A feasible rounding approach for mixed-integer optimization problems

Neumann, Cornelius; Stein, Oliver; Sudermann‐Merx, Nathan

doi:10.1007/s10589-018-0042-y

Cited by 6 publications

(12 citation statements)

References 18 publications

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“…In the following let us recall some constructions which were presented in [19,20] for the case of mixed-integer linear optimization problems (MILPs) and mixed-integer nonlinear optimization problems (MINLPs), respectively. We shall denote the feasible set of the NLP relaxation MI NLP of MI NLP by…”

Section: Granularitymentioning

confidence: 99%

“…We remark that [19,20] provide several examples for granular problems in the linear as well as in the nonlinear case.…”

Section: Definitionmentioning

confidence: 99%

“…In this paper we study the quality of feasible points of mixed-integer linear and nonlinear optimization problems (MILPs and MINLPs, resp.) as they are constructed by the feasible rounding approaches from [19,20]. These approaches are based on a property of the feasible set which we call granularity and which states that a certain inner parallel set of the continuously relaxed feasible set is nonempty.…”

Section: Introductionmentioning

confidence: 99%

“…Our present analysis is motivated by the successful application of this algorithmic approach to mixed-integer linear and nonlinear problems in [19] and [20], respectively. In these papers computational studies of problems from the MIPLIB and MINLPLib libraries show that granularity may be expected and exploited in various real world applications.…”

Section: Introductionmentioning

confidence: 99%

“…The remainder of this article is structured as follows. Section 2 recalls some preliminaries from [19,20], such as the main construction needed for the definition of granularity, an explicit description of a subset of the appearing inner parallel set, and enlargement ideas which promote the performance of the resulting feasible rounding approach. Section 3 provides a-priori bounds on the deviation of the objective value of the generated feasible point from the optimal value, before Section 4 applies these bounds to explain and quantify the positive effect of fine integer grids on the performance of the feasible rounding approach.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Bounds on the Objective Value of Feasible Roundings

2020

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Section: Granularitymentioning

confidence: 99%

“…We remark that [19,20] provide several examples for granular problems in the linear as well as in the nonlinear case.…”

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Bounds on the Objective Value of Feasible Roundings

2020

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Granularity in Nonlinear Mixed-Integer Optimization

Neumann

Stein

Sudermann‐Merx³

2019

J Optim Theory Appl

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A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization

Eichfelder,

Stein,

Warnow

2023

J Optim Theory Appl

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This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures.

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A feasible rounding approach for mixed-integer optimization problems

Cited by 6 publications

References 18 publications

Bounds on the Objective Value of Feasible Roundings

Bounds on the Objective Value of Feasible Roundings

Granularity in Nonlinear Mixed-Integer Optimization

A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization

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