On the Biobjective Adjacent Only Quadratic Spanning Tree Problem

The adjacent only quadratic minimum spanning tree problem is an NP-hard version of the minimum spanning tree where the costs of interaction effects between every pair of adjacent edges are included in the objective function. This paper addresses the biobjective version of this problem. A Pareto local search algorithm is proposed. The algorithm is applied to a set of 108 benchmark instances. The results are compared to the optimal Pareto front generated by a branch and bound algorithm, which is a multiobjective adaptation of a well known algorithm for the mono-objective case.     

Solution methods for the bi-objective (cost-coverage) unconstrained facility location problem with an illustrative example

The Colombian coffee supply network, managed by the Federación Nacional de Cafeteros de Colombia (Colombian National Coffee-Growers Federation), requires slimming down operational costs while continuing to provide a high level of service in terms of coverage to its affiliated coffee growers. We model this problem as a biobjective (cost-coverage) uncapacitated facility location problem (BOUFLP). We designed and implemented three different algorithms for the BOUFLP that are able to obtain a good approximation of the Pareto frontier. We designed an algorithm based on the Nondominated Sorting Genetic Algorithm; an algorithm based on the Pareto Archive Evolution Strategy; and an algorithm based on mathematical programming. We developed a random problem generator for testing and comparison using as reference the Colombian coffee supply network with 29 depots and 47 purchasing centers. We compared the algorithms based on the quality of the approximation to the Pareto frontier using a nondominated space metric inspired on Zitzler and Thiele's. We used the mathematical programming-based algorithm to identify unique tradeoff opportunities for the reconfiguration of the Colombian coffee supply network. Finally, we illustrate an extension of the mathematical programming-based algorithm to perform scenario analysis for a set of uncapacitated location problems found in the literature.     

Two-phase Pareto local search for the biobjective traveling salesman problem

In this work, we present a method, called Two-Phase Pareto Local Search, to find a good approximation of the efficient set of the biobjective traveling salesman problem. In the first phase of the method, an initial population composed of a good approximation of the extreme supported efficient solutions is generated. We use as second phase a Pareto Local Search method applied to each solution of the initial population. We show that using the combination of these two techniques: good initial population generation plus Pareto Local Search gives better results than state-of-the-art algorithms. Two other points are introduced: the notion of ideal set and a simple way to produce near-efficient solutions of multiobjective problems, by using an efficient single-objective solver with a data perturbation technique.     

The biobjective travelling purchaser problem

The purpose of this article is to present and solve the Biobjective Travelling Purchaser Problem, which consists in determining a route through a subset of markets in order to collect a set of products, minimizing the travel distance and the purchasing cost simultaneously. The most convenient purchase of the product in the visited markets is easily computed once the route has been determined. Therefore, this problem contains a finite set of solutions (one for each route) and the problem belongs to the field of the Biobjective Combinatorial Optimization. It is here formulated as a Biobjective Mixed Integer Linear Programming model with an exponential number of valid inequalities, and this model is used within a cutting plane algorithm to generate the set of all supported and non-supported efficient points in the objective space. A variant of the algorithm computes only supported efficient points. For each efficient point in the objective space exactly one Pareto optimal solution in the decision space is computed by solving a single-objective problem. Each of these single-objective problems, in turn, is solved by a specific branch-and-cut approach. A heuristic improvement based on saving previously generated cuts in a common cut-pool structure has also been developed with the aim of speeding up the algorithm performance. Results based on benchmark instances from literature show that the common cut-pool heuristic is very useful, and that the proposed algorithm manages to solve instances containing up to 100 markets and 200 different products. The general procedure can be extended to address other biobjective combinatorial optimization problems whenever a branch-and-cut algorithm is available to solve a single-objective linear combination of these criteria.     

Nondominated Nash points: application of biobjective mixed integer programming

We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem.     

Approximation with a fixed number of solutions of some multiobjective maximization problems

We investigate the problem of approximating the Pareto set of some multiobjective optimization problems with a given number of solutions. Our purpose is to exploit general properties that many well studied problems satisfy. We derive existence and constructive approximation results for the biobjective versions of Max Submodular Symmetric Function (and special cases), Max Bisection, and Max Matching and also for the k-objective versions of Max Coverage, Heaviest Subgraph, Max Coloring of interval graphs.     

Conical Algorithm in Global Optimization for Optimizing over Efficient Sets

The problem of optimizing some contiuous function over the efficient set of a multiple objective programming problem can be formulated as a nonconvex global optimization problem with special structure. Based on the conical branch and bound algorithm in global optimization, we establish an algorithm for optimizing over efficient sets and discuss about the implementation of this algorithm for some interesting special cases including the case of biobjective programming problems.     

求多目标优化问题Pareto最优解集的方法

主要讨论了无约束多目标优化问题Pareto最优解集的求解方法,其中问题的目标函数是C1连续函数.给出了Pareto最优解集的一个充要条件,定义了α强有效解,并结合区间分析的方法,建立了求解无约束多目标优化问题Pareto最优解集的区间算法,理论分析和数值结果均表明该算法是可靠和有效的.     

Evaluation of the multiobjective ant colony algorithm performances on biobjective quadratic assignment problems

The difficulty of resolving the multiobjective combinatorial optimization problems with traditional methods has directed researchers to investigate new approaches which perform better. In recent years some algorithms based on ant colony optimization (ACO) metaheuristic have been suggested to solve these multiobjective problems. In this study these algorithms have been reported and programmed both to solve the biobjective quadratic assignment problem (BiQAP) instances and to evaluate the performances of these algorithms. The robust parameter sets for each 12 multiobjective ant colony optimization (MOACO) algorithms have been calculated and BiQAP instances in the literature have been solved within these parameter sets. The performances of the algorithms have been evaluated by comparing the Pareto fronts obtained from these algorithms. In the evaluation step, a multi significance test is used in a non hierarchical structure, and a performance metric (P metric) essential for this test is introduced. Through this study, decision makers will be able to put in the biobjective algorithms in an order according to the priority values calculated from the algorithms’ Pareto fronts. Moreover, this is the first time that MOACO algorithms have been compared by solving BiQAPs.     

On the exactness of the ε-constraint method for biobjective nonlinear integer programming

The ε-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with biobjective nonlinear integer problems. Under specific assumptions, we prove that the number of subproblems that the method needs to address to detect the complete Pareto front is finite. We report numerical results on portfolio optimization instances built on real-world data and show a comparison with an existing criterion space algorithm.     

A new Pareto set generating method for multi-criteria optimization problems

This paper proposes a new classical method to capture the complete Pareto set of a multi-criteria optimization problem (MOP) even without having any prior information about the location of Pareto surface. The solutions obtained through the proposed method are globally Pareto optimal. Moreover, each and every global Pareto optimal point is within the attainable range. This paper also suggests a procedure to ensure the proper Pareto optimality of the outcomes if slight modifications are allowed in the constraint set of the MOP under consideration. Among the set of all outcomes, the proposed method can effectively detect the regions of unbounded trade-offs between the criteria, if they exist.     

Solvability of the vector problem by the linear criteria convolution algorithm

Conditions are found under which a multicriteria problem with a finite set of vector estimates is solvable by means of the linear criteria convolution (LCC) algorithm, that is, any Pareto optimum for the problem can be obtained as an optimum solution to a one-criterion problem with an aggregated criterion defined as an LCC. Also, an algorithm is suggested that is polynomial in dimension and reduces any problem with minimax and minimin criteria to an equivalent vector problem with the same Pareto set solvable by the LCC algorithm. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 502–509, October, 1997. Translated by V. N. Dubrovsky     

Constructing a Pareto front approximation for decision making

An approach to constructing a Pareto front approximation to computationally expensive multiobjective optimization problems is developed. The approximation is constructed as a sub-complex of a Delaunay triangulation of a finite set of Pareto optimal outcomes to the problem. The approach is based on the concept of inherent nondominance. Rules for checking the inherent nondominance of complexes are developed and applying the rules is demonstrated with examples. The quality of the approximation is quantified with error estimates. Due to its properties, the Pareto front approximation works as a surrogate to the original problem for decision making with interactive methods.     

A Post-Optimality Analysis Algorithm for Multi-Objective Optimization

Algorithms for multi-objective optimization problems are designed to generate a single Pareto optimum (non-dominated solution) or a set of Pareto optima that reflect the preferences of the decision-maker. If a set of Pareto optima are generated, then it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optima using an unbiased technique of filtering solutions. This suggests the need for an efficient selection procedure to identify such a preferred subset that reflects the preferences of the decision-maker with respect to the objective functions. Selection procedures typically use a value function or a scalarizing function to express preferences among objective functions. This paper introduces and analyzes the Greedy Reduction (GR) algorithm for obtaining subsets of Pareto optima from large solution sets in multi-objective optimization. Selection of these subsets is based on maximizing a scalarizing function of the vector of percentile ordinal rankings of the Pareto optima within the larger set. A proof of optimality of the GR algorithm that relies on the non-dominated property of the vector of percentile ordinal rankings is provided. The GR algorithm executes in linear time in the worst case. The GR algorithm is illustrated on sets of Pareto optima obtained from five interactive methods for multi-objective optimization and three non-linear multi-objective test problems. These results suggest that the GR algorithm provides an efficient way to identify subsets of preferred Pareto optima from larger sets.     

Maximizing upgrading and downgrading margins for ordinal regression

In ordinal regression, a score function and threshold values are sought to classify a set of objects into a set of ranked classes. Classifying an individual in a class with higher (respectively lower) rank than its actual rank is called an upgrading (respectively downgrading) error. Since upgrading and downgrading errors may not have the same importance, they should be considered as two different criteria to be taken into account when measuring the quality of a classifier. In Support Vector Machines, margin maximization is used as an effective and computationally tractable surrogate of the minimization of misclassification errors. As an extension, we consider in this paper the maximization of upgrading and downgrading margins as a surrogate of the minimization of upgrading and downgrading errors, and we address the biobjective problem of finding a classifier maximizing simultaneously the two margins. The whole set of Pareto-optimal solutions of such biobjective problem is described as translations of the optimal solutions of a scalar optimization problem. For the most popular case in which the Euclidean norm is considered, the scalar problem has a unique solution, yielding that all the Pareto-optimal solutions of the biobjective problem are translations of each other. Hence, the Pareto-optimal solutions can easily be provided to the analyst, who, after inspection of the misclassification errors caused, should choose in a later stage the most convenient classifier. The consequence of this analysis is that it provides a theoretical foundation for a popular strategy among practitioners, based on the so-called ROC curve, which is shown here to equal the set of Pareto-optimal solutions of maximizing simultaneously the downgrading and upgrading margins.     

A Proximal Point-Type Method for Multicriteria Optimization

In this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem.     

Variants of the $$ \vare psilon $$-constraint method for biobjective integer programming problems: a p plication to p-median-cover problems

We conduct an in-depth analysis of the \(\varepsilon \)-constraint method (ECM) for finding the exact Pareto front for biobjective integer programming problems. We have found up to six possible different variants of the ECM. We first discuss the complexity of each of these variants and their relationship with other exact methods for solving biobjective integer programming problems. By extending some results of Neumayer and Schweigert (OR Spektrum 16:267–276, 1994), we develop two variants of the ECM, both including an augmentation term and requiring \(N+1\) integer programs to be solved, where N is the number of nondominated points. In addition, we present another variant of the ECM, based on the use of elastic constraints and also including an augmentation term. This variant has the same complexity, namely \(N+1\), which is the minimum reached for any exact method. A comparison of the different variants is carried out on a set of biobjective location problems which we call p-median-cover problems; these include the objectives of the p-median and the maximal covering problems. As computational results show, for this class of problems, the augmented ECM with elastic constraint is the most effective variant for finding the Pareto front in an exact manner.     

Enumeration of Pareto optimal multi-criteria spanning trees – a proof of the incorrectness of Zhou and Gen's proposed algorithm

The minimum spanning tree (MST) problem is a well-known optimization problem of major significance in operational research. In the multi-criteria MST (mc-MST) problem, the scalar edge weights of the MST problem are replaced by vectors, and the aim is to find the complete set of Pareto optimal minimum-weight spanning trees. This problem is NP-hard and so approximate methods must be used if one is to tackle it efficiently. In an article previously published in this journal, a genetic algorithm (GA) was put forward for the mc-MST. To evaluate the GA, the solution sets generated by it were compared with solution sets from a proposed (exponential time) algorithm for enumerating all Pareto optimal spanning trees. However, the proposed enumeration algorithm that was used is not correct for two reasons: (1) It does not guarantee that all Pareto optimal minimum-weight spanning trees are returned. (2) It does not guarantee that those trees that are returned are Pareto optimal. In this short paper we prove these two theorems.     

Shortest Paths with Shortest Detours

This paper is concerned with a biobjective routing problem, called the shortest path with shortest detour problem, in which the length of a route is minimized as one criterion and, as second, the maximal length of a detour route if the chosen route is blocked is minimized. Furthermore, the relation to robust optimization is pointed out, and we present a new polynomial time algorithm, which computes a minimal complete set of efficient paths for the shortest path with shortest detour problem. Moreover, we show that the number of nondominated points is bounded by the number of arcs in the graph.