A transformation-based discretization method for solving general semi-infinite optimization problems
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A transformation-based discretization method for solving general semi-infinite optimization problems Jan Schwientek1
· Tobias Seidel1 · Karl-Heinz Küfer1
Received: 6 August 2019 / Revised: 26 June 2020 © The Author(s) 2020
Abstract Discretization methods are commonly used for solving standard semi-infinite optimization (SIP) problems. The transfer of these methods to the case of general semi-infinite optimization (GSIP) problems is difficult due to the x-dependence of the infinite index set. On the other hand, under suitable conditions, a GSIP problem can be transformed into a SIP problem. In this paper we assume that such a transformation exists globally. However, this approach may destroy convexity in the lower level, which is very important for numerical methods. We present in this paper a solution approach for GSIP problems, which cleverly combines the above mentioned two techniques. It is shown that the convergence results for discretization methods in the case of SIP problems can be transferred to our transformation-based discretization method under suitable assumptions on the transformation. Finally, we illustrate the operation of our approach as well as its performance on several examples, including a problem of volume-maximal inscription of multiple variable bodies into a larger fixed body, which has never before been considered as a GSIP test problem. Keywords Semi-infinite optimization · Discretization · Coordinate transformation · Design centering · Inscribing Mathematics Subject Classification 90C34 · 90C30 · 65K05
B 1
Jan Schwientek [email protected] Fraunhofer Institute for Industrial Mathematics (ITWM), Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany
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1 Introduction In the present paper we consider general semi-infinite optimization problems of the following form: GSIP :
min f (x)
x∈M
with M := {x ∈ X ⊆ Rm | gi (x, y) ≤ 0for all y ∈ Y (x), i ∈ I }, where Y : Rm ⇒ Rn is a set-valued mapping, |Y (x)| = ∞ for at least some x ∈ X , I := {1, ..., p} is a finite index set and f , gi , i ∈ I , are real-valued and at least continuous functions. By X we summarize the finite restrictions on the decision variables. Moreover, we assume that X is non-empty and closed and for every x ∈ X the so-called infinite index set Y (x) is closed and compact. The infinite index set can, e.g., be given as the solution set of finitely many inequalities: Y (x) := {y ∈ Rn | u j (x, y) ≤ 0, j ∈ J }, where J := {1, . . . , q} is a finite index set and u j , j ∈ J , are again real-valued and at least continuous functions. For the special case, where Y (x) does not depend on x, the problem above is called a standard or ordinary semi-infinite optimization problem and is abbreviated by SIP. One of the keys for both the theoretical and numerical treatment of semi-infinite optimization problems lies in their bi-level structure. The parametric lower level problems of a semi-infinite problem are given by Qi (x) :
max gi (x, y) s.t. y ∈ Y (x).
y∈Rn
(1)
The decision variables x of t
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