A hybrid discretization algorithm with guaranteed feasibility for the global solution of semi-infinite programs

Djelassi, Hatim; Mitsos, Alexander

doi:10.1007/s10898-016-0476-7

Cited by 22 publications

(67 citation statements)

References 24 publications

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“…The necessary bisection of the SIC parameter domains is performed transparently, and helps improving the efficiency of the method. This allows taking full benefits of existing efficient implementations of branchand-bound algorithms: Experimental results in this paper are obtained using the NLP solver IBEX [14], and they show that the proposed branch-and-bound algorithm is competitive with the distributed implementation of [46,19] based on BARON [54], hence re-establishing the interest of branch-and-bound algorithms for solving SIPs.…”

Section: Introductionmentioning

confidence: 81%

“…We first propose in Section 4.1 an experiment inspired from [29] focusing on the clustering effect around optima of SIPs. Next in Section 4.2.2, we propose a general comparison to the state-of-the-art alternative from [19], which is to our knowledge the best general SIP solving algorithm to date 5 . Finally, in Section 4.4 we present the resolution of a recently proposed difficult SIP model for telescope design.…”

Section: Methodsmentioning

confidence: 99%

“…We now propose a comparison of the performance of our branch-and-bound method with the Hybrid, Blankenship-based, approach in [19]. The GAMS implementation of this method is that provided by the authors as supplementary material with their paper 6 .…”

Section: Comparative Benchmarkingmentioning

confidence: 99%

“…It has been recently improved [19] using in addition the "oracle" technique previously proposed in [58]. A complete comparison of [46,19] with [23,11] is difficult, but some clues for a very favorable comparison were provided in [46], supported by some experiments on a small set of benchmarks and different machines. Nevertheless, [19] is currently the best implementation for solving globally non-convex SIPs.…”

Section: Introductionmentioning

confidence: 99%

“…A complete comparison of [46,19] with [23,11] is difficult, but some clues for a very favorable comparison were provided in [46], supported by some experiments on a small set of benchmarks and different machines. Nevertheless, [19] is currently the best implementation for solving globally non-convex SIPs. However, it presents the drawback of solving globally several instances of the discretized SIP, which can turn out to be computationally very expensive for hard problems.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A standard branch-and-bound approach for nonlinear semi-infinite problems

Marendet

Goldsztejn

Chabert

et al. 2020

European Journal of Operational Research

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This paper considers nonlinear semi-infinite problems, which contain at least one semi-infinite constraint (SIC). The standard branch-and-bound algorithm is adapted to such problems by extending usual upper and lower bounding techniques for nonlinear inequality constraints to SICs. This is achieved by defining the interval evaluation of parametrized functions and their generalized gradients, by also adapting numerical constraint programming techniques to quantified inequalities, and by introducing linear relaxations and restrictions for SICs. The overall efficiency of our algorithm is demonstrated on a standard set of benchmarks from the literature, in comparison with the best state of the art alternative.

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Section: Introductionmentioning

confidence: 81%

Section: Methodsmentioning

confidence: 99%

Section: Comparative Benchmarkingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A standard branch-and-bound approach for nonlinear semi-infinite problems

Marendet

Goldsztejn

Chabert

et al. 2020

European Journal of Operational Research

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Solving continuous set covering problems by means of semi-infinite optimization

et al. 2022

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This article introduces the new class of continuous set covering problems. These optimization problems result, among others, from product portfolio design tasks with products depending continuously on design parameters and the requirement that the product portfolio satisfies customer specifications that are provided as a compact set. We show that the problem can be formulated as semi-infinite optimization problem (SIP). Yet, the inherent non-smoothness of the semi-infinite constraint function hinders the straightforward application of standard methods from semi-infinite programming. We suggest an algorithm combining adaptive discretization of the infinite index set and replacement of the non-smooth constraint function by a two-parametric smoothing function. Under few requirements, the algorithm converges and the distance of a current iterate can be bounded in terms of the discretization and smoothing error. By means of a numerical example from product portfolio optimization, we demonstrate that the proposed algorithm only needs relatively few discretization points and thus keeps the problem dimensions small.

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Adaptive discretization-based algorithms for semi-infinite programs with unbounded variables

2022

Self Cite

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The proof of convergence of adaptive discretization-based algorithms for semi-infinite programs (SIPs) usually relies on compact host sets for the upper- and lower-level variables. This assumption is violated in some applications, and we show that indeed convergence problems can arise when discretization-based algorithms are applied to SIPs with unbounded variables. To mitigate these convergence problems, we first examine the underlying assumptions of adaptive discretization-based algorithms. We do this paradigmatically using the lower-bounding procedure of Mitsos [Optimization 60(10–11):1291–1308, 2011], which uses the algorithm proposed by Blankenship and Falk [J Optim Theory Appl 19(2):261–281, 1976]. It is noteworthy that the considered procedure and assumptions are essentially the same in the broad class of adaptive discretization-based algorithms. We give sharper, slightly relaxed, assumptions with which we achieve the same convergence guarantees. We show that the convergence guarantees also hold for certain SIPs with unbounded variables based on these sharpened assumptions. However, these sharpened assumptions may be difficult to prove a priori. For these cases, we propose additional, stricter, assumptions which might be easier to prove and which imply the sharpened assumptions. Using these additional assumptions, we present numerical case studies with unbounded variables. Finally, we review which applications are tractable with the proposed additional assumptions.

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A hybrid discretization algorithm with guaranteed feasibility for the global solution of semi-infinite programs

Cited by 22 publications

References 24 publications

A standard branch-and-bound approach for nonlinear semi-infinite problems

A standard branch-and-bound approach for nonlinear semi-infinite problems

Solving continuous set covering problems by means of semi-infinite optimization

Adaptive discretization-based algorithms for semi-infinite programs with unbounded variables

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