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Deep learning the efficient frontier of convex vector optimization problems

JOURNAL OF GLOBAL optimization 2024年第2期90卷 429-458页

作者： Feinstein, Zachary Rudloff, Birgit Stevens Inst Technol Sch Business Hoboken NJ 07030 USA Vienna Univ Econ & Business Inst Stat & Math A-1020 Vienna Austria

In this paper, we design a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems (CVOP) satisfying Slater's condition. The proposed machine learning methodology provides both an inner and outer approximation of the weakly efficient frontier, as well as an upper bound to the error at each approximated efficient point. In numerical case studies we demonstrate that the proposed algorithm is effectively able to approximate the true weakly efficient frontier of CVOPs. This remains true even for large problems (i.e., many objectives, variables, and constraints) and thus overcoming the curse of dimensionality.

关键词： convex vector optimization convex multi-objective optimization Machine learning Deep learning Neural networks Efficient frontier

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CONVERGENCE ANALYSIS OF A NORM MINIMIZATION-BASED convex vector optimization ALGORITHM

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SIAM JOURNAL ON optimization 2024年第3期34卷 2700-2728页

作者： Ararat, Cagin Ulus, Firdevs Umer, Muhammad Bilkent Univ Dept Ind Engn Ankara Turkiye Natl Univ Sci & Technol NUST Coll Aeronaut Engn CAE Dept Ind Engn Islamabad Pakistan

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.

关键词： convex vector optimization multiobjective optimization approximation algorithm convergence rate convex compact set Hausdorff distance

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ALGORITHMS TO SOLVE UNBOUNDED convex vector optimization PROBLEMS

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SIAM JOURNAL ON optimization 2023年第4期33卷 2598-2624页

作者： Wagner, Andrea Ulus, Firdevs Rudloff, Birgit Kovacova, Gabriela Hey, Niklas Vienna Univ Econ & Business Inst Stat & Math A-1020 Vienna Austria Bilkent Univ Dept Ind Engn TR-06800 Ankara Turkiye

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. Ararat, F. Ulus, and M. Umer, J. Optim. Theory Appl., 194 (2022), pp. 681-712], [D. Dorfler, A. Lohne, C. Schneider, and B. Wei ss ing, Optim. Methods Softw., 37 (2022), pp. 1006-1026], [A. Lohne, 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 epsilon. 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 (epsilon, delta)-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 delta-outer and delta-inner approximations of the recession cone of the upper image. In combination with the results of [A. L"ohne, 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 (epsilon, delta)-solutions of (potentially unbounded) CVOPs. Numerical examples are provided.

关键词： convex vector optimization unbounded problems approximation algorithm approximation of cones

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GEOMETRIC DUALITY RESULTS AND APPROXIMATION ALGORITHMS FOR convex vector optimization PROBLEMS\ast

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SIAM JOURNAL ON optimization 2023年第1期33卷 116-146页

作者： Ararat, Cagin Tekgul, Imay Ulus, Firdevs Bilkent Univ Dept Ind Engn TR-06800 Ankara Turkiye Univ Edinburgh Sch Math Edinburgh EH9 3FD Scotland

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.

关键词： convex vector optimization multiobjective optimization approximation algorithm scalarization geometric duality hypervolume indicator

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Outer approximation algorithms for convex vector optimization problems

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optimization METHODS & SOFTWARE 2023年第4期38卷 723-755页

作者： Keskin, Irem Nur Ulus, Firdevs Duke Univ Fuqua Sch Business Durham NC USA Bilkent Univ Dept Ind Engn Ankara Turkiye Bilkent Univ Dept Ind Engn TR-06800 Ankara Turkiye Bilkent Univ Dept Ind Engn Ankara Turkiye

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.

关键词： Multiobjective optimization convex vector optimization approximation algorithms Pascoletti-Serafini scalarization

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A Norm Minimization-Based convex vector optimization Algorithm

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JOURNAL OF optimization THEORY AND APPLICATIONS 2022年第2期194卷 681-712页

作者： Ararat, Cagin Ulus, Firdevs Umer, Muhammad Bilkent Univ Dept Ind Engn Ankara Turkey

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.

关键词： convex vector optimization Multiobjective optimization Approximation algorithm Scalarization Norm minimization

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Certainty equivalent and utility indifference pricing for incomplete preferences via convex vector optimization

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MATHEMATICS AND FINANCIAL ECONOMICS 2021年第2期15卷 397-430页

作者： Rudloff, Birgit Ulus, Firdevs Vienna Univ Econ & Business Inst Stat & Math A-1020 Vienna Austria Bilkent Univ Dept Ind Engn TR-06800 Ankara Turkey

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.

关键词： Utility maximization Indifference price bounds Certainty equivalent Incomplete preferences convex vector optimization

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Computing the recession cone of a convex upper image via convex projection

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JOURNAL OF GLOBAL optimization 2024年第4期89卷 975-994页

作者： Kovacova, Gabriela Ulus, Firdevs Univ Calif Los Angeles Dept Math Los Angeles CA 90095 USA Bilkent Univ Dept Ind Engn TR-06800 Ankara Turkiye

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.

关键词： convex vector optimization Linear vector optimization Unbounded vector optimization Recession cone convex projection

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convex projection and convex multi-objective optimization

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JOURNAL OF GLOBAL optimization 2022年第2期83卷 301-327页

作者： Kovacova, Gabriela Rudloffl, Birgit Vienna Univ Econ & Business Inst Stat & Math A-1020 Vienna Austria

In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Lohne and Weissing (Math Methods Oper Res 84(2):411-426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Lohne and Weissing (Math Methods Oper Res 84(2):411-426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations.

关键词： convex projection convex vector optimization convex multi-objective optimization

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Further study on the Levitin-Polyak well-posedness of constrained convex vector optimization problems

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第3期75卷 1341-1347页

作者： Huang, X. X. Yang, X. Q. Hong Kong Polytech Univ Dept Appl Math Hong Kong Hong Kong Peoples R China Chongqing Univ Sch Econ & Business Adm Chongqing 400030 Peoples R China

In this paper, we first establish characterizations of the nonemptiness and compactness of the set of weakly efficient solutions of a convex vector optimization problem with a general ordering cone (with or without a cone constraint) defined in a finite dimensional space. Using one of the characterizations, we further establish for a convex vector optimization problem with a general ordering cone and a cone constraint defined in a finite dimensional space the equivalence between the nonemptiness and compactness of its weakly efficient solution set and the generalized type I Levitin-Polyak well-posednesses. Finally, for a cone-constrained convex vector optimization problem defined in a Banach space, we derive sufficient conditions for guaranteeing the generalized type I Levitin-Polyak well-posedness of the problem. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： convex vector optimization Cone-constrained optimization Weakly efficient solution set Well-posedness Ekeland's variational principle

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