Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints

Djelassi, Hatim; Glass, Moll Helene; Mitsos, Alexander

doi:10.1007/s10898-019-00764-3

Cited by 22 publications

(21 citation statements)

References 55 publications

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“…The disjunctive programs resulting from the discretization are solved globally using a solver for mixed integer nonlinear problems. 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.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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

2020

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“…The GSIP algorithm in Reference requires continuity of all functions and equations and the existence of a Slater point arbitrarily close to a GSIP optimum in the objective. As shown in Reference the assumption of an ε ‐optimal GSIP Slater point is always satisfied for any feasible GSIP resulting from the reformulation of a min–max program.…”

Section: Brute Force Solution Methodsmentioning

confidence: 99%

“…At each sampling point v m in the host set V we formulate and then solve the bilevel problem identical to the second‐level problem Equations with the exception that now v comes from a discrete set and is not continuous. The bilevel problem Equations can be reformulated as a GSIP and solved using the algorithm proposed by Djelassi et al with a specialization of the algorithm to min–max problems as proposed in Reference . The smallest worst‐case process cost

Φ_{U}^{*} ((), {bold-italicv}_{normalm}^{*})

is then chosen as the optimal solution value among all the discrete worst‐case realizations.…”

Section: Brute Force Solution Methodsmentioning

confidence: 99%

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Optimal experimental design for optimal process design: A trilevel optimization formulation

2019

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Typical optimal experimental design (OED) methods aim at minimizing the covariance matrix of the estimated parameters regardless of the intended application of the model that is being estimated. This can unnecessarily increase the experimental costs. Herein, we propose a new OED method, which tailors the designed experiments to the model application. The method is demonstrated for model-based process design and aims at mitigating a worst-case realization of the process design. The proposed formulation results in a min-max-min problem and is based on boundederror OED. The method is illustrated via an ad hoc solution method using two examples, a simple illustrative example and the van de Vusse reaction, that show the differences between typical and the new tailored OED method: experimental designs can be considered good using the latter method, while the same design would be considered bad with the former methods.

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“…The deterministic approach of Mitsos et al [27] and the approximation method of Tsoukalas et al [41] apply to very general nonlinear bilevel problems, restricted solely by the absence of inner equality constraints. Recently, both these approaches were extended as a new discretization-based algorithm for bilevel problems with coupling equality constraints [17].…”

Section: Introductionmentioning

confidence: 99%

New bounding schemes and algorithmic options for the Branch-and-Sandwich algorithm

Paulavičius

Adjiman

2020

J Glob Optim

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We consider the global solution of bilevel programs involving nonconvex functions. Deterministic global optimization algorithms for the solution of this challenging class of optimization problems have started to emerge over the last few years. We present new schemes to generate valid bounds on the solution of nonconvex inner and outer problems and examine new strategies for branching and node selection. We integrate these within the Branch-and-Sandwich algorithm (Kleniati and Adjiman in J Glob Opt 60:425-458, 2014), which is based on a branch-and-bound framework and enables the solution of a wide range of problems, including those with nonconvex inequalities and equalities in the inner problem. The impact of the proposed modifications is demonstrated on an illustrative example and 10 nonconvex bilevel test problems from the literature. It is found that the performance of the algorithm is improved for all but one problem (where the CPU time is increased by 2%), with an average reduction in CPU time of 39%. For the two most challenging problems, the CPU time required is decreased by factors of over 3 and 10.

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Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints

Cited by 22 publications

References 55 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

Optimal experimental design for optimal process design: A trilevel optimization formulation

New bounding schemes and algorithmic options for the Branch-and-Sandwich algorithm

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