) : ψ(x, y) ≤ ψ(x0 , y ε ) + ε.
Moreover, M (·) is l.r., y ε ∈ M (x0 ) and N (y ε ) is a neighborhood of y ε imply that M −1 (N (y ε )) is a semi-neighborhood of x0 . Hence, Q := N (x0 ) ∩ M −1 (N (y ε )), by Proposition 6.3(2), is a semineighborhood of x0 , too. Thus, we have for all x ∈ Q, ye ∈ M (x) ∩ N (y ε ) ϕ(x)
= infy∈M (x) ψ(x, y) ≤ ψ(x, ye)
≤ ψ(x0 , y ε ) + ε < ϕ(x0 ) + 2ε.
inf
ξ∈M (x)
r (ρ(x, ξ))
is u.r. Proof. The functions ¡ρ and r are continuous and ¢ inf ξ∈M (x) r (ρ(x, ξ)) = r inf ξ∈M (x) ρ(x, ξ) . ¤
This corollary guarantees that, when r is as above, ψ : X × X → R+ , ψ(x, ξ) := r (kx − ξk) and the mapping M (·) is l.r., then the marginal function ϕ of the (PSIP) is u.r. In general (see Remark 5.2), the upper semi-continuity assumption on ψ cannot be replaced by upper robustness.
6.2
Robust partition of X - decomposition
Again following [17] we define piecewise semicontinuity and analogously, as a small extension, also piecewise robustness properties for functions and setvalued mappings. Here we have the main behavior that piecewise robustness imply robustness being not true for semi-continuity. Thus some suitable decomposition is possible under the weaker robustness assumptions. We state and prove only those results which are to be used in our framework. Let X and Y be two topological spaces. We say that X1 , X2 , . . . , Xr is a partition of X iff the sets Xi are pairwise disjoint and X is the union of all Xi . The partition is called robust iff each Xi is robust w.r.t. X. M : X ⇒ Y is said to be piecewise l.s.c. (l.r.)[u.s.c.] iff there exists a robust partition X1 , X2 , . . . , Xr of X such that for all i ∈ {1, . . . , r} the restriction of M (·) to Xi is l.s.c. (l. r.)[u.s.c.] with respect to the relative topology on Xi induced by the topological space X. ϕ : X → R is called piecewise u.r. iff there exists a robust partition X1 , X2 , . . . , Xr of X such that for all i ∈ {1, . . . , r} the restriction of ϕ to Xi is u.r. with respect to the relative topology of Xi induced by the topological space X. Theorem 6.8 ([17]) If M (·) is piecewise l.s.c., then M (·) is l.r.
7
Lemma 6.9 Let X be a topological space and A be a non-empty robust subset of X. If B ⊂ A such that intA B 6= ∅, then intX B 6= ∅, where intA B is interior of B relative to the topology of A induced by X. Proof. Clearly, intA B is an open set in A. Hence, there exists O ⊂ X open in X such that B ⊃ intA B = O ∩ A. Since A is robust in X and O∩A 6= ∅ (while intA B 6= ∅ and A 6= ∅) we have that O ∩ intX A 6= ∅. This yields B ⊃ intA B = O ∩ A ⊃ O ∩ intX A 6= ∅ and intX B ⊃ O ∩ intX A 6= ∅ which completes the proof. ¤
Proof. Since M (·) is piecewise l.s.c. (piecewise l.r.), there is a robust partition X1 , ..., Xr of X such that for each i ∈ I := {1, ..., r}, Xi is robust in X and the restriction of M (·) to Xi is l.s.c. (l.r.). Thus, using [1, Thm. 4, p.51] (or Theorem 6.6) , we see that ϕ is u.s.c. (ϕ is u.r. ) on Xi , which implies that ϕ is u.r. on Xi for each i ∈ I. Therefore, by Thm. 6.10, ϕ is u.r. on X. ¤
Special structure of M (x)
6.3
If the set A ⊂ X is not assumed to be robust, then the above implication fails to be true. Take e.g. X = R, A = B = Q( Q: set of rational numbers). Observe that intA B 6= ∅. However, intX B = ∅ and A = Q is not robust in X = R.
Recall that M (x) := {ξ ∈ X | G(ξ, t) ≥ 0, ∀ t ∈ B(x)}. Until now we assumed the mapping M (·) to be l.r. Hence, based on the structure given to M (·) we would like next to give some conditions on G and B(·) that guarantee the required lower robustness of M (·). Following [8], we consider a right hand side perturbation
Theorem 6.10 Let X be a topological space and ϕ : X → R. If ϕ is piecewise u.r., then ϕ is u.r.
G(ξ, t)
Proof. Let c ∈SR, such that Fc := {x ∈ X | ϕ(x) < c}. Then Fc = i∈I (Xi ∩ {x ∈ X | ϕ(x) < c}) . Assume now x ∈ Fc and N (x) be any open neighborhood of x w.r.t. X. Then we get x ∈ Xi ∩ {x ∈ X | ϕ(x) < c} for some i ∈ I, and, hence, N (x) ∩ Xi is a neighborhood of x relative to Xi . Since ϕ is u.r. w.r.t. the relative topology on Xi , we get intXi [Xi ∩ {x ∈ X | ϕ(x) < c} ∩ N (x)] 6= ∅ and intXi [Xi ∩ {x ∈ X | ϕ(x) < c} ∩ N (x)] ⊂ N (x) ∩ Xi . Since N (x) is open and N (x) ∩ Xi is robust in X, Lemma 6.9 yields:
of the system defining M (·), where b ∈ C(T, R) and C(T, R) is the Banach space of all continuous functions from T in R. Let X ⊂ Rn be a compact metric space with a metric induced by some norm in Rn , ω := (x, b) ∈ X × C(T, R). Assume Σ(ω) ⊂ X is the set of solutions of the system (10), θ is the zero + function in C(T, R) and let [y] := max {0, y} for y ∈ R.
intX [{x
∈ X | ϕ(x) < c} ∩ N (x)] ⊃
intX [Xi ∩ {x ∈ X | ϕ(x) < c} ∩ N (x)] 6= ∅, from which follows that x is a robust point of {x ∈ X | ϕ(x) < c}. Consequently, by Prop. 6.1 1., we have that the set {x ∈ X | ϕ(x) < c} is robust and, therefore, ϕ is u.r. on X. ¤
ξ
≥ b(t), ∀ t ∈ B(x) ∈
(10)
X
Definition 6.13 (Local Metric Regularity (LMR), Klatte/Henrion [8]) Let ω = (x, b(·)), ω 0 = (x0 , θ). We say that the system (10) is metrically regular at (ω 0 , ξ 0 ) with respect to X if there exists a neighborhood (V × W )×U of (ω 0 , ξ 0 ) in (X × C(T, R)) × X and a real number γ > 0 such that dist(ξ, Σ(ω)) ≤ γ max [b(t) − G(ξ, t)]+ t∈B(x)
Theorem 6.11 If M (·) is piecewise l.r., then M (·) is l.r. Proof. Replace the property piecewise l.s.c. by piecewise l.r. and repeat the proof of [17] by using again the Lemma 6.9. ¤
for all ξ ∈ U and for all ω = (x, b) ∈ V × W . Assumption (A3): X has a robust partition (Xi )i∈I , I = {0, 1, 2, ..., r}, such that X0 := {x ∈ X |B (x) = ∅ }. Assumption (A4): B |Xi , i = 1, 2, ..., r, is u.s.c. w.r.t. the relative topology on Xi .
Theorem 6.12 Let ψ : X × Y → R be an u.s.c. function and let M : X ⇒ Y be a piecewise l.s.c. (l.r.) SVM on X. Then the marginal function ϕ is u.r. on X.
Theorem 6.14 If (A1), (A3) and (A4) are satisfied and (LMR) holds at (ω 0 , ξ 0 ) for each x0 ∈ X and each ξ 0 ∈ Σ(ω 0 ), then M (·) is l.r. on X.
8 Proof. For all x ∈ X0 we have M (x) = X, i.e. M (·) is continuous on X0 . Now let i > 0, let x0 ∈ Xi , ξ 0 ∈ M (x0 ) and let U be a neighborhood of ξ 0 in X. © ªFirst, we want to show that for each sequence xk with xk → x0 in Xi the intersection M (xk ) ∩ U 6= ∅ for all sufficiently large k. Since, (LMR) holds w.r.t. X, there are neighe of ξ 0 , and V × W of (x0 , θ) and a real borhoods U number γ > 0 such that dist(ξ, Σ(ω)) ≤ γ max [b(x) − G(ξ, t)]+ t∈B(x)
(11)
e , ω = (x, b) ∈ V × W . Thus, from (11) for all ξ ∈ U e ∩ U, ∀x ∈ V it follows that ∀ξ ∈ U dist(ξ, M (x)) ≤ γ max [−G(ξ, t)]+ . t∈B(x)
Set g(ξ, x) := [− inf t∈B(x) G(ξ, t)]+ , then we have that g(ξ, ·) is an u.s.c. function on Xi w.r.t. the relative topology on Xi . Furthermore, Xi is a robust set and x0 ∈ V implies that V ∩ int(Xi ) 6= ∅. Thus, let {xk } ⊂ Xi be any sequence such that xk → x0 (in the relative topology of Xi ). Then there is an integer k0 such that xk ∈ V ∩ Xi , ∀k ≥ k0 . Consequently, we have dist(ξ 0 , M (xk )) ≤ γg(ξ 0 , xk ), ∀k ≥ k0 . But M (xk ) is compact for each k ≥ k0 by the compactness of X. This implies that for each k ≥ k0 , there is ξ k ∈ M (xk ) such that kξ 0 − ξ k k ≤ γg(ξ 0 , xk ) + kxk − x0 k, ∀k ≥ k0 . However, limk→∞ kxk − x0 k = 0 and limk→∞ g(ξ 0 , xk ) ≤ g(ξ 0 , x0 ) = 0 (both with the relative topology on Xi ). Consequently, we have that limk→∞ kξ 0 − ξ k k = 0, from which fole ∩ U 6= ∅, ∀k ≥ k0 . That is lows that M (xk ) ∩ U k M (x ) ∩ U 6= ∅, ∀k ≥ k0 . Taking into consideration, x0 ∈ V ∩ Xi has been chosen arbitrarily we get that M (·) is l.s.c. with the relative topology on Xi , in fact for each i ∈ I. Consequently, M (·) is l.r. on X by Thm. 6.11 ¤ Observe that the upper semi-continuity of B on the whole of X is not assumed, except on each of the partitioning sets Xi of X.
6.4
Robustness of the admissible set M of GSIP
Recall that M := {x ∈ X | G(x, t) ≥ 0, ∀ t ∈ B(x)}. For B(x) := {t ∈ T | ai (x, t) ≤ 0, i ∈ I} where I is a finite index set and all the functions G and ai , i ∈ I are affine w.r.t. (x, t), it has been shown in [15] that M could be the union of a finite number of closed and open half spaces. Obviously in this case, M is a robust set. We want to give some topological condition for the robustness of M in a more general sense.
Take next a continuous function g : Rn → R and consider the set S := {x ∈ Rn | g(x) ≥ 0} and let S0 := {x ∈ Rn | g(x) = 0}. Strong slater condition (SSC): For each x0 ∈ S0 and each neighborhood N (x0 ), there is some x ∈ N (x0 ) such that g(x) > g(x0 ). Lemma 6.15 The set S is robust and closed under (SSC) and the continuity of g. Proof. Clearly {x ∈ Rn | g(x) > 0} ⊂ int(S) and S = {x ∈ Rn | g(x) > 0} ∪ {x ∈ R | g(x) = 0}. It then follows that S = int(S) ∪ {x ∈ X | g(x) = 0} = int(S) ∪ S0 . Obviously, every element of int(S) is a robust point of S. And if x0 ∈ S0 , by assumption, for every neighborhood N (x0 ) we have N (x0 ) ∩ int(S) 6= ∅. Then x0 is a robust point of S. Therefore, the set S is robust by Prop.6.1(1)) and closed by the (upper semi-)continuity of g. ¤ Assumption (A5): B |Xi is l.s.c. on Xi in the relative topology of Xi for i = 1, 2, ..., r. Theorem 6.16 If (A1), (A3), (A4), (A5) are satisfied and g : X \ X0 → R with g(x) := inf G(x, t) t∈B(x)
fulfils the (SSC) on each partition Xi , i = 1, 2, .., r with respect to the relative topology on Xi then the admissible set M of (GSIP) is robust. Proof. M can be equivalently written as M
:= S {x ∈ X | g(x) ≥ 0} ∪ X0 r = j=1 {x ∈ Xj | g(x) ≥ 0} ∪ X0 .
From assumption (A3) we have the robustness of X0 . (A1)(A4)(A5) ensure the continuity of g on the partition sets Xi w.r.t. the relative topology by using [1, Thm. 6, p. 53]. The (SSC) on each Xi yields that {x ∈ Xj | g(x) ≥ 0} is a robust set of Xi in the relative topology of Xi . By Lemma 6.9 (Xi is robust in X) this set is also robust in X. Finally M is robust as a union of robust sets. ¤ Sufficient for the (SSC) are conditions of the type ”extended Mangasarian Fromowitz Constraint qualification (EMFCQ)” whenever differentiability assumptions for G are made and the constraints describing the set-valued mapping B (x) satisfy some regularity conditions on each Xi . What we need to stress here again is that: the known strong conditions for nice behavior of a (GSIP) are expected to hold on each component of a suitable robust partition. For the usage of the IGOM such a partition need not be explicitly known. Only the existence of
9
such a partition is important. What we need apriori in any case is the robustness of the set of all x where the image of B is empty. In the known examples of (GSIP) with ill behavior given in [7, 19, 18] the conditions (A1), (A3)-(A5) can be principally satisfied. However, for a few of these examples a free choice of some functions must be properly done so that our basic assumptions hold.
7
Measurability functions
of
marginal
We consider the measurability of set-valued mappings according to [12] and cite the results which are in line with our investigations. Only a few results concerning again the partitioning of X must be shown. W.r.t. functions and sets in Rn we use the ordinary notion of Lebesgue measurability. We simply say measurable instead of Lebesgue measurable.
7.1
Basic definitions and results
Let X, Y ⊂ Rn be closed sets. M : X ⇒ Y is called measurable iff M is closed valued and M −1 (C) is measurable for each closed set C ⊂ Y . Let f : X × Y → R ∪ {−∞, +∞} =: R. Then the mapping Ef (x) := {(y, α) ∈ Y × R | f (x, y) ≤ α} is called the epigraphical map associated with f . The extended real-valued function f is a normal integrand iff for each x ∈ X the function y → f (x, y) → Y × R is is l.s.c. and epigraphical map Ef : X − → measurable. Theorem 7.1 [12] Let ϕ be the marginal function (9). If ψ : X × Y → R is a normal integrand and M : X ⇒ Y is measurable, then ϕ is measurable.
7.2
Special structure of M (x)
Next we give conditions on the index SVM B(·) and the function G to guarantee the measurability of M (·), of the marginal and of the penalty functions, and of the feasible set M of (GSIP). We say that X0 , X1 , ..., Xr is a (robust and measurable ) measurable partition of X iff X0 , X1 , ..., Xr is a partition of X and all parts Xi are (robust and measurable) measurable. We assume further that X0 := {x ∈ X |B (x) = ∅ } belongs to this partition. Proposition 7.2 Let X be some measurable subset of Rn and let ϕ : X → R be a function. Suppose also that X0 , X1 , ..., Xr is a measurable partition of X. If, for each i ∈ I, ϕ is measurable on Xi , then ϕ is measurable on X.
Proof. Let {Oα }α∈Λ be the family of measurable sets in Rn . Then, for each i ∈ I, the family of sets {Xi ∩ Oα }α∈Λ is the family of measurable sets w.r.t. the relative topology on Xi . Thus, the σ-algebra σ({Xi ∩ Oα }α∈Λ ) defines the measure on Xi . As ϕ is measurable w.r.t. Xi , then for any measurable set D ⊂ R we have that ϕ−1 (D) ∩ Xi is measurable in Xi . Since Xi are measurable in X, we have ϕ−1 (D) ∩ Xi is measurable in X. Therefore, since I is finite (countable infinite is here also possible) we conclude that [ (ϕ−1 (D) ∩ Xi ) = ϕ−1 (D) i∈I
is measurable in X. Consequently, ϕ is measurable on X. ¤ Hence based on this proposition, we are only required to verify the measurability of the ϕ of (PSIP) on each of the partitioning sets Xi of X. Defining g(ξ, x) := inf t∈B(x) G(ξ, t), we get M (x) = {ξ ∈ X | g(ξ, x) ≥ 0}. Proposition 7.3 Let Xi be measurable, i = 1, 2, .., r. If assumptions (A1), (A3), (A4), (A5) hold, then −g is a normal integrand on X × Xi . Proof. Observe that, by the assumptions, the function g : X × Xi → is continuous ([1]) and hence a normal integrand on X × Xi (see [12, p. 661ff.]). ¤
Proposition 7.4 [12] Let Xi be measurable. Then if −g is a normal integrand on X × Xi , then M (·) is measurable on Xi . Proof. Follows from [12].
¤
Theorem 7.5 (Measurability of the marginal function) Let the marginal function (9) be given and let (Xi )i∈I be a (robust) measurable partition of X. If ψ is a normal integrand on X × X and assumptions (A1), (A3), (A4), (A5) hold, then ϕ is measurable on each Xi ; hence, measurable on X. Proof. Clearly ψ is a normal integrand and M (·) is closed valued and measurable, by Prop. 7.4, on Xi . Moreover, by [12, Thm. 14.37] ϕ is measurable w.r.t. Xi . Therefore, by Prop. 7.2., ϕ is measurable. ¤
Corollary 7.6 If assumptions (A1), (A3), (A4), (A5) hold, then the feasible set M of (GSIP) is measurable. Hence, ϕd is also measurable.
10 Proof. By Prop. 3.2. M = ϕ−1 (0). However, under assumptions (A1), (A3), (A4), (A5), ϕ is measurable, by Thm. 7.5. Consequently, M is measurable. Furthermore, for the function ϕd (see Section 4), given c ∈ R, we have (8) and from which follows the measurability of ϕd due to the measurability of ϕ and that of M . ¤
8
Conclusions
We attempted to announce in this paper that robust analysis together with the analysis of measurability of functions and set-valued mappings is able to handle the so-called ill-behaved (GSIP). At least we can hope to get coarse approximations of a solution or a minimizing sequence by using some suitable global optimization procedure. The main ideas are the following: First to formulate the (GSIP) equivalently as a global optimization problem where the objective is closely connected with a discontinuous penalty of the feasible region of (GSIP). Second to apply a software which is able to solve global optimization problems with discontinuous but upper robust objective functions. This job could be done by the IGOM, which works with integration over level sets. Third to show that the penalty term is an upper robust function - under assumptions which do not reduce the class of treatable problems to the class of ”nice” (GSIP). This could be achieved by the marginal analysis of robust functions and set-valued mappings which has been principally developed in this paper. The measurability follows from standard arguments under similar assumptions. Some of the conditions such as the local metric regularity (LMR) and the strong slater condition (SSC) may be replaced for detailed investigations by suitable and sufficient criteria. We think that the EMFCQ (extended Mangasarian Fromowitz condition) or its modifications should be taken into consideration. Our investigations till now indicate that this approach works theoretically. Numerical treatments are under investigation. It is also in our belief that the results of this paper could be extended to more than one but finitely many so-called ”semi-infinite” constraints. However the set X should be a robust subset of Rn . Nevertheless, equality constraints are found to generate troubles if they partially describe the set X. Instead, one might need to add equality constraints as penalty to the objective.
A similar approach with discontinuous penalty function is given by the following problem f (x) + λ[−ψ(x)]+ → inf x ∈ X where ψ(x) = inf t∈B(x) G(x, t). Under the assumptions made above, we get, in a similar manner, the upper robustness of [−ψ(x)]+ = max{0, −ψ(x)} and also its measurability. Last, but not least, we would like to express our utmost indebtedness and thankfulness to our two anonymous referees for their valuable hints and comments; and above all, for their suggestion of new directions of investigations.
References [1] J.-P. Aubin, A. Cellina, Differential inclusions, Springer Verlag, Berlin, 1984. [2] J.-P. Aubin, H. Frankowska, Set valued analysis, Birkh¨auser, Basel, 1990. [3] S. Chew, Q. Zheng, Integral Global Optimization, Springer-Verlag, Berlin, 1988. [4] J. Hichert, Methoden zur Bestimmung des wesentlichen Supremums mit Anwendung in der globalen Optimierung, Dissertation, TUIlmenau, 1999, Berichte aus der Mathematik, Shaker-Verlag, Aachen, 2001. [5] J. Hichert, A. Hoffmann, H. X. Ph´ u, Convergence speed of an integral method for computing essential supremum, in: I. M. Bomze, T. Csendes, R. Horst, P. M. Pardalos, Developments in Global Optimization, Kluwer Academic Publishers, Dodrecht, Boston, London, 1997, pp. 153-170. [6] J. Hichert, A. Hoffmann, H. X. Ph´ u, R. Reinhardt, A primal-dual integral method in global optimzation, Discussiones Mathematicae, Differential Inclusions, Control and Optimization 20(2)(2000)257-278. [7] H. Th. Jongen, J.-J. R¨ uckmann, O. Stein, Generalized semi-infinite optimization, A first order optimality condition and examples, Mathematical Programming 83A(1)(1998)145 - 158. [8] D. Klatte, R. Henrion, Regularity and stability in non-linear semi-infinite optimization, in: R. Reemtsen, J.-J.R¨ uckmann (eds.), SemiInfinite Programming, Kluwer Academic Publishers, 1998, pp. 69-102.
11
[9] E. Levitin, R. Tichatschke, A Branch-andbound approach for solving a class of generalized semi-infinite programming problems, Journal of Global Optimization 13(3)(1998)299-315. [10] H. X. Ph´ u, A. Hoffmann, Essential supremum and supremum of summable functions, Numerical Functional Analysis and Optimization 17(1&2)(1996)167-180. [11] St. Pickl, G.-W. Weber, An algorithmic approach by linear programming problems in generalized semi-infinite optimization, Journal of Computer Science and Technology 5(3)(2000)62-82. [12] R. T. Rockafellar, R. J.-B. Wets, Varaitional Analysis, Springer Verlag, 1998. [13] J.-J. R¨ uckmann, A. Shapiro, Second order optimality conditions in generalized semiinfinite programming, Set-Valued Analysis 9(12)(2001)169-186. [14] J.-J. R¨ uckmann, A. Shapiro, First order optimality conditions in generalized semi-infinite programming, Journal of Optimization Theory and Applications 101(3)(1999)677-691. [15] J.-J. R¨ uckmann, O. Stein, On linear and linearized generalized semi-infinite optimization problems, Annals of Operation Research 101(2001)191-208. [16] S. Shi, Q. Zheng, D. Zhuang, Discontinuous robust mappings are approximatable, Transactions of the American Mathematical Society 347(12)(1995)4943-4957. [17] S. Shi, Q. Zheng, D. Zhuang, Set valued robust mappings and approximatable mappings, Journal of Mathematical Analysis and Applications 183(3)(1994)706 - 726. [18] O. Stein, The feasible set in generalized semiinfinite optimization, in: M. Lassonde (Ed.), Approximation, Optimization and Mathematical Economics, Physica, Heidelberg, 2001, pp. 309-327. [19] O. Stein, The reduction ansatz in the absence of lower semi-continuity, in: J. Guddat, R.
Hirabayashi and H. Th. Jongen (Eds.), Parametric Optimization and Related Topics Vol. V, pp. 165-178, Peter Lang Verlag, 2000. [20] O. Stein, On level sets of marginal functions, Optimization, 48(1)(2000)43-67. [21] O. Stein, G. Still, On optimality conditions for generalized semi-infinite programming, Journal of Optimization Theory and Applications 104(2)(2000)443-458. [22] O. Stein, G. Still, On generalized semiinfinite optimization and bilevel optimization, European Journal of Operations Research 142(3)(2002)444-462. [23] G. Still, Generalized semi-infinite programming: Theory and methods, European Journal of Operational Research, 119(2)(1999)301-313. [24] G. Still, Generalized semi-infinite programming: Numerical aspects, University of Twente, Faculty of Mathematical Sciences, Memorandum No. 1470, 1998. [25] G.-W. Weber, Generalized semi-infinite optimization: on some foundations, Journal of Computer Science and Technology 4(3)(1999)41-61. [26] G.-W. Weber, Generalized Semi-Infinite Optimization and Related Topics, habilitation thesis, Darmstadt University of Technology, Department of Mathematics, 1999/2000, pp.xii+359. [27] G.-W. Weber, Generalized semi-infinite optimization: theory and applications in optimal control and discrete optimization, in: Cambini, A., and Martein, L. (Eds.), Optimality Conditions, Generalized Convexity and Duality in Vector Optimization, Journal of Statistics & Management Systems,5 (2002), (to appear). [28] Q. Zheng, Integral global optimization of robust discontinuous functions, Dissertation, Clemson University, December 1992. [29] Q. Zheng, L. Zhang, Global minimization of constrained problems with discontinuous penalty functions, Computers & Mathematics with Applications 37(4-5)(1999)41-58.
Continuous Optimization
A conceptual method for solving generalized semi-infinite programming problems via global optimization by exact discontinuous penalization Abebe Geletu, Armin Hoffmann Technical University of Ilmenau, Institute of Mathematics, PF 100565, D98684 Ilmenau, Germany, [email protected], [email protected]
Abstract We consider a generalized semi-infinite programming problem (GSIP) with one semi-infinite constraint where the index set depends on the variable to be minimized. Keeping in mind the integral global optimization method of Zheng & Chew and its modifications we would like to outline theoretical considerations for determining coarse approximations of a solution of (GSIP) via global optimization of an exact discontinuous penalty approach. We consider an auxiliary parametric semi-infinite programming problem and the behavior of its marginal functional. In so doing we extend the theory of robust analysis to study robustness of marginal functions and robustness of set valued mappings with given structures. Keywords: Global optimization, Penalty methods, Generalized semi-infinite programming, Robustness and sensitivity analysis, Piecewise lower (upper) semi-continuous. MSC 2000: 90C34, 49J52, 49J53, 90C31
1
Introduction
programming problem (PSIP) characterizing the admissible set of the (GSIP) and enabeling us to formulate the (GSIP) via some discontinuous penalty [29] equivalently as a global optimization problem. For satisfying the assumptions of (IGOM) we do not need the lower or upper semi-continuity of the index mapping with respect to the full domain X. We have to ensure the upper robustness and measurability of the penalty term which mainly consists of the marginal function of (PSIP). Hence we give in this paper a possible frame work how to use the theory of robust analysis [28, 17, 16] to warranty the upper robustness of marginal functions. Additionally, we give conditions working for some kind of (GSIP). Roughly spoken, we use the continuity of the semi-infinite constraint function over its compact domain, some piecewise continuity of the index mapping and some metric regularity of the constraint function w.r.t. the index mapping in a modified sense. The paper is organized as follows. In Section 2 we formulate the problem and the main troubles associated with (GSIP). In Section 3 we give an obvious characterization of the feasible set of (GSIP) to be used in Section 4 for the con-
We attempt to discuss the possibility of numerical solvability of a class of generalized semi-infinite programming problems (GSIP) by the integral global optimization method (IGOM) which was proposed by Zheng and Chew [3] and was further elaborated and extended by Zheng et al [3, 16, 28, 29, 17], Hoffmann and Ph´ u [10, 5, 6], Hichert [4]. The (IGOM) has been found computationally viable in case when the data of the optimization problem posses some relevant discontinuity properties, which are characterized by the notions of robustness [28]. Moreover, the last ten years have seen an avid interest on the theoretical and numerical investigation of the class of problems known as (GSIP). Among major publications on theoretical aspects we find those works by R¨ uckmann and Shapiro [14, 13] Still, Stein and Still [23, 24, 21], Weber [25] etc. Recently, we also find first attempts on the numerical aspects of (GSIP), see Still [24], Stein & Still [22], Levitin & Tichatschke [9], Pickl & Weber [11] and Weber [26, 27]. In our case, we define an auxiliary parametric semi-infinite 1
2
struction of a global optimization problem which is equivalent to (GSIP) under mild conditions. In Section 5 we give a brief review over the (IGOM) and discuss the main assumptions being necessary for its application. In the remaining two sections we develop the main results to ensure the robustness and measurability properties of the penalty function.
2
Problem and motivation
3
We consider the problem f (x) → G(x, t) x
≥ ∈
inf 0, ∀ t ∈ B(x), X,
where we make the Assumption (A1): The sets X ⊂ Rn , T ⊂ Rm are compact and non empty, the functions f : X → R and G : X × T → R are upper semi continuous (u.s.c.) on X and continuous on X×T , respectively. The set-valued mapping (SVM) B : X ⇒ T is at least compact valued but may have empty values for some x ∈ X. We assume neither the upper nor the lower semicontinuity of B on X. Naturally, in case of continuity of B, we feel that usual treatments are more preferable than the one we suggested here. The admissible set of the (GSIP) defined by M := {x ∈ X | G(x, t) ≥ 0, ∀ t ∈ B(x)} may posses strange properties [18, 19, 20, 23]. Especially, M might be not closed, not connected, not locally convex, etc. which is a large handicap for solving such problems numerically. The following example could give an impression about the situation. 2
Example 2.1 [7] Let X = [−10, 10] and put ½ √ √ [− x1 , x1 ] if x1 ≥ 0, B(x) := ∅ if x1 < 0.
Obviously, B is compact valued, and continuous1 w.r.t. the relative topology when restricted to each of the sets {x ∈ X | x1 ≥ 0} and {x ∈ X | x1 < 0}, u.s.c but not l.s.c. on the whole set X. Already with one linear semi-infinite constraint M := {x ∈ X | G(x, t) = t + x2 ≥ 0 ∀ t ∈ B(x)}, we easily find that M = {x ∈ X | x1 < 0} ∪ {x ∈ X | x1 ≥ 0,
√
Problem of feasibility
We define now some kind of distance function p from some open superset of X − X = {y − x | y, x ∈ X} into the nonnegative reals with the following additional properties.
(GSIP)
which is not a closed set.
Jongen et al [7] have shown that if B is a continuous set valued mapping, then M will be a closed set. Nevertheless, we make no such assumption over the whole set X, except on some subset of it, as could be seen later. Consequently, the (GSIP) may not posses a solution. Even then, we may assume that there is a minimizing sequence (see Section 4) for (GSIP), the existence of which may be assured with some further assumptions on the problem data.
x1 ≤ x2 },
Assumptions (A2): i) The function p : W → R+ is u.s.c. on W and continuous in some neighborhood of x = 0; and ii) p (x) = 0 if and only if x = 0. We consider, for each parameter x ∈ X, the problem of feasibility s.t.
p(x − ξ)
→
inf
G(ξ, t) ξ
≥ ∈
0, ∀ t ∈ B(x), X.
(PSIP)
Problem (PSIP) is a parametric semi-infinite programming problem, in which, if we fix x ∈ X, the resulting problem is an ordinary semi-infinite programming problem (SIP). If we let M (x) := {ξ ∈ X | G(ξ, t) ≥ 0, ∀ t ∈ B(x)},
it can be seen that the sets M (x), x ∈ X, of (PSIP) and M of (GSIP) have, in general, entirely different structures. The set valued map M : X ⇒ X has closed values by the continuity of the function G. For instance, for the example given above we have M (x) = {ξ ∈ X | G(ξ, t) = t + ξ2 ≥ 0, ∀ t ∈ B(x)},
which yields ½ √ {ξ ∈ X | x1 ≤ ξ2 } if x1 ≥ 0, M (x) = X if x1 < 0.
We now define a (generally discontinuous) marginal value function ( inf p(x − ξ) if M (x) 6= ∅, ξ∈M (x) ϕ(x) := +∞ if M (x) = ∅.
An obvious but important property of the function ϕ is given in the following proposition.
1 From the usual definitions of the semi-continuity (see e.g. [1, 2]) it follows that a set valued map is also l.s.c. at those points where it has empty image and u.s.c. at points with empty image whenever the image is empty in some neighborhood of such points.
3
Proposition 3.1 Assume that (A1) and (A2) are satisfied. Then we obtain the equivalences x ∈ M ⇐⇒ x ∈ M (x) ⇐⇒ ϕ(x) = 0. Proof. Obviously, without using the assumptions, we find by definition of M and M (x) the implications x ∈ M ⇐⇒ G(x, t) ≥ 0, ∀ t ∈ B(x) ⇐⇒ x ∈ M (x) =⇒ ϕ(x) = 0. Now let ϕ(x) = 0. Then 0 = inf ξ∈M (x) p(x − ξ) yields by (A1) a convergent sequence (ξn )n∈N with limit x ¯ ∈ M (x) and by (A2) such one that 0 = limn→∞ p(x − ξn ) = p (x − x ¯). Hence x ∈ M (x). ¤ Remark 3.2 As a special case of the function p we may take p(x − ξ) := kx − ξk, or p(x − ξ) := r (kx − ξk) , where r : R+ → R+ is continuous on R+ and zero only at zero. Consequently, we have in the first case ϕ(x) = inf ξ∈M (x) kx − ξk = dist(x, M (x)), for M (x) 6= ∅ where dist (x, M (x)) is the well known distance from the point x to the set M (x).
4
Exact penalty approach
In this section we introduce a discontinuous penalty function and verify, under some assumptions, the penalized problem and (GSIP) posses the same set of minimizing sequences. Following Proposition 3.1 we have, for fixed d > 0, the discontinuous penalty function ½ 0 if x ∈ M, ϕd (x) := ϕ(x) + d if x ∈ / M, of the admissible set M of (GSIP) and consider the associated penalty problem f (x) + λϕd (x) → x ∈
inf, X.
(PPλ d)
A sequence {xn } ⊂ X is called a minimizing sequence of (GSIP) iff 1. ∃ n0 ∈ N : ∀ n ≥ n0 : xn ∈ M , 2. limn→∞ f (xn ) = inf x∈M f (x) =: α∗ . A sequence {yn } ⊂ X is called a minimizing sequence of (PPλd ) iff limn→∞ {f (yn ) + λϕd (yn )} = inf y∈X {f (y) + λϕd (y)} =: β∗ . Theorem 4.1 Let M be non-empty, f be Lipschitzcontinuous with module L, D be the diameter of X and let d > 0. If {xn } in X is a minimizing sequence of (PPλd ) and λd > DL, then {xn } is a minimizing sequence of (GSIP).
Proof. Let {xn } be a minimizing sequence of (PPλd ) and ε > 0 be given. Then, there is some n0 (ε) such that for all n ≥ n0 (ε)
β∗ ≤ f (xn ) + λϕd (xn ) ≤ β∗ + ε
(1)
kxnε − xn k ≥ dist(xn , M ) ≥ kxnε − xn k − ε.
(2)
and corresponding xnε ∈ M such that It follows that f (xn )
+ λϕd (xn ) − ε ≤ β∗
≤ f (xnε ) + λϕd (xnε ) = f (xnε )
(3)
and |f (xnε ) − f (xn )| + ε λ L ε ≤ kxnε − xn k + λ λ ε L (dist(xn ,M ) + ε) + (4) ≤ λ λ L ε ≤ (D + ε) + , λ λ since dist(xn ,M ) ≤ D. Hence for xn ∈ / M, n ≥ n0 (ε) and ϕd (xn ) ≤
(d − L λ D) λ =: ε0 L+1 we get then the contradiction ε
0 and if there is some γ > 0 such that ϕ(x) ≥ γ dist(x, M ), ∀x ∈ X
and γ > L λ , then {xn } is a minimizing sequence of (GSIP). Furthermore, {xn } is a minimizing sequence for each other d > 0 with the above parameter λ.
4 Proof. Let {xn } be a minimizing sequence of (PPλd ) and ε > 0 be given. Then, there is some n0 (ε) such that for all n ≥ n0 (ε) and associated xnε ∈ M we have again (1) – (4). Hence for xn ∈ / M, n ≥ n0 (ε) and
Consequently, {xn } is a minimizing sequence of (PPλd ). ¤
ε
0. ¤
0
< d−(
Corollary 4.4 Let any assumption be satisfied, which ensures that whenever {xn } is a minimizing sequence for (PPλd ) thus {xn } is a minimizing sequence of (GSIP). Then each minimizing sequence of (GSIP) is also a minimizing sequence of (PPλd ). Proof. Let {xn } be a minimizing sequence of (GSIP). Then it follows by definition that ∃ n0 : ∀ n ≥ n0 : ϕ(xn ) = 0 and lim f (xn ) = inf f (x) = α∗ . n→∞
x∈M
Let ε > 0. Then for some n1 (ε) with n ≥ n1 > n0 and for a minimizing sequence {yn } of (PPλd ) we have α∗ + ε
≥ f (xn ) = f (xn ) + λϕd (xn )
≥
Coarse global optimization approach
Assuming that (PSIP) could be handled for fixed x by known algorithms of semi-infinite programming problems, we want to determine a coarse approximation to the solution of (GSIP) by solving (PPλd ) with suitable λ and d keeping in mind the integral global optimization method (IGOM). However, (IGOM) has its roots in robust analysis, measure and integration theory. Hence, we briefly mention next some important properties. We take X ⊂ Rn , with int(X) 6= ∅, as a topological space; moreover, X is Lebesgue measurable, bounded in Rn and µ the Lebesgue measure in Rn .
5.1
Essential infimum and integral global optimization
Let f : X → R be Lebesgue measurable. Then α ∈ R is an essential lower bound of f iff f (x) ≥ α almost everywhere (a.e.) in X, i.e. µ{x ∈ X | f (x) < α} = 0. The essential infimum of f over X is the supremum over all essential lower bounds of f [10], i.e. ess inf f = sup {α ∈ R | µ{x ∈ X | f (x) < α} = 0 } . The (IGOM) theoretically uses iterations of the form
αk+1 :=
R
[f ≤αk ]
f (x)dµ(x)
µ[f ≤ αk ]
,
where [f ≤ αk ] := {x ∈ X | f (x) ≤ αk } are the level sets of f at the level αk . (IGOM) determines the essential infimum of f over X. We have
inf [f (x) + λϕd (x)]
x∈X
≥ f (yn ) + λϕd (yn ) − ε ≥ f (yn ) − ε ≥ α∗ − ε.
Theorem 5.1 ([3]) If f ∈ L∞ (X) and µ(X) < ∞, then limk→∞ αk = ess inf x∈X f (x).
5
Hichert [4] has developed a software package, known as BARLO, as an implementation of the IGOM. The coded routines of BARLO include: Monte-Carlo Sampling methods and Mean-value/Rieman sum methods, which are developed by Zheng [28] for computation of integrals; and branch and bound methods for level set approximation (Hichert [4]). Furthermore, the algorithms suggested by Chew/Zheng [3] are improved and sped up using some duality and Newton techniques (Hichert et al. [5, 6]) through the volume function introduced by Ph´ u/Hoffmann [10]. To apply this package for our purpose we have to ensure that the main assumptions of this method are valid and that the infimum and the essential infimum are equal in order to find minimizing sequences of (GSIP).
5.2
When is min = inf = ess inf ?
Consider X as a topological space and let D ⊂ X. Then D is called a robust set [28] iff cl D = cl(int D), where cl D and int D denote the closure and the interior of D, resp., in the topology of X. Remark 5.2 In [28] we find that ∅, X and open sets are robust, the union of an arbitrary collection of robust sets is again robust and the intersection of two robust sets may be not robust. However, the intersection of an open and a robust set is again robust. We say [28] that a function f : X → R is upper robust (u.r.) on X iff for all c ∈ R the set Fc := {x ∈ X | f (x) < c} =: [f < c] is a robust set. Among a lots of properties of upper robust functions we find the following statements.
Proposition 5.7 Let ϕ and ϕd be as above. If ϕ is an upper robust and measurable function and M is a robust and measurable set, then ∀d > 0 : ϕd is also upper robust and measurable. Proof. Given c ∈ R {x ∈ X | ϕd (x) < c} (8) if c < 0 ∅ M if 0 < c ≤ d : = {x ∈ X | ϕ(x) < c − d} if c > d. ¤
Consequently, it remains for us to show upper robustness and measurability of the marginal function ϕ of (PSIP) and the robustness and measurability of the admissible set M of (GSIP), under suitable assumptions. This is the remaining subject of this paper.
6
Upper robustness of marginal functions
6.1
General ideas
We first cite further relevant definitions and properties of robustness. Let D ⊂ X. x ∈ cl(D) is said to be a robust point to D [28] if N (x) ∩ int(D) 6= ∅ for each neighborhood N (x) of x. If, further, x ∈ D, then x is said to be a robust point of D. Proposition 6.1 ([28])
Corollary 5.3 (Zheng [28]) Let f : X → R. If X is a robust set and f is u.s.c., then f is u.r.
1. D is a robust subset of X if and only if each point x ∈ D is a robust point of D.
Theorem 5.4 (Ph´ u/Hoffmann [10]) Let f ∈ L∞ (X). If X is robust, Lebesgue measurable and f is u.r., then ess inf x∈X f (x) = inf x∈X f (x).
2. Any accumulation point of a set of robust points to D is also a robust point to D.
Corollary 5.5 Let f ∈ L∞ (X). If X is robust, Lebesgue measurable, f is u.r. and l.s.c., then ess inf x∈X f (x) = minx∈X f (x). For example, a function f of one variable is u.r. and l.s.c. whenever f is a l.s.c. regularization. That means the concept of upper robustness and lower semi-continuity allows us to minimize special kinds of discontinuous functions using the IGOM. However the set of u.r. functions is not a linear space. The sum of two or more u.r. functions generally need not to be u.r. Proposition 5.6 (Zheng [28]) Let f and ϕd be as above. If f is u.s.c. and ϕd is upper robust, then f + λϕd is upper robust, for every λ > 0.
Similarly, as each open set is a neighborhood of all its points, the robustness of a set is connected with a weaker notion of a neighborhood. D is called a semi-neighborhood of x iff x is a robust point of the set D. Corollary 6.2 A robust set D neighborhood of each of its points.
is
a
semi-
We also have the following properties, which we would frequently make use of in our discussions. Proposition 6.3 1. If D is a semi-neighborhood of x and int(D) ⊂ A, then A is also a semi-neighborhood of x.
6 2. If D is a semi-neighborhood of x and x ∈ O, where O is an open set, then D ∩ O is also a semi-neighborhood of x ([28]).
Choosing ε > 0 such that 0 < 2ε < c − ϕ(x0 ), we have Q ⊂ Φc . Hence Φc is a semi-neighborhood of x0 . ¤
Let M : X ⇒ Y be a set valued mapping. For y ∈ Y and C ⊂ Y , we define S M −1 (y) = {x ∈ X | y ∈ −1 M (x)} and M (C) = y∈C M −1 (y). Thus, M (·) is lower robust at x ∈ X iff for each y ∈ M (x) and each neighborhood U (y) ⊂ Y of y, M −1 (U (y)) is a semi-neighborhood of x in X. M (·) is lower robust (l.r.) iff M (·) is lower robust at x, for all x ∈ X.
Corollary 6.7 Let (X, ρ) be a metric space and M : X ⇒ Y be a set valued mapping. If M (·) is l.r., r : R+ → R+ is continuous and strictly increasing on R+ , then the function ϕ(x) = r (d(x, M (x))) :=
−1
Corollary 6.4 M (·) is l.r. iff M (U ) is a robust set in X for every nonempty open set U ⊂ Y . Corollary 6.5 If M : X ⇒ Y is l.s.c., then M (·) is l.r. A function f : X → R is called upper robust (u.r.) at x ∈ X iff for each c > f (x) the level set [f < c] is a semi-neighborhood of x. It is easy to see that f is u.r. on X if and only if f is u.r. at x for all x ∈ X. The set-valued mapping [1, 4] if x > 0 {4} if x = 0 M (x) := [2, 3] if x < 0
is a simple example of a set valued mapping which is l.r., but not l.s.c. at x = 0. Let us now come back to our original intention of investigating the behavior of marginal functions w.r.t. robustness properties of the data. Hence, we consider the marginal function ϕ defined by ϕ(x) :=
inf
y∈M (x)
ψ(x, y).
(9)
Theorem 6.6 (upper robustness of infimum) Let ψ : X × Y → R be u.s.c. on {x0 } × M (x0 ), where M : X ⇒ Y is a l.r. set valued mapping, then ϕ is u.r. at x0 . Proof. Let c ∈ R and x0 ∈ Φc := {x | ϕ(x) < c}. We have to show that Φc is a semi-neighborhood of x0 . Let ε > 0 be arbitrary, then there exists y ε ∈ M (x0 ) such that ψ(x0 , y ε ) < ϕ(x0 ) + ε. Since ψ is u.s.c. on {x0 } × M (x0 ), there exist open neighborhoods N (x0 ) of x0 and N (y ε ) of y ε such that ∀x ∈ N (x0 ), ∀y ∈ N (y ε ) : ψ(x, y) ≤ ψ(x0 , y ε ) + ε.
Moreover, M (·) is l.r., y ε ∈ M (x0 ) and N (y ε ) is a neighborhood of y ε imply that M −1 (N (y ε )) is a semi-neighborhood of x0 . Hence, Q := N (x0 ) ∩ M −1 (N (y ε )), by Proposition 6.3(2), is a semineighborhood of x0 , too. Thus, we have for all x ∈ Q, ye ∈ M (x) ∩ N (y ε ) ϕ(x)
= infy∈M (x) ψ(x, y) ≤ ψ(x, ye)
≤ ψ(x0 , y ε ) + ε < ϕ(x0 ) + 2ε.
inf
ξ∈M (x)
r (ρ(x, ξ))
is u.r. Proof. The functions ¡ρ and r are continuous and ¢ inf ξ∈M (x) r (ρ(x, ξ)) = r inf ξ∈M (x) ρ(x, ξ) . ¤
This corollary guarantees that, when r is as above, ψ : X × X → R+ , ψ(x, ξ) := r (kx − ξk) and the mapping M (·) is l.r., then the marginal function ϕ of the (PSIP) is u.r. In general (see Remark 5.2), the upper semi-continuity assumption on ψ cannot be replaced by upper robustness.
6.2
Robust partition of X - decomposition
Again following [17] we define piecewise semicontinuity and analogously, as a small extension, also piecewise robustness properties for functions and setvalued mappings. Here we have the main behavior that piecewise robustness imply robustness being not true for semi-continuity. Thus some suitable decomposition is possible under the weaker robustness assumptions. We state and prove only those results which are to be used in our framework. Let X and Y be two topological spaces. We say that X1 , X2 , . . . , Xr is a partition of X iff the sets Xi are pairwise disjoint and X is the union of all Xi . The partition is called robust iff each Xi is robust w.r.t. X. M : X ⇒ Y is said to be piecewise l.s.c. (l.r.)[u.s.c.] iff there exists a robust partition X1 , X2 , . . . , Xr of X such that for all i ∈ {1, . . . , r} the restriction of M (·) to Xi is l.s.c. (l. r.)[u.s.c.] with respect to the relative topology on Xi induced by the topological space X. ϕ : X → R is called piecewise u.r. iff there exists a robust partition X1 , X2 , . . . , Xr of X such that for all i ∈ {1, . . . , r} the restriction of ϕ to Xi is u.r. with respect to the relative topology of Xi induced by the topological space X. Theorem 6.8 ([17]) If M (·) is piecewise l.s.c., then M (·) is l.r.
7
Lemma 6.9 Let X be a topological space and A be a non-empty robust subset of X. If B ⊂ A such that intA B 6= ∅, then intX B 6= ∅, where intA B is interior of B relative to the topology of A induced by X. Proof. Clearly, intA B is an open set in A. Hence, there exists O ⊂ X open in X such that B ⊃ intA B = O ∩ A. Since A is robust in X and O∩A 6= ∅ (while intA B 6= ∅ and A 6= ∅) we have that O ∩ intX A 6= ∅. This yields B ⊃ intA B = O ∩ A ⊃ O ∩ intX A 6= ∅ and intX B ⊃ O ∩ intX A 6= ∅ which completes the proof. ¤
Proof. Since M (·) is piecewise l.s.c. (piecewise l.r.), there is a robust partition X1 , ..., Xr of X such that for each i ∈ I := {1, ..., r}, Xi is robust in X and the restriction of M (·) to Xi is l.s.c. (l.r.). Thus, using [1, Thm. 4, p.51] (or Theorem 6.6) , we see that ϕ is u.s.c. (ϕ is u.r. ) on Xi , which implies that ϕ is u.r. on Xi for each i ∈ I. Therefore, by Thm. 6.10, ϕ is u.r. on X. ¤
Special structure of M (x)
6.3
If the set A ⊂ X is not assumed to be robust, then the above implication fails to be true. Take e.g. X = R, A = B = Q( Q: set of rational numbers). Observe that intA B 6= ∅. However, intX B = ∅ and A = Q is not robust in X = R.
Recall that M (x) := {ξ ∈ X | G(ξ, t) ≥ 0, ∀ t ∈ B(x)}. Until now we assumed the mapping M (·) to be l.r. Hence, based on the structure given to M (·) we would like next to give some conditions on G and B(·) that guarantee the required lower robustness of M (·). Following [8], we consider a right hand side perturbation
Theorem 6.10 Let X be a topological space and ϕ : X → R. If ϕ is piecewise u.r., then ϕ is u.r.
G(ξ, t)
Proof. Let c ∈SR, such that Fc := {x ∈ X | ϕ(x) < c}. Then Fc = i∈I (Xi ∩ {x ∈ X | ϕ(x) < c}) . Assume now x ∈ Fc and N (x) be any open neighborhood of x w.r.t. X. Then we get x ∈ Xi ∩ {x ∈ X | ϕ(x) < c} for some i ∈ I, and, hence, N (x) ∩ Xi is a neighborhood of x relative to Xi . Since ϕ is u.r. w.r.t. the relative topology on Xi , we get intXi [Xi ∩ {x ∈ X | ϕ(x) < c} ∩ N (x)] 6= ∅ and intXi [Xi ∩ {x ∈ X | ϕ(x) < c} ∩ N (x)] ⊂ N (x) ∩ Xi . Since N (x) is open and N (x) ∩ Xi is robust in X, Lemma 6.9 yields:
of the system defining M (·), where b ∈ C(T, R) and C(T, R) is the Banach space of all continuous functions from T in R. Let X ⊂ Rn be a compact metric space with a metric induced by some norm in Rn , ω := (x, b) ∈ X × C(T, R). Assume Σ(ω) ⊂ X is the set of solutions of the system (10), θ is the zero + function in C(T, R) and let [y] := max {0, y} for y ∈ R.
intX [{x
∈ X | ϕ(x) < c} ∩ N (x)] ⊃
intX [Xi ∩ {x ∈ X | ϕ(x) < c} ∩ N (x)] 6= ∅, from which follows that x is a robust point of {x ∈ X | ϕ(x) < c}. Consequently, by Prop. 6.1 1., we have that the set {x ∈ X | ϕ(x) < c} is robust and, therefore, ϕ is u.r. on X. ¤
ξ
≥ b(t), ∀ t ∈ B(x) ∈
(10)
X
Definition 6.13 (Local Metric Regularity (LMR), Klatte/Henrion [8]) Let ω = (x, b(·)), ω 0 = (x0 , θ). We say that the system (10) is metrically regular at (ω 0 , ξ 0 ) with respect to X if there exists a neighborhood (V × W )×U of (ω 0 , ξ 0 ) in (X × C(T, R)) × X and a real number γ > 0 such that dist(ξ, Σ(ω)) ≤ γ max [b(t) − G(ξ, t)]+ t∈B(x)
Theorem 6.11 If M (·) is piecewise l.r., then M (·) is l.r. Proof. Replace the property piecewise l.s.c. by piecewise l.r. and repeat the proof of [17] by using again the Lemma 6.9. ¤
for all ξ ∈ U and for all ω = (x, b) ∈ V × W . Assumption (A3): X has a robust partition (Xi )i∈I , I = {0, 1, 2, ..., r}, such that X0 := {x ∈ X |B (x) = ∅ }. Assumption (A4): B |Xi , i = 1, 2, ..., r, is u.s.c. w.r.t. the relative topology on Xi .
Theorem 6.12 Let ψ : X × Y → R be an u.s.c. function and let M : X ⇒ Y be a piecewise l.s.c. (l.r.) SVM on X. Then the marginal function ϕ is u.r. on X.
Theorem 6.14 If (A1), (A3) and (A4) are satisfied and (LMR) holds at (ω 0 , ξ 0 ) for each x0 ∈ X and each ξ 0 ∈ Σ(ω 0 ), then M (·) is l.r. on X.
8 Proof. For all x ∈ X0 we have M (x) = X, i.e. M (·) is continuous on X0 . Now let i > 0, let x0 ∈ Xi , ξ 0 ∈ M (x0 ) and let U be a neighborhood of ξ 0 in X. © ªFirst, we want to show that for each sequence xk with xk → x0 in Xi the intersection M (xk ) ∩ U 6= ∅ for all sufficiently large k. Since, (LMR) holds w.r.t. X, there are neighe of ξ 0 , and V × W of (x0 , θ) and a real borhoods U number γ > 0 such that dist(ξ, Σ(ω)) ≤ γ max [b(x) − G(ξ, t)]+ t∈B(x)
(11)
e , ω = (x, b) ∈ V × W . Thus, from (11) for all ξ ∈ U e ∩ U, ∀x ∈ V it follows that ∀ξ ∈ U dist(ξ, M (x)) ≤ γ max [−G(ξ, t)]+ . t∈B(x)
Set g(ξ, x) := [− inf t∈B(x) G(ξ, t)]+ , then we have that g(ξ, ·) is an u.s.c. function on Xi w.r.t. the relative topology on Xi . Furthermore, Xi is a robust set and x0 ∈ V implies that V ∩ int(Xi ) 6= ∅. Thus, let {xk } ⊂ Xi be any sequence such that xk → x0 (in the relative topology of Xi ). Then there is an integer k0 such that xk ∈ V ∩ Xi , ∀k ≥ k0 . Consequently, we have dist(ξ 0 , M (xk )) ≤ γg(ξ 0 , xk ), ∀k ≥ k0 . But M (xk ) is compact for each k ≥ k0 by the compactness of X. This implies that for each k ≥ k0 , there is ξ k ∈ M (xk ) such that kξ 0 − ξ k k ≤ γg(ξ 0 , xk ) + kxk − x0 k, ∀k ≥ k0 . However, limk→∞ kxk − x0 k = 0 and limk→∞ g(ξ 0 , xk ) ≤ g(ξ 0 , x0 ) = 0 (both with the relative topology on Xi ). Consequently, we have that limk→∞ kξ 0 − ξ k k = 0, from which fole ∩ U 6= ∅, ∀k ≥ k0 . That is lows that M (xk ) ∩ U k M (x ) ∩ U 6= ∅, ∀k ≥ k0 . Taking into consideration, x0 ∈ V ∩ Xi has been chosen arbitrarily we get that M (·) is l.s.c. with the relative topology on Xi , in fact for each i ∈ I. Consequently, M (·) is l.r. on X by Thm. 6.11 ¤ Observe that the upper semi-continuity of B on the whole of X is not assumed, except on each of the partitioning sets Xi of X.
6.4
Robustness of the admissible set M of GSIP
Recall that M := {x ∈ X | G(x, t) ≥ 0, ∀ t ∈ B(x)}. For B(x) := {t ∈ T | ai (x, t) ≤ 0, i ∈ I} where I is a finite index set and all the functions G and ai , i ∈ I are affine w.r.t. (x, t), it has been shown in [15] that M could be the union of a finite number of closed and open half spaces. Obviously in this case, M is a robust set. We want to give some topological condition for the robustness of M in a more general sense.
Take next a continuous function g : Rn → R and consider the set S := {x ∈ Rn | g(x) ≥ 0} and let S0 := {x ∈ Rn | g(x) = 0}. Strong slater condition (SSC): For each x0 ∈ S0 and each neighborhood N (x0 ), there is some x ∈ N (x0 ) such that g(x) > g(x0 ). Lemma 6.15 The set S is robust and closed under (SSC) and the continuity of g. Proof. Clearly {x ∈ Rn | g(x) > 0} ⊂ int(S) and S = {x ∈ Rn | g(x) > 0} ∪ {x ∈ R | g(x) = 0}. It then follows that S = int(S) ∪ {x ∈ X | g(x) = 0} = int(S) ∪ S0 . Obviously, every element of int(S) is a robust point of S. And if x0 ∈ S0 , by assumption, for every neighborhood N (x0 ) we have N (x0 ) ∩ int(S) 6= ∅. Then x0 is a robust point of S. Therefore, the set S is robust by Prop.6.1(1)) and closed by the (upper semi-)continuity of g. ¤ Assumption (A5): B |Xi is l.s.c. on Xi in the relative topology of Xi for i = 1, 2, ..., r. Theorem 6.16 If (A1), (A3), (A4), (A5) are satisfied and g : X \ X0 → R with g(x) := inf G(x, t) t∈B(x)
fulfils the (SSC) on each partition Xi , i = 1, 2, .., r with respect to the relative topology on Xi then the admissible set M of (GSIP) is robust. Proof. M can be equivalently written as M
:= S {x ∈ X | g(x) ≥ 0} ∪ X0 r = j=1 {x ∈ Xj | g(x) ≥ 0} ∪ X0 .
From assumption (A3) we have the robustness of X0 . (A1)(A4)(A5) ensure the continuity of g on the partition sets Xi w.r.t. the relative topology by using [1, Thm. 6, p. 53]. The (SSC) on each Xi yields that {x ∈ Xj | g(x) ≥ 0} is a robust set of Xi in the relative topology of Xi . By Lemma 6.9 (Xi is robust in X) this set is also robust in X. Finally M is robust as a union of robust sets. ¤ Sufficient for the (SSC) are conditions of the type ”extended Mangasarian Fromowitz Constraint qualification (EMFCQ)” whenever differentiability assumptions for G are made and the constraints describing the set-valued mapping B (x) satisfy some regularity conditions on each Xi . What we need to stress here again is that: the known strong conditions for nice behavior of a (GSIP) are expected to hold on each component of a suitable robust partition. For the usage of the IGOM such a partition need not be explicitly known. Only the existence of
9
such a partition is important. What we need apriori in any case is the robustness of the set of all x where the image of B is empty. In the known examples of (GSIP) with ill behavior given in [7, 19, 18] the conditions (A1), (A3)-(A5) can be principally satisfied. However, for a few of these examples a free choice of some functions must be properly done so that our basic assumptions hold.
7
Measurability functions
of
marginal
We consider the measurability of set-valued mappings according to [12] and cite the results which are in line with our investigations. Only a few results concerning again the partitioning of X must be shown. W.r.t. functions and sets in Rn we use the ordinary notion of Lebesgue measurability. We simply say measurable instead of Lebesgue measurable.
7.1
Basic definitions and results
Let X, Y ⊂ Rn be closed sets. M : X ⇒ Y is called measurable iff M is closed valued and M −1 (C) is measurable for each closed set C ⊂ Y . Let f : X × Y → R ∪ {−∞, +∞} =: R. Then the mapping Ef (x) := {(y, α) ∈ Y × R | f (x, y) ≤ α} is called the epigraphical map associated with f . The extended real-valued function f is a normal integrand iff for each x ∈ X the function y → f (x, y) → Y × R is is l.s.c. and epigraphical map Ef : X − → measurable. Theorem 7.1 [12] Let ϕ be the marginal function (9). If ψ : X × Y → R is a normal integrand and M : X ⇒ Y is measurable, then ϕ is measurable.
7.2
Special structure of M (x)
Next we give conditions on the index SVM B(·) and the function G to guarantee the measurability of M (·), of the marginal and of the penalty functions, and of the feasible set M of (GSIP). We say that X0 , X1 , ..., Xr is a (robust and measurable ) measurable partition of X iff X0 , X1 , ..., Xr is a partition of X and all parts Xi are (robust and measurable) measurable. We assume further that X0 := {x ∈ X |B (x) = ∅ } belongs to this partition. Proposition 7.2 Let X be some measurable subset of Rn and let ϕ : X → R be a function. Suppose also that X0 , X1 , ..., Xr is a measurable partition of X. If, for each i ∈ I, ϕ is measurable on Xi , then ϕ is measurable on X.
Proof. Let {Oα }α∈Λ be the family of measurable sets in Rn . Then, for each i ∈ I, the family of sets {Xi ∩ Oα }α∈Λ is the family of measurable sets w.r.t. the relative topology on Xi . Thus, the σ-algebra σ({Xi ∩ Oα }α∈Λ ) defines the measure on Xi . As ϕ is measurable w.r.t. Xi , then for any measurable set D ⊂ R we have that ϕ−1 (D) ∩ Xi is measurable in Xi . Since Xi are measurable in X, we have ϕ−1 (D) ∩ Xi is measurable in X. Therefore, since I is finite (countable infinite is here also possible) we conclude that [ (ϕ−1 (D) ∩ Xi ) = ϕ−1 (D) i∈I
is measurable in X. Consequently, ϕ is measurable on X. ¤ Hence based on this proposition, we are only required to verify the measurability of the ϕ of (PSIP) on each of the partitioning sets Xi of X. Defining g(ξ, x) := inf t∈B(x) G(ξ, t), we get M (x) = {ξ ∈ X | g(ξ, x) ≥ 0}. Proposition 7.3 Let Xi be measurable, i = 1, 2, .., r. If assumptions (A1), (A3), (A4), (A5) hold, then −g is a normal integrand on X × Xi . Proof. Observe that, by the assumptions, the function g : X × Xi → is continuous ([1]) and hence a normal integrand on X × Xi (see [12, p. 661ff.]). ¤
Proposition 7.4 [12] Let Xi be measurable. Then if −g is a normal integrand on X × Xi , then M (·) is measurable on Xi . Proof. Follows from [12].
¤
Theorem 7.5 (Measurability of the marginal function) Let the marginal function (9) be given and let (Xi )i∈I be a (robust) measurable partition of X. If ψ is a normal integrand on X × X and assumptions (A1), (A3), (A4), (A5) hold, then ϕ is measurable on each Xi ; hence, measurable on X. Proof. Clearly ψ is a normal integrand and M (·) is closed valued and measurable, by Prop. 7.4, on Xi . Moreover, by [12, Thm. 14.37] ϕ is measurable w.r.t. Xi . Therefore, by Prop. 7.2., ϕ is measurable. ¤
Corollary 7.6 If assumptions (A1), (A3), (A4), (A5) hold, then the feasible set M of (GSIP) is measurable. Hence, ϕd is also measurable.
10 Proof. By Prop. 3.2. M = ϕ−1 (0). However, under assumptions (A1), (A3), (A4), (A5), ϕ is measurable, by Thm. 7.5. Consequently, M is measurable. Furthermore, for the function ϕd (see Section 4), given c ∈ R, we have (8) and from which follows the measurability of ϕd due to the measurability of ϕ and that of M . ¤
8
Conclusions
We attempted to announce in this paper that robust analysis together with the analysis of measurability of functions and set-valued mappings is able to handle the so-called ill-behaved (GSIP). At least we can hope to get coarse approximations of a solution or a minimizing sequence by using some suitable global optimization procedure. The main ideas are the following: First to formulate the (GSIP) equivalently as a global optimization problem where the objective is closely connected with a discontinuous penalty of the feasible region of (GSIP). Second to apply a software which is able to solve global optimization problems with discontinuous but upper robust objective functions. This job could be done by the IGOM, which works with integration over level sets. Third to show that the penalty term is an upper robust function - under assumptions which do not reduce the class of treatable problems to the class of ”nice” (GSIP). This could be achieved by the marginal analysis of robust functions and set-valued mappings which has been principally developed in this paper. The measurability follows from standard arguments under similar assumptions. Some of the conditions such as the local metric regularity (LMR) and the strong slater condition (SSC) may be replaced for detailed investigations by suitable and sufficient criteria. We think that the EMFCQ (extended Mangasarian Fromowitz condition) or its modifications should be taken into consideration. Our investigations till now indicate that this approach works theoretically. Numerical treatments are under investigation. It is also in our belief that the results of this paper could be extended to more than one but finitely many so-called ”semi-infinite” constraints. However the set X should be a robust subset of Rn . Nevertheless, equality constraints are found to generate troubles if they partially describe the set X. Instead, one might need to add equality constraints as penalty to the objective.
A similar approach with discontinuous penalty function is given by the following problem f (x) + λ[−ψ(x)]+ → inf x ∈ X where ψ(x) = inf t∈B(x) G(x, t). Under the assumptions made above, we get, in a similar manner, the upper robustness of [−ψ(x)]+ = max{0, −ψ(x)} and also its measurability. Last, but not least, we would like to express our utmost indebtedness and thankfulness to our two anonymous referees for their valuable hints and comments; and above all, for their suggestion of new directions of investigations.
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