Browsing by Subject "Multiobjective optimization"

Now showing 1 - 12 of 12

Open Access
Algorithms for on-line vertex enumeration problem
(2017-09) Kaya, İrfan Caner
Vertex enumeration problem is to enumerate all vertices of a polyhedron P which is given by intersection of finitely many halfspaces. It is a basis for many algorithms designed to solve problems from various application areas and there are many algorithms to solve these problems in the literature. On the one hand, there are iterative algorithms which solve the so called 'on-line' vertex enumeration problem in each iteration. In other words, in each iteration of these algorithms, the current polyhedron is intersected with an additional halfspace that defines P. On the other hand, there are simplex-type algorithms which takes the set off all halfspaces as its input from the beginning. One of the usages of the vertex enumeration problem is the Benson-type multiobjective optimization algorithms. The aim of these algorithms is to generate or approximate the Pareto frontier (the set of nondominated points in the objective space). In each iteration of the Benson's algorithm, a polyhedron which contains the Pareto frontier is intersected with an additional halfspace in order tofind a finer outer approximation. The vertex enumeration problem to be used within this algorithm has a special structure. Namely, the polyhedron to be generated is known to be unbounded with a recession cone which is equal to the positive orthant. In this thesis, we consider the double description method which is a method to solve an on-line vertex enumeration problem where the starting polyhedron is bounded. (1) We generate an iterative algorithm to solve the vertex enumeration problem from the scratch where polyhedron P is allowed to be bounded or unbounded. (2) Then, we slightly modify this algorithm to be more efficient while it only works for problems where the recession cone of P is known to be the positive orthant. (3) Finally, we generate an additional algorithm for these problems. For this one, we modify the double description method such that it uses the extreme directions of the recession cone more effectively. We provide an illustrative example to explain the algorithms in detail. We implement the algorithms using MATLAB; employ each of them as a function of a Benson-type multiobjective optimization algorithm; and test the performances of the algorithms for randomly generated linear multiobjective optimization problems. Accordingly, for two dimensional problems, there is no clear distinction between the run-time performances of these algorithms. However, as the dimension of the vertex enumeration problem increases, the last algorithm (Algorithm 3) gets more efficient than the others.
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
Investigation of multi-objective optimization criteria for RNA design
(IEEE, 2017-12) Hampson, D. J. D.; Sav, Sinem; Tsang, H. H.
RNA design is the inverse of RNA folding and it appears to be NP-hard. In RNA design, a secondary structure is given and the goal is to find a nucleotide sequence that will fold into this structure. To find such sequence(s) involves exploring the exponentially large sequence space. In literature, heuristic algorithms are the standard technique for tackling the RNA design. Heuristic algorithms enable effective and efficient exploration of the high-dimensional sequence-structure space when searching for candidates that fold into a given target structure. The main goal of this paper is to investigate the use of multi-objective criteria in SIMARD and Quality Pre-selection Strategy (QPS). The objectives that we optimize are Hamming distance (between designed structure and target structure) and thermodynamic free energy. We examine the different combinations of optimization criteria, and attempt to draw conclusions about the relationships between them. We find that energy is a poor primary objective but makes an excellent secondary objective. We also find that using multi-objective pre-selection produces viable solutions in far fewer steps than was previously possible with SIMARD. © 2016 IEEE.
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 new geometric duality and approximation algorithms for convex vector optimization problems
(2021-07) Tekgül, Simay
In the literature, there are different algorithms for solving convex vector optimization problems, in the sense of approximating the set of all minimal points in the objective space. One of the main approaches is to provide outer approximations to this set and improve the approximation iteratively by solving scalarization models. In addition to the outer approximation algorithms, which are referred to as primal algorithms, there are also geometric dual algorithms which work on a dual space and approximate the set of all maximal elements of a geometric dual problem. In most of the primal and dual algorithms in the literature, the scalarization methods, the solution concepts and the design of the algorithms depend on a fixed direction vector from the ordering cone. Recently, a primal algorithm that does not depend on a direction parameter is proposed in (Ararat et al., 2021). Using the primal algorithm in (Ararat et al., 2021), we construct a new geometric dual algorithm based on a new geometric duality relation between the primal and dual images. This relation is shown by providing an inclusion reversing one-to-one correspondence between weakly minimal proper faces of the primal image and maximal proper faces of the dual image. For a primal problem with a q-dimensional objective space, we present a dual problem with a q+1-dimensional objective space. Consequently, the resulting dual image is a convex cone. The primal algorithm in (Ararat et al., 2021) is modified to give a finite epsilon-solution to the dual problem as well as a finite weak epsilon-solution to the primal problem. The constructed geometric dual algorithm gives a finite epsilon-solution to the dual problem; moreover, it gives a finite weak delta-solution to the primal problem, where delta is determined by epsilon and the structure of the underlying ordering cone. We implement primal and dual algorithms using MATLAB and test the performance of the algorithms for randomly generated convex vector optimization problems. The tests are performed with different dimensions of the objective and decision spaces, different ordering cones, different ell-p-norms, and different stopping criteria. It is observed that the dual algorithm gives a fraction of the allowed approximation error, epsilon, resulting in a longer runtime with epsilon stopping criterion. When runtime is used as stopping criterion, the dual algorithm returns a closer approximation for higher dimensions of the objective space.
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.
Open Access
Outer approximation algorithms for convex vector optimization problems
(2021-07) Keskin, Irem Nur
There are diﬀerent outer approximation algorithms in the literature that are de-signed to solve convex vector optimization problems in the sense that they approx-imate the upper image using polyhedral sets. At each iteration, these algorithms solve vertex enumeration and scalarization problems. The vertex enumeration problem is used to ﬁnd the vertex representation of the current outer approxima-tion. The scalarization problem is used in order to generate a weakly C-minimal element of the upper image as well as a supporting halfspace that supports the upper image at that point. In this study, we present a general framework of such algorithm in which the Pascoletti-Seraﬁni scalarization is used. 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. The reference point and the direction vector are the parameters for this scalarization. The motivation of this study is to come up with eﬃcient methods to select the parameters of the Pascoletti-Serraﬁni scalarization and analyze the eﬀects of these parameter selections on the performance of the algorithm. We ﬁrst propose three rules to choose the direction parameter at each iteration. We conduct a preliminary computational study to observe the eﬀects of these rules under various, rather simple rules for vertex selection. Depending on the results of the preliminary analysis, we ﬁx a direction selection rule to continue with. Moreover, we observe that vertex selection also has a signiﬁcant impact on the performance, as expected. Then, we propose additional vertex selection rules, which are slightly more complicated than the previous ones, and are designed with the motivation that they generate a well-distributed points on the boundary of the upper image. Diﬀerent from the existing vertex selection rules from the literature, they do not require to solve additional single-objective optimization problems. Using some test problems, we conduct a computational study where three dif-ferent measures set as the stopping criteria: the approximation error, the runtime, and the cardinality of the solution set. We compare the proposed variants and some algorithms from the literature in terms of these measures that are used as the stopping criteria as well as an additional proximity measure, hypervolume gap. We observe that the proposed variants have satisfactory results especially in terms of runtime. When the approximation error is chosen as the stopping criteria, the proposed variants require less CPU time compared to the algorithms from the literature. Under ﬁxed runtime, they return better proximity measures in general. Under ﬁxed cardinality, the algorithms from the literature yield bet-ter proximity measures, but they require signiﬁcantly more CPU time than the proposed variants.
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
A parametric simplex algorithm for linear vector optimization problems
(Springer, 2017) Rudloff, B.; Ulus, F.; Vanderbei, R.
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. © 2016, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.
Open Access
A problem space genetic algorithm in multiobjective optimization
(Springer New York LLC, 2003) Türkcan, A.; Aktürk, M. S.
In this study, a problem space genetic algorithm (PSGA) is used to solve bicriteria tool management and scheduling problems simultaneously in flexible manufacturing systems. The PSGA is used to generate approximately efficient solutions minimizing both the manufacturing cost and total weighted tardiness. This is the first implementation of PSGA to solve a multiobjective optimization problem (MOP). In multiobjective search, the key issues are guiding the search towards the global Pareto-optimal set and maintaining diversity. A new fitness assignment method, which is used in PSGA, is proposed to find a well-diversified, uniformly distributed set of solutions that are close to the global Pareto set. The proposed fitness assignment method is a combination of a nondominated sorting based method which is most commonly used in multiobjective optimization literature and aggregation of objectives method which is popular in the operations research literature. The quality of the Pareto-optimal set is evaluated by using the performance measures developed for multiobjective optimization problems.
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.
Embargo
Using equitable optimization for the hazmat transport network design problem
(2024-08) Çakır, Yunus Emre
The shipment of hazardous materials is challenging due to the risk the population centers face during transportation. The policy makers often have concerns for ensuring a balanced allocation of the risks to the population centers. We structure this problem as an equitable optimization problem with multiple objectives aiming to minimize the risk exposure of different neighborhoods. The resulting multiobjective mixed integer linear programming problem is solved to provide equitably nondominated solutions, each with different levels of efficiency (total risk) and fairness (allocation of risk). Moreover, a robust programming extension is discussed and shown to be valuable under uncertainty.