Browsing by Subject "Polyhedral approximation"

Now showing 1 - 3 of 3

Open Access
Approximation algorithms for difference of convex (DC) programming problems
(2023-07) Pirani, Fahaar Mansoor
This thesis is concerned with Difference of Convex (DC) programming problems and approximation algorithms to solve them. There is an existing exact algorithm that solves DC programming problems if one component of the DC function is polyhedral convex [1]. Motivated by this, first, we propose an algorithm (Algorithm 1) for generating an ϵ-polyhedral underestimator of a convex function g. The algorithm starts with a polyhedral underestimator of g and the epigraph of the current underestimator is intersected with a single halfspace in each iteration to obtain a better approximation. We prove the correctness and establish the convergence rate of Algorithm 1. We also propose a modified variant (Algorithm 2) in which multiple halfspaces are used to update the epigraph of current approximation in each iteration. In addition to its correctness, we prove that Algorithm 2 terminates after finitely many iterations. We show that after obtaining an ϵ-polyhedral underestimator of the first component of a DC function, the algorithm from [1] can be applied to compute an ϵ-solution of the DC programming problem. We also propose an algorithm (Algorithm 3) for solving DC programming problems directly. In each iteration, Algorithm 3 updates the polyhedral underestimator of g locally while searching for an ϵ-solution to the DC problem directly. We prove that the algorithm stops after finitely many iterations and it returns an ϵ-solution to the DC programming problem. Moreover, the sequence {xk}k≥0 outputted by Algorithm 3 converges to a global minimizer of the DC problem when ϵ is set to zero. The computational results, obtained using some test examples from [2], show comparable performance of Algorithms 1, 2 and 3 with respect to two DC programming algorithms from the literature.
Open Access
An iterative vertex enumeration method for objective space based vector optimization algorithms
(EDP Sciences, 2021-03-02) Kaya, İrfan Caner; Ulus, Firdevs
An application area of vertex enumeration problem (VEP) is the usage within objective space based linear/convex vector optimization algorithms whose aim is to generate (an approximation of) the Pareto frontier. In such algorithms, VEP, which is defined in the objective space, is solved in each iteration and it has a special structure. Namely, the recession cone of the polyhedron to be generated is the ordering cone. We consider and give a detailed description of a vertex enumeration procedure, which iterates by calling a modified “double description (DD) method” that works for such unbounded polyhedrons. We employ this procedure as a function of an existing objective space based vector optimization algorithm (Algorithm 1); and test the performance of it for randomly generated linear multiobjective optimization problems. We compare the efficiency of this procedure with another existing DD method as well as with the current vertex enumeration subroutine of Algorithm 1. We observe that the modified procedure excels the others especially as the dimension of the vertex enumeration problem (the number of objectives of the corresponding multiobjective problem) increases.
Open Access
Tractability of convex vector optimization problems in the sense of polyhedral approximations
(Springer New York LLC, 2018) Ulus, Firdevs
There are different solution concepts for convex vector optimization problems (CVOPs) and a recent one, which is motivated from a set optimization point of view, consists of finitely many efficient solutions that generate polyhedral inner and outer approximations to the Pareto frontier. A CVOP with compact feasible region is known to be bounded and there exists a solution of this sense to it. However, it is not known if it is possible to generate polyhedral inner and outer approximations to the Pareto frontier of a CVOP if the feasible region is not compact. This study shows that not all CVOPs are tractable in that sense and gives a characterization of tractable problems in terms of the well known weighted sum scalarization problems.