A conceptual method for solving generalized semi-infinite programming problems via global optimization by exact discontinuous penalization
Introduction
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], [17], [28], [29], Hoffmann and Phú [5], [6], [10], 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 Rückmann and Shapiro [13], [14], Still [23], [24], Stein and Still [21], Weber [25] etc. Recently, we also find first attempts on the numerical aspects of (GSIP), see [9], [11], [22], [24], [26], [27]. In our case, we define an auxiliary parametric semi-infinite programming problem (PSIP) characterizing the admissible set of the (GSIP) and enabaling 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 [16], [17], [28] 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 construction 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.
Section snippets
Problem and motivation
We consider the problemwhere we make the Assumption (A1) The sets , are compact and nonempty, the functions and are upper semi-continuous (u.s.c.) on X and continuous on X×T, respectively. The set-valued mapping (SVM) is at least compact valued but may have empty values for some x∈X.
We assume neither the upper nor the lower semi-continuity of B on X. Naturally, in case of continuity of B, we feel that usual treatments are more preferable than the
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. Assumption (A2) The function is continuous on W and p(x)=0 if and only if x=0.
We consider, for each parameter x∈X, the problem of feasibilityProblem (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).
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 functionof the admissible set M of (GSIP) and consider the associated penalty problemA sequence {xn}⊂X is called a minimizing sequence of (GSIP) iff
- 1.
,
- 2.
.
A
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 , with , as a topological space; moreover, X is Lebesgue
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 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 D is a robust subset of X if and only if each point x∈D is a robust point of D. Any accumulation point of a set of robust points to D is also a robust point to D.[28]
Similarly, as each open set is a neighborhood of all its points, the robustness of a set is
Measurability of marginal functions
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 we use the ordinary notion of Lebesgue measurability. We simply say measurable instead of Lebesgue measurable.
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
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