Browsing by Author "Ulus, Firdevs"

Now showing 1 - 11 of 11

Open Access
Algorithms to solve unbounded convex vector optimization problems
(Society for Industrial and Applied Mathematics Publications, 2023-10-12) Wagner, A.; Ulus, Firdevs; Rudloff, B.; Kováčová, G.; Hey, N.
This paper is concerned with solution algorithms for general convex vector optimization problems (CVOPs). So far, solution concepts and approximation algorithms for solving CVOPs exist only for bounded problems [\c C. Ararat, F. Ulus, and M. Umer, J. Optim. Theory Appl., 194 (2022), pp. 681-712], [D. Dörfler, A. Löhne, C. Schneider, and B. Weißing, Optim. Methods Softw., 37 (2022), pp. 1006-1026], [A. Löhne, B. Rudloff, and F. Ulus, J. Global Optim., 60 (2014), pp. 713-736]. They provide a polyhedral inner and outer approximation of the upper image that have a Hausdorff distance of at most ε. However, it is well known (see [F. Ulus, J. Global Optim., 72 (2018), pp. 731-742]), that for some unbounded problems such polyhedral approximations do not exist. In this paper, we will propose a generalized solution concept, called an (ε,δ)-solution, that allows one to also consider unbounded CVOPs. It is based on additionally bounding the recession cones of the inner and outer polyhedral approximations of the upper image in a meaningful way. An algorithm is proposed that computes such δ-outer and δ-inner approximations of the recession cone of the upper image. In combination with the results of [A. Löhne, B. Rudloff, and F. Ulus, J. Global Optim., 60 (2014), pp. 713-736] this provides a primal and a dual algorithm that allow one to compute (ε,δ)-solutions of (potentially unbounded) CVOPs. Numerical examples are provided.
Unknown
Certainty equivalent and utility indifference pricing for incomplete preferences via convex vector optimization
(Springer Science and Business Media Deutschland GmbH, 2020) Rudloff, B.; Ulus, Firdevs
For incomplete preference relations that are represented by multiple priors and/or multiple—possibly multivariate—utility functions, we define a certainty equivalent as well as the utility indifference price bounds as set-valued functions of the claim. Furthermore, we motivate and introduce the notion of a weak and a strong certainty equivalent. We will show that our definitions contain as special cases some definitions found in the literature so far on complete or special incomplete preferences. We prove monotonicity and convexity properties of utility buy and sell prices that hold in total analogy to the properties of the scalar indifference prices for complete preferences. We show how the (weak and strong) set-valued certainty equivalent as well as the indifference price bounds can be computed or approximated by solving convex vector optimization problems. Numerical examples and their economic interpretations are given for the univariate as well as for the multivariate case.
Open Access
Computing the recession cone of a convex upper image via convex projection
(Springer New York LLC, 2024-03-01) Kovácová, G.; Ulus, Firdevs
It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples.
Open Access
Convergence analysis of a norm minimization-based convex vector optimization algorithm
(Society for Industrial and Applied Mathematics, 2024-07-25) Ararat, Çağın; Ulus, Firdevs; Umer, Muhammad
In this work, we propose an outer approximation algorithm for solving bounded convex vector optimization problems (CVOPs). The scalarization model solved iteratively within the algorithm is a modification of the norm-minimizing scalarization proposed in [\c C. Ararat, F. Ulus, and we prove that the algorithm terminates after finitely many iterations, and it returns a polyhedral outer approximation to the upper image of the CVOP such that the Hausdorff distance between the two is less than \epsilon . We show that for an arbitrary norm used in the scalarization models, the approximation error after k iterations decreases by the order of O(k1/(1-q)), where q is the dimension of the objective space. An improved convergence rate of O(k2/(1-q)) is proved for the special case of using the Euclidean norm.
Open Access
An exact algorithm for biobjective integer programming problems
(Elsevier Ltd, 2021-08) Doğan, Saliha Ferda; Karsu, Özlem; Ulus, Firdevs
We propose an exact algorithm for solving biobjective integer programming problems, which arise in various applications of operations research. The algorithm is based on solving Pascoletti-Serafini scalarizations to search specified regions (boxes) in the objective space and returns the set of nondominated points. We implement the algorithm with different strategies, where the choices of the scalarization model parameters and splitting rule differ. We then derive bounds on the number of scalarization models solved; and demonstrate the performances of the variants through computational experiments both as exact algorithms and as solution approaches under time restriction. The experiments demonstrate that different strategies have advantages in different aspects: while some are quicker in finding the whole set of nondominated solutions, others return good-quality solutions in terms of representativeness when run under time restriction. We also compare the proposed approach with existing algorithms. The results of our experiments show the satisfactory behaviour of our algorithm, especially when run under time limit, as it achieves better coverage of the whole frontier with a smaller number of solutions compared to the existing algorithms.
Open Access
Geometric duality results and approximation algorithms for convex vector optimization problems
(Society for Industrial and Applied Mathematics Publications, 2023-01-27) Ararat, Çağın; Tekgül, S.; Ulus, Firdevs
We study geometric duality for convex vector optimization problems. For a primal problem with a q-dimensional objective space, we formulate a dual problem with a (q+1)-dimensional objective space. Consequently, different from an existing approach, the geometric dual problem does not depend on a fixed direction parameter, and the resulting dual image is a convex cone. We prove a one-to-one correspondence between certain faces of the primal and dual images. In addition, we show that a polyhedral approximation for one image gives rise to a polyhedral approximation for the other. Based on this, we propose a geometric dual algorithm which solves the primal and dual problems simultaneously and is free of direction-biasedness. We also modify an existing direction-free primal algorithm in such a way that it solves the dual problem as well. We test the performance of the algorithms for randomly generated problem instances by using the so-called primal error and hypervolume indicator as performance measures. © 2023 Society for Industrial and Applied Mathematics.
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
A Norm Minimization-Based Convex Vector Optimization Algorithm
(Springer New York LLC, 2022-06-04) Ararat, Çağın; Ulus, Firdevs; Umer, Muhammad
We propose an algorithm to generate inner and outer polyhedral approximations to the upper image of a bounded convex vector optimization problem. It is an outer approximation algorithm and is based on solving norm-minimizing scalarizations. Unlike Pascoletti–Serafini scalarization used in the literature for similar purposes, it does not involve a direction parameter. Therefore, the algorithm is free of direction-biasedness. We also propose a modification of the algorithm by introducing a suitable compact subset of the upper image, which helps in proving for the first time the finiteness of an algorithm for convex vector optimization. The computational performance of the algorithms is illustrated using some of the benchmark test problems, which shows promising results in comparison to a similar algorithm that is based on Pascoletti–Serafini scalarization.
Open Access
Outer approximation algorithms for convex vector optimization problems
(Taylor and Francis Ltd., 2023-02-09) Keskin, İrem Nur; Ulus, Firdevs
In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarization finds the minimum ‘distance’ from a reference point, which is usually taken as a vertex of the current outer approximation, to the upper image through a given direction. We propose efficient methods to select the parameters (the reference point and direction vector) of the PS scalarization and analyse the effects of these on the overall performance of the algorithm. Different from the existing vertex selection rules from the literature, the proposed methods do not require solving additional single-objective optimization problems. Using some test problems, we conduct an extensive computational study where three different measures are set as the stopping criteria: the approximation error, the runtime, and the cardinality of the solution set. We observe that the proposed variants have satisfactory results, especially in terms of runtime compared to the existing variants from the literature. © 2023 Informa UK Limited, trading as Taylor & Francis Group.
Open Access
Split algorithms for multiobjective integer programming problems
(Elsevier, 2022-04) Karsu, Özlem; Ulus, Firdevs
We consider split algorithms that partition the objective function space into p or p−1 dimensional regions so as to search for nondominated points of multiobjective integer programming problems, where p is the number of objectives. We provide a unified approach that allows different split strategies to be used within the same algorithmic framework with minimum change. We also suggest an effective way of making use of the information on subregions when setting the parameters of the scalarization problems used in the p-split structure. We compare the performances of variants of these algorithms both as exact algorithms and as solution approaches under time restriction, considering the fact that finding the whole set may be computationally infeasible or undesirable in practice. We demonstrate through computational experiments that while the (p−1)-split structure is superior in terms of overall computational time, the p-split structure provides significant advantage under time/cardinality limited settings in terms of representativeness, especially with adaptive parameter setting and/or a suitably chosen order for regions to be explored.
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.