A new geometric duality and approximation algorithms for convex vector optimization problems
buir.advisor | Ulus, Firdevs | |
dc.contributor.author | Tekgül, Simay | |
dc.date.accessioned | 2021-08-09T06:01:14Z | |
dc.date.available | 2021-08-09T06:01:14Z | |
dc.date.copyright | 2021-07 | |
dc.date.issued | 2021-07 | |
dc.date.submitted | 2021-08-02 | |
dc.description | Cataloged from PDF version of article. | en_US |
dc.description | Thesis (Master's): Bilkent University, Department of Industrial Engineering, İhsan Doğramacı Bilkent University, 2021. | en_US |
dc.description | Includes bibliographical references (leaves 67-70). | en_US |
dc.description.abstract | In the literature, there are different algorithms for solving convex vector optimization problems, in the sense of approximating the set of all minimal points in the objective space. One of the main approaches is to provide outer approximations to this set and improve the approximation iteratively by solving scalarization models. In addition to the outer approximation algorithms, which are referred to as primal algorithms, there are also geometric dual algorithms which work on a dual space and approximate the set of all maximal elements of a geometric dual problem. In most of the primal and dual algorithms in the literature, the scalarization methods, the solution concepts and the design of the algorithms depend on a fixed direction vector from the ordering cone. Recently, a primal algorithm that does not depend on a direction parameter is proposed in (Ararat et al., 2021). Using the primal algorithm in (Ararat et al., 2021), we construct a new geometric dual algorithm based on a new geometric duality relation between the primal and dual images. This relation is shown by providing an inclusion reversing one-to-one correspondence between weakly minimal proper faces of the primal image and maximal proper faces of the dual image. For a primal problem with a q-dimensional objective space, we present a dual problem with a q+1-dimensional objective space. Consequently, the resulting dual image is a convex cone. The primal algorithm in (Ararat et al., 2021) is modified to give a finite epsilon-solution to the dual problem as well as a finite weak epsilon-solution to the primal problem. The constructed geometric dual algorithm gives a finite epsilon-solution to the dual problem; moreover, it gives a finite weak delta-solution to the primal problem, where delta is determined by epsilon and the structure of the underlying ordering cone. We implement primal and dual algorithms using MATLAB and test the performance of the algorithms for randomly generated convex vector optimization problems. The tests are performed with different dimensions of the objective and decision spaces, different ordering cones, different ell-p-norms, and different stopping criteria. It is observed that the dual algorithm gives a fraction of the allowed approximation error, epsilon, resulting in a longer runtime with epsilon stopping criterion. When runtime is used as stopping criterion, the dual algorithm returns a closer approximation for higher dimensions of the objective space. | en_US |
dc.description.provenance | Submitted by Betül Özen (ozen@bilkent.edu.tr) on 2021-08-09T06:01:14Z No. of bitstreams: 1 10410114.pdf: 2390372 bytes, checksum: 642f2521e02912f53ec0330c033692a4 (MD5) | en |
dc.description.provenance | Made available in DSpace on 2021-08-09T06:01:14Z (GMT). No. of bitstreams: 1 10410114.pdf: 2390372 bytes, checksum: 642f2521e02912f53ec0330c033692a4 (MD5) Previous issue date: 2021-07 | en |
dc.description.statementofresponsibility | by Simay Tekgül | en_US |
dc.format.extent | xii, 70 leaves : charts ; 30 cm. | en_US |
dc.identifier.itemid | B151909 | |
dc.identifier.uri | http://hdl.handle.net/11693/76412 | |
dc.language.iso | English | en_US |
dc.rights | info:eu-repo/semantics/openAccess | en_US |
dc.subject | Convex vector optimization | en_US |
dc.subject | Multiobjective optimization | en_US |
dc.subject | Approxima-tion algorithm | en_US |
dc.subject | Scalarization | en_US |
dc.title | A new geometric duality and approximation algorithms for convex vector optimization problems | en_US |
dc.title.alternative | Dışbükey vektör eniyileme problemleri için yeni bir geometrik çifteşlik teorisi ve dış yaklaşıklama algoritmaları | en_US |
dc.type | Thesis | en_US |
thesis.degree.discipline | Industrial Engineering | |
thesis.degree.grantor | Bilkent University | |
thesis.degree.level | Master's | |
thesis.degree.name | MS (Master of Science) |