The concept of local upper bounds plays an important role in numerical algorithms for nonconvex, integer, and mixed-integer multiobjective optimization with respect to the componentwise partial ordering, that is, where the ordering cone is the nonnegative orthant. In this paper, we answer the question of whether and how this concept can be extended to arbitrary ordering cones. We define local upper bounds with respect to a closed pointed solid convex cone and study their properties. We show that for special polyhedral ordering cones the concept of local upper bounds can be as practical as it is for the nonnegative orthant.

We consider difference of convex (DC) programming problems and propose three algorithms to solve them globally. The main working mechanism of the proposed algorithms is to generate polyhedral underestimators to convex functions. Two of these algorithms generate a ‘fine’ polyhedral approximation of the first convex component over the compact feasible region of the DC programming problem. We prove the finiteness of these algorithms, establish the convergence rate of one of them. Moreover, we show that using the polyhedral approximation of the first component, it is possible to compute an approximate global solution of the corresponding DC programming problem without further computational effort. The third algorithm also computes a polyhedral underestimator of the first component of the DC function. Different from the first two algorithms, the third algorithm approximates it locally until finding an approximate global solution to the DC programming problem. It is shown that for any positive approximation error, the third algorithm stops after finitely many iterations. Computational results based on some test instances from the literature are provided.

We consider criterion space algorithms for biobjective mixed integer programs. The algorithms solve scalarization models in order to explore predetermined regions of the objective space called boxes, defined by two nondominated points. When exploring, the algorithm exploits information on its corner points and chooses the scalarization problem accordingly so as to detect line segments quickly, without having to solve many scalarizations. We propose three algorithms: The first one creates new boxes immediately when it finds a nondominated point, whereas the second algorithm conducts additional operations after obtaining a nondominated point by the Pascoletti-Serafini scalarization. The third algorithm is another variant that uses the computational advantage of dichotomic search whenever possible. Our computational experiments demonstrate the computational feasibility of the algorithms and show that the number of mixed integer linear programming models is significantly lower compared to similar approaches in the literature. The results further validate the utilization of Pascoletti-Serafini scalarization, aimed at enhancing the representativeness of solutions under time and cardinality limits. We observe that the third variant is particularly effective in finding a representative subset of the nondominated solutions under such limits.

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 Ararat et al. (2022). For a predetermined tolerance $\epsilon>0$, 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 $\mathcal{O}(k^{{1}/{(1-q)}})$, where $q$ is the dimension of the objective space. An improved convergence rate of $\mathcal{O}(k^{{2}/{(1-q)}})$ is proved for the special case of using the Euclidean norm.

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.

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.

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 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.

There is an existing exact algorithm that solves DC programming problems if one component of the DC function is polyhedral convex (Loehne, Wagner, 2017). Motivated by this, first, we consider two cutting-plane algorithms for generating an $\epsilon$-polyhedral underestimator of a convex function g. The algorithms start with a polyhedral underestimator of g and the epigraph of the current underestimator is intersected with either a single halfspace (Algorithm 1) or with possibly multiple halfspaces (Algorithm 2) in each iteration to obtain a better approximation. We prove the correctness and finiteness of both algorithms, establish the convergence rate of Algorithm 1, and show that after obtaining an $\epsilon$-polyhedral underestimator of the first component of a DC function, the algorithm from (Loehne, Wagner, 2017) can be applied to compute an $\epsilon$ solution of the DC programming problem without further computational effort. We then propose an algorithm (Algorithm 3) for solving DC programming problems by iteratively generating a (not necessarily $\epsilon$-) polyhedral underestimator of g. We prove that Algorithm 3 stops after finitely many iterations and it returns an $\epsilon$-solution to the DC programming problem. Moreover, the sequence $\{x_k\}_{k\geq 0} outputted by Algorithm 3 converges to a global minimizer of the DC problem when $\epsilon$ is set to zero. Computational results based on some test instances from the literature are provided.

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 [Ararat et al. 2022, Doerfler et al. 2021, Loehne et al. 2014]. They provide a polyhedral inner and outer approximation of the upper image that have a Hausdorff distance of at most $\varepsilon$. However, it is well known (see [Ulus, 2018]), that for some unbounded problems such polyhedral approximations do not exist. In this paper, we will propose a generalized solution concept, called an $(\varepsilon,\delta)$--solution, that allows also to 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 [Loehne et al. 2014] this provides a primal and a dual algorithm that allow to compute $(\varepsilon,\delta)$--solutions of (potentially unbounded) CVOPs. Numerical examples are provided.

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.

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.

An application area of vertex enumeration problem (VEP) is the usage withinobjective space based linear/convex vector optimization algorithms whose aimis to generate (an approximation of) the Pareto frontier. In such algorithms,VEP, which is defined in the objective space, is solved in each iteration andit has a special structure. Namely, the recession cone of the polyhedron to begenerated is the ordering cone. We consider and give a detailed descriptionof a vertex enumeration procedure, which iterates by calling a modified`double description (DD) method' that works for such unbounded polyhedrons. Weemploy this procedure as a function of an existing objective space basedvector optimization algorithm (Algorithm 1); and test the performance of itfor randomly generated linear multiobjective optimization problems. We comparethe efficiency of this procedure with another existing DD method as well aswith the current vertex enumeration subroutine of Algorithm 1. We observe thatthe modified procedure excels the others especially as the dimension of thevertex enumeration problem (the number of objectives of the correspondingmultiobjective problem) increases.

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.

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.

Dışbükey çok amaçlı eniyileme problemlerini Pareto kümeye iç ve dış yaklaşık kümeler bulmak anlamında ‘çözen’ bir Benson tipi algoritma ele alınmıştır. Algoritma her yinelemede o anki dış yaklaşık kümenin herhangi bir köşesi için Pascoletti-Serafini skalerizasyon modeli çözer. Bu şekilde bu köşenin Pareto kümeye yeterince yakın olup olmadığı anlaşılır. Eğer yeterince yakın değilse o anki dış yaklaşık küme bir kesit eklenerek güncellenir. Bu uygulama tüm köşeler Pareto kümeye yeterince yakın oluncaya kadar tekrarlanır. Dış yaklaşık kümenin güncellemesi işlemi, Pareto kümeye yeterince yakın olmayan ilk köşe bulunduktan sonra yapılabileceği gibi tüm köşeler kontrol edildikten sonra da yapılabilmektedir. Bu seçim algoritmanın çalışma performansını etkilemektedir. Bu çalışma ile algoritmaya bu iki uç varyanta ek olarak farklı varyantlar önerilmiş ve tüm varyantların performansları bilgisayımsal testler yolu ile karşılaştırılmıştır.

We consider a Benson type algorithm to ‘solve’ convex multiobjective optimization problems in the sense that it generates inner and outer approximations to the Pareto frontier. In each iteration of the algorithm, a Pascoletti-Serafini scalarization is solved for an arbitrary vertex of the current outer approximation. In this way, it is possible to determine if the vertex is close enough to the Pareto frontier. If not, then the current outer approximation is updated by a cut. This procedure continues until all the vertices are close enough. The update of the outer approximation can be done right after finding the first vertex that is not close enough to the Pareto frontier; or after checking all the vertices. This choice affects the performance of the algorithm. With this study, additional variants that are different than these two extreme ones are proposed and the performances of all these variants are compared via computational tests. Araştırma makalesi Başvuru: 12/09/2019 Düzeltme: 10/02/2020 Kabul: 26/02/2020

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.

In this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (the Evans–Steuer) algorithm (Math Program 5(1):54–72, 1973 ). Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions. Instead, it finds a solution in the sense of Löhne (Vector optimization with infimum and supremum. Springer, Berlin, 2011 ), that is, it finds a subset of efficient solutions that allows to generate the whole efficient frontier. In that sense, it can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modified to solve two objective bounded LVOPs with the positive orthant as the ordering cone in Ruszczyński and Vanderbei (Econometrica 71(4):1287–1297, 2003 ). The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone C and for bounded as well as unbounded problems. Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm [Benson’s algorithm in Hamel et al. (J Global Optim 59(4):811–836, 2014 )], that also provides a solution in the sense of Löhne ( 2011 ), and with the Evans–Steuer algorithm ( 1973 ). The results show that for non-degenerate problems the proposed algorithm outperforms Benson’s algorithm and is on par with the Evans–Steuer algorithm. For highly degenerate problems Benson’s algorithm (Hamel et al. 2014 ) outperforms the simplex-type algorithms; however, the parametric simplex algorithm is for these problems computationally much more efficient than the Evans–Steuer algorithm.

The authors present a benchmarking study on the companies in the Turkish food industry based on their financial data. The aim is to develop a comprehensive benchmarking framework using Data Envelopment Analysis (DEA) and information visualization. Besides DEA, a traditional tool for financial benchmarking based on financial ratios is also incorporated. The consistency/inconsistency between the two methodologies is investigated using information visualization tools. In addition, k-means clustering, a fundamental method from machine learning, is applied. Finally, other relevant data, apart from the financial data, is introduced to the analysis through information visualization to discover new insights into DEA results. The results show that the framework developed is a comprehensive and effective strategy for benchmarking; it can be applied in other industries as well. The study contributes to the literature with a novel methodology that integrates the various benchmarking methods from the fields of operations research, machine learning, and financial analysis.

Two approximation algorithms for solving convex vector optimization problems (CVOPs) are provided. Both algorithms solve the CVOP and its geometric dual problem simultaneously. The first algorithm is an extension of Benson’s outer approximation algorithm, and the second one is a dual variant of it. Both algorithms provide an inner as well as an outer approximation of the (upper and lower) images. Only one scalar convex program has to be solved in each iteration. We allow objective and constraint functions that are not necessarily differentiable, allow solid pointed polyhedral ordering cones, and relate the approximations to an appropriate ϵ -solution concept. Numerical examples are provided.