The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets

Stein, Oliver; Steuermann, Paul

doi:10.1007/s10107-012-0556-5

Cited by 26 publications

(13 citation statements)

References 25 publications

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“…The focus herein is on deterministic global optimization solvers without any convexity assumptions that can provide such points in finitely many iterations. Over the last decade a series of algorithms have been proposed [12][13][14][15][16][17][18][19] that tackle SIPs. Some of these can guarantee global solution of the SIP while others focus on local solution.…”

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confidence: 99%

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Mitsos

Tsoukalas

2014

J Glob Optim

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The algorithm proposed in Mitsos (Optimization 60(10-11):1291(Optimization 60(10-11): -1308(Optimization 60(10-11): , 2011 for the global optimization of semi-infinite programs is extended to the global optimization of generalized semi-infinite programs. No convexity or concavity assumptions are made. The algorithm employs convergent lower and upper bounds which are based on regular (in general nonconvex) nonlinear programs (NLP) solved by a (black-box) deterministic global NLP solver. The lower bounding procedure is based on a discretization of the lower-level host set; the set is populated with Slater points of the lower-level program that result in constraint violations of prior upper-level points visited by the lower bounding procedure. The purpose of the lower bounding procedure is only to generate a certificate of optimality; in trivial cases it can also generate a global solution point. The upper bounding procedure generates candidate optimal points; it is based on an approximation of the feasible set using a discrete restriction of the lower-level feasible set and a restriction of the right-hand side constraints (both lower and upper level). Under relatively mild assumptions, the algorithm is shown to converge finitely to a truly feasible point which is approximately optimal as established from the lower bound. Test cases from the literature are solved and the algorithm is shown to be computationally efficient.

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confidence: 99%

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Mitsos

Tsoukalas

2014

J Glob Optim

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“…Under extended Mangasarian-Fromovitz constraint qualification (EMFCQ), John's theorem in [29] coincides with the finite-dimensional representation of KKT conditions [19,49], which states that for a local minimizer of SIP there exists at most n active indices of the minimizer to characterize its KKT (first-order) necessary optimality conditions of SIP. In Assumption 1, we assume by analogy that there exists at most n active indices of (PCDP) for u s , or in other words, that there exists an approximately optimal point for which the path constraint is not active over an interval.…”

Section: Local Optimization Algorithm Of Path-constrained Dynamic Promentioning

confidence: 99%

“…For theoretical developments and applications of SIP, we refer to reviews [25,40] and latest results [34,36,49]. In the context of path-constrained dynamic optimization, SIP formulations arise naturally if time is viewed as the (single) parameter of SIP [33,42].…”

Section: Introductionmentioning

confidence: 99%

Local optimization of dynamic programs with guaranteed satisfaction of path constraints

Faust

Chachuat

et al. 2015

Automatica

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An algorithm is proposed for locating a feasible point satisfying the KKT conditions to a specified tolerance of feasible inequalitypath-constrained dynamic programs (PCDP) within a finite number of iterations. The algorithm is based on iteratively approximating the PCDP by restricting the right-hand side of the path constraints and enforcing the path constraints at finitely many time points. The main contribution of this article is an adaptation of the semi-infinite program (SIP) algorithm proposed in [Mitsos, Optimization, 2011, 60(10-11):1291-1308] to PCDP. It is proved that the algorithm terminates finitely with a guaranteed feasible point which satisfies the first-order KKT conditions of the PCDP to a specified tolerance. The main assumptions are: i) availability of a nonlinear program (NLP) local solver that generates a KKT point of the constructed approximation to PCDP at each iteration if this problem is indeed feasible; ii) existence of a Slater point of the PCDP that also satisfies the first-order KKT conditions of the PCDP to a specified tolerance; iii) all KKT multipliers are nonnegative and uniformly bounded with respect to all iterations. The performance of the algorithm is analyzed through two numerical case studies.

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“…Similarly, Bhattacharjee et al [17] propose to use interval bounds on the constraint function in a deterministic setting to provide a sequence of feasible points converging to a solution of the SIP. As an alternative to interval extensions, approaches employing convex and concave relaxations to overestimate the semi-infinite constraint in (G)SIPs are proposed by Floudas and Stein [18], Mitsos et al [19], and Stein and Steuermann [20].…”

Section: Introductionmentioning

confidence: 99%

Global Solution of Semi-infinite Programs with Existence Constraints

Djelassi

Mitsos

2021

J Optim Theory Appl

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We consider what we term existence-constrained semi-infinite programs. They contain a finite number of (upper-level) variables, a regular objective, and semi-infinite existence constraints. These constraints assert that for all (medial-level) variable values from a set of infinite cardinality, there must exist (lower-level) variable values from a second set that satisfy an inequality. Existence-constrained semi-infinite programs are a generalization of regular semi-infinite programs, possess three rather than two levels, and are found in a number of applications. Building on our previous work on the global solution of semi-infinite programs (Djelassi and Mitsos in J Glob Optim 68(2):227–253, 2017), we propose (for the first time) an algorithm for the global solution of existence-constrained semi-infinite programs absent any convexity or concavity assumptions. The algorithm is guaranteed to terminate with a globally optimal solution with guaranteed feasibility under assumptions that are similar to the ones made in the regular semi-infinite case. In particular, it is assumed that host sets are compact, defining functions are continuous, an appropriate global nonlinear programming subsolver is used, and that there exists a Slater point with respect to the semi-infinite existence constraints. A proof of finite termination is provided. Numerical results are provided for the solution of an adjustable robust design problem from the chemical engineering literature.

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The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets

Cited by 26 publications

References 25 publications

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Local optimization of dynamic programs with guaranteed satisfaction of path constraints

Global Solution of Semi-infinite Programs with Existence Constraints

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