Global optimization of generalized semi-infinite programs using disjunctive programming

Kirst, Peter; Stein, Oliver

doi:10.1007/s10898-018-0690-6

Cited by 4 publications

(3 citation statements)

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“…We tested our new algorithm on the other 15 examples collected by Lemonidis (2008), too. This test set was also used in the recent publications (Mitsos and Tsoukalas 2015;Kirst and Stein 2019). However, to apply Algorithm 1 we need to ensure that the central Assumption 1 is satisfied.…”

Section: Some Native Examplesmentioning

confidence: 99%

“…In Djelassi et al (2019) the strategy is extended to problems with equality constraints in the lower level. Instead of using a reformulation to a mixed integer program, in Kirst and Stein (2019) the authors develop tailored strategies to solve the disjunctive problems. An early attempt to globally solve GSIP problems was followed by Lemonidis (2008).…”

Section: Introductionmentioning

confidence: 99%

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A transformation-based discretization method for solving general semi-infinite optimization problems

2020

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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 $$\mathbf {x}$$ 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.

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Section: Some Native Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A transformation-based discretization method for solving general semi-infinite optimization problems

2020

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“…In moving to the general case, we obtain a GSIP (recall problem ( 1)). We refer to recent work in [25,19] for adapting this basic solution method for SIP to GSIP, and leave their application to equilibrium problems for future work.…”

Section: Solution Methods For Minimum Disequilibriummentioning

confidence: 99%

Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

Harwood¹,

Trespalacios²,

Papageorgiou³

et al. 2021

Preprint

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This paper introduces the concept of optimization equilibrium as an equivalently versatile definition of a generalized Nash equilibrium for multi-agent non-cooperative games. Through this modified definition of equilibrium, we draw precise connections between generalized Nash equilibria, feasibility for bilevel programming, the Nikaido-Isoda function, and classic arguments involving Lagrangian duality and social welfare maximization. Significantly, this is all in a general setting without the assumption of convexity. Along the way, we introduce the idea of minimum disequilibrium as a solution concept that reduces to traditional equilibrium when equilibrium exists. The connections with bilevel programming and related semi-infinite programming permit us to adapt global optimization methods for those classes of problems, such as constraint generation or cutting plane methods, to the problem of finding a minimum disequilibrium solution. We show that this method works, both theoretically and with a numerical example, even when the agents are modeled by mixed-integer programs.

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Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

Harwood,

Trespalacios,

Papageorgiou

et al. 2023

Comput Optim Appl

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Global optimization of generalized semi-infinite programs using disjunctive programming

Cited by 4 publications

References 35 publications

A transformation-based discretization method for solving general semi-infinite optimization problems

A transformation-based discretization method for solving general semi-infinite optimization problems

Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

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