Vector Optimization with Infimum and Supremum

Löhne, Andreas

doi:10.1007/978-3-642-18351-5

Cited by 140 publications

(244 citation statements)

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“…Conjugate maps and Farkas-type results are crucial in any duality theory, and this is why they constitute the main tool and the main objective, respectively, of our research. The state of the art in vector optimization is described, e.g., in [3], [16], [18], [23], and references therein.…”

Section: Introductionmentioning

confidence: 99%

Farkas-Type Results for Vector-Valued Functions with Applications

Dinh

Goberna

López

et al. 2017

J Optim Theory Appl

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The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space de…ned by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

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Section: Introductionmentioning

confidence: 99%

Farkas-Type Results for Vector-Valued Functions with Applications

Dinh

Goberna

López

et al. 2017

J Optim Theory Appl

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“…According to [14] and [15], the primal algorithm can be improved when solving linear MPP (P X ). Actually only one linear programme needs to be solved for finding the cut at each iteration step instead of two as the previous cut and bound algorithm indicates.…”

Section: Primal Algorithm For Solving Linear Mppsmentioning

confidence: 99%

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Shao

Ehrgott

2015

Optimization

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Multiplicative programming problems (MPPs) are global optimization problems known to be NP-hard. In this paper, we employ algorithms developed to compute the entire set of nondominated points of multi-objective linear programmes (MOLPs) to solve linear MPPs. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear programme in each iteration, instead of two as the previous version indicates. We call this algorithm, which is based on Benson's outer approximation algorithm for MOLPs, the primal objective space algorithm. Then, based on the dual variant of Benson's algorithm, we propose a dual objective space algorithm for solving linear MPPs. The dual algorithm also requires solving only one linear programme in each iteration. We prove the correctness of the dual algorithm and use computational experiments comparing our algorithms to a recent global optimization algorithm for linear MPPs from the literature as well as two general global optimization solvers to demonstrate the superiority of the new algorithms in terms of computation time. Thus, we demonstrate that the use of multi-objective optimization techniques can be beneficial to solve difficult single objective global optimization problems.

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“…A MOLP-solver such as Bensolve [28,30] computes not only a solution to (MOLP) in the sense of [21,27] but also a V-representation of the so-called upper image. The upper image of (MOLP) is the set…”

Section: Polyhedral Projection Via Multiple Objective Linear Programmingmentioning

confidence: 99%

Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver

Löhne¹,

Wagner²

2017

J Glob Optim

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A class of non-convex optimization problems with DC objective function is studied, where DC stands for being representable as the difference f = g − h of two convex functions g and h. In particular, we deal with the special case where one of the two convex functions g or h is polyhedral. In case g is polyhedral, we show that a solution of the DC program can be obtained from a solution of an associated polyhedral projection problem. In case h is polyhedral, we prove that a solution of the DC program can be obtained by solving a polyhedral projection problem and finitely many convex programs. Since polyhedral projection is equivalent to multiple objective linear programming (MOLP), a MOLP solver (in the second case together with a convex programming solver) can be used to solve instances of DC programs with polyhedral component. Numerical examples are provided, among them an application to locational analysis.

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Vector Optimization with Infimum and Supremum

Cited by 140 publications

References 0 publications

Farkas-Type Results for Vector-Valued Functions with Applications

Farkas-Type Results for Vector-Valued Functions with Applications

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver

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