Browsing by Author "Umer, Muhammad"

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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
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
Norm minimization-based convex vector optimization algorithms
(2022-08) Umer, Muhammad
This thesis is concerned with convex vector optimization problems (CVOP). We propose an outer approximation algorithm (Algorithm 1) for solving CVOPs. In each iteration, the algorithm solves a norm-minimizing scalarization for a reference point in the objective space. The idea is inspired by some Benson-type algorithms in the literature that are based on Pascoletti-Seraﬁni scalarization. Since this scalarization needs a direction parameter, the eﬃciency of these algorithms depend on the selection of the direction parameter. In contrast, our algorithm is free of direction biasedness since it solves a scalarization that is based on minimizing a norm. However, the structure of such algorithms, including ours, has some built-in limitation which makes it diﬃcult to perform convergence analysis. To overcome this, we modify the algorithm by introducing a suitable compact subset of the upper image. After the modiﬁcation, we have Algorithm 2 in which norm-minimizing scalarizations are solved for points in the compact set. To the best of our knowledge, Algorithm 2 is the ﬁrst algorithm for CVOPs, which is proven to be ﬁnite. Finally, we propose a third algorithm for the purposes of con-vergence analysis (Algorithm 3), where a modiﬁed norm-minimizing scalarization is solved in each iteration. This scalarization includes an additional constraint which ensures that the algorithm deals with only a compact subset of the upper image from the beginning. Besides having the ﬁniteness result, Algorithm 3 is the ﬁrst CVOP algorithm with an estimate of a convergence rate. The experimental results, obtained using some benchmark test problems, show comparable performance of our algorithms with respect to an existing CVOP algorithm based on Pascoletti-Seraﬁni scalarization.