A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

Diehl, Moritz; Houska, Boris; Stein, Oliver; Steuermann, Paul

doi:10.1007/s10589-012-9489-4

Cited by 19 publications

(19 citation statements)

References 25 publications

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“…Problems in these categories can generally only be solved using semiinfinite programming techniques [46][47][48] , which are limited to small scale problems, or approximate robust optimization schemes [49][50][51][52] . Category 2 has received limited attention in the robust optimization and semiinfinite programming communities [53][54][55] . A particular subclass, (non-convex) MILP problems, has been studied extensively in the process systems engineering literature, e.g., several robust planning and scheduling applications 2,13,56,57 .…”

Section: Robust Optimizationmentioning

confidence: 99%

Robust Optimization for the Pooling Problem

Wiebe

Cecı́lio

Misener

2019

Ind. Eng. Chem. Res.

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The pooling problem has applications, e.g., in petrochemical refining, water networks, and supply chains and is widely studied in global optimization. To date, it has largely been treated deterministically, neglecting the influence of parametric uncertainty. This paper applies two robust optimization approaches, reformulation and cutting planes, to the non-linear, non-convex pooling problem. Most applications of robust optimization have been either convex or mixed-integer linear problems. We explore the suitability of robust optimization in the context of global optimization problems which are concave in the uncertain parameters by considering the pooling problem with uncertain inlet concentrations. We compare the computational efficiency of reformulation and cutting plane approaches for three commonly-used uncertainty set geometries on 14 pooling problem instances and demonstrate how accounting for uncertainty changes the optimal solution.

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Section: Robust Optimizationmentioning

confidence: 99%

Robust Optimization for the Pooling Problem

Wiebe

Cecı́lio

Misener

2019

Ind. Eng. Chem. Res.

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“…One notes that indeed NLP (14) does not have the complementarity constraints that make MPCC (10) numerically unfavorable. Proposition 10 is similar to Corollary 2.4 in [18]. The difference is that the latter result assumes that −g and h(x, •) are convex on all of Y = R ny .…”

Section: Wolfe Dualmentioning

confidence: 62%

“…Unfortunately, there are numerical difficulties involved in solving MPCCs. This relates to the fact that the Mangasarian-Fromovitz Constraint Qualification is violated everywhere in its feasible set [18]. This motivates the reformulation of §4.1.3.…”

Section: Kkt Conditionsmentioning

confidence: 99%

“…To obtain a more numerically tractable reformulation than the KKT reformulation (10), one follows the ideas in [18] to obtain a reformulation of (GSIP) which does not have complementarity constraints. This follows by looking at the dual function q i (x, •) and noting that if Y is a nonempty open convex set and g i and −h(x, •) are concave and differentiable, then for µ ≥ 0, the supremum defining the dual function is achieved at y if and only if ∇g i (y) − ∇ y h(x, y)µ = 0.…”

Section: Wolfe Dualmentioning

confidence: 99%

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How to solve a design centering problem

Harwood

Barton

2017

Math Meth Oper Res

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This work considers the problem of design centering. Geometrically, this can be thought of as inscribing one shape in another. Theoretical approaches and reformulations from the literature are reviewed; many of these are inspired by the literature on generalized semi-infinite programming, a generalization of design centering. However, the motivation for this work relates more to engineering applications of robust design. Consequently, the focus is on specific forms of design spaces (inscribed shapes) and the case when the constraints of the problem may be implicitly defined, such as by the solution of a system of differential equations. This causes issues for many existing approaches, and so this work proposes two restriction-based approaches for solving robust design problems that are applicable to engineering problems. Another feasible-point method from the literature is investigated as well. The details of the numerical implementations of all these methods are discussed. The discussion of these implementations in the particular setting of robust design in engineering problems is new. keywords: gsip, design centering, lower level duality, global optimization * Current version as of: May 4, 2017. This research was funded by Novartis Pharmaceuticals as part of the Novartis-MIT Center for Continuous Manufacturing.

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“…Similar approaches have been used in robust optimization and lead to a systematic treatment of GSIPs with smooth and convex lower-level problems in [29]. There, the function ϕ is rewritten as the optimal value function of the corresponding Wolfe dual of the lower-level problem, and the additional dual variables are included by lifting the feasible set.…”

Section: The Lower-level Duality Reformulationmentioning

confidence: 99%

Solving Disjunctive Optimization Problems by Generalized Semi-infinite Optimization Techniques

Kirst

Stein

2016

J Optim Theory Appl

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We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. In contrast to existing methods in disjunctive programming, our approach does not expect any special formulation of the underlying logical expression. Applying existing lower-level reformulations for the corresponding semi-infinite program, we derive conjunctive nonlinear problems without any logical expressions, which can be locally solved by standard nonlinear solvers. Our preliminary numerical results on some small-scale examples indicate that our reformulation procedure is a reasonable method to solve disjunctive optimization problems.

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A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

Cited by 19 publications

References 25 publications

Robust Optimization for the Pooling Problem

Robust Optimization for the Pooling Problem

How to solve a design centering problem

Solving Disjunctive Optimization Problems by Generalized Semi-infinite Optimization Techniques

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