Synchronous approach in interactive multiobjective optimization

We introduce a new approach in the methodology development for interactive multiobjective optimization. The presentation is given in the context of the interactive NIMBUS method, where the solution process is based on the classification of objective functions. The idea is to formulate several scalarizing functions, all using the same preference information of the decision maker. Thus, opposed to fixing one scalarizing function (as is done in most methods), we utilize several scalarizing functions in a synchronous way. This means that we as method developers do not make the choice between different scalarizing functions but calculate the results of different scalarizing functions and leave the final decision to the expert, the decision maker. Simultaneously, (s)he obtains a better view of the solutions corresponding to her/his preferences expressed once during each iteration.In this paper, we describe a synchronous variant of the NIMBUS method. In addition, we introduce a new version of its implementation WWW-NIMBUS operating on the Internet. WWW-NIMBUS is a software system capable of solving even computationally demanding nonlinear problems. The new version of WWW-NIMBUS can handle versatile types of multiobjective optimization problems and includes new desirable features increasing its user-friendliness.     

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

Lagrange multipliers for set-valued optimization problems associated with coderivatives

In this paper we investigate a vector optimization problem (P) where objective and constraints are given by set-valued maps. We show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of (P) as solutions of a scalar optimization problem (SP) with single-valued objective and constraint functions. Then applying some classical or recent results in optimization theory to (SP) and using estimates of subdifferentials of marginal functions, we obtain optimality conditions for (P) expressed in terms of Lagrange or sequential Lagrange multipliers associated with various coderivatives of the set-valued data.     

A conic scalarization method in multi-objective optimization

This paper presents the conic scalarization method for scalarization of nonlinear multi-objective optimization problems. We introduce a special class of monotonically increasing sublinear scalarizing functions and show that the zero sublevel set of every function from this class is a convex closed and pointed cone which contains the negative ordering cone. We introduce the notion of a separable cone and show that two closed cones (one of them is separable) having only the vertex in common can be separated by a zero sublevel set of some function from this class. It is shown that the scalar optimization problem constructed by using these functions, enables to characterize the complete set of efficient and properly efficient solutions of multi-objective problems without convexity and boundedness conditions. By choosing a suitable scalarizing parameter set consisting of a weighting vector, an augmentation parameter, and a reference point, decision maker may guarantee a most preferred efficient or properly efficient solution.     

Use of Substitute Scalarizing Functions to Guide a Local Search Based Heuristic: The Case of moTSP

Solving the Tchebycheff program means optimizing a particular scalarizing function. When dealing with combinatorial problems, however, it is due to computational intractability often necessary to apply heuristics and settle for approximations to the optimal solution. The experiments in this paper suggest that for the multiobjective traveling salesman problem (moTSP) instances considered, heuristic optimization of the Tchebycheff program gives better results when using a substitute scalarizing function instead of the Tchebycheff based one to guide the local search path. Two families of substitute scalarizing functions are considered.     

Nonlinear scalarizing functions in set optimization problems

In this paper, we obtain Hölder continuity of the nonlinear scalarizing function for l-type less order relation, which is introduced by Hernández and Rodríguez-Marín (J. Math. Anal. Appl. 2007;325:1–18). Moreover, we introduce the nonlinear scalarizing function for u-type less order relation and establish continuity, convexity and Hölder continuity of the nonlinear scalarizing function for u-type less order relation. As applications, we firstly obtain Lipschitz continuity of solution mapping to the parametric equilibrium problems and then establish Lipschitz continuity of strongly approximate solution mappings for l-type less order relation, u-type less order relation and set less order relation to the parametric set optimization problems by using convexity and Hölder continuity of the nonlinear scalarizing functions.     

g-dominance: Reference point based dominance for multiobjective metaheuristics

One of the main tools for including decision maker (DM) preferences in the multiobjective optimization (MO) literature is the use of reference points and achievement scalarizing functions [A.P. Wierzbicki, The use of reference objectives in multiobjective optimization, in: G. Fandel, T. Gal (Eds.), Multiple-Criteria Decision Making Theory and Application, Springer-Verlag, New York, 1980, pp. 469–486.]. The core idea in these approaches is converting the original MO problem into a single-objective optimization problem through the use of a scalarizing function based on a reference point. As a result, a single efficient point adapted to the DM’s preferences is obtained. However, a single solution can be less interesting than an approximation of the efficient set around this area, as stated for example by Deb in [K. Deb, J. Sundar, N. Udaya Bhaskara Rao, S. Chaudhuri, Reference point based multiobjective optimization using evolutionary algorithms, International Journal of Computational Intelligence Research, 2(3) (2006) 273–286]. In this paper, we propose a variation of the concept of Pareto dominance, called g-dominance, which is based on the information included in a reference point and designed to be used with any MO evolutionary method or any MO metaheuristic. This concept will let us approximate the efficient set around the area of the most preferred point without using any scalarizing function. On the other hand, we will show how it can be easily used with any MO evolutionary method or any MO metaheuristic (just changing the dominance concept) and, to exemplify its use, we will show some results with some state-of-the-art-methods and some test problems.     

一个标量泛函的研究及其在集值优化问题中的应用

本文在K条件下,研究了所给标量泛函的连续性和拟凸性,并利用该标量泛函,将集值优化问题转化为均衡问题,进而研究了含约束的集值优化问题弱充分解的存在性和拟集值优化问题强逼近解映射的上半连续性与下半连续性.与最近的文献相比,我们的方法是新的,条件和结论也更具一般性.     

Gap functions for vector variational inequalities

In this article, we intend to study several scalar-valued gap functions for Stampacchia and Minty-type vector variational inequalities. We first introduce gap functions based on a scalarization technique and then develop a gap function without any scalarizing parameter. We then develop its regularized version and under mild conditions develop an error bound for vector variational inequalities with strongly monotone data. Further, we introduce the notion of a partial gap function which satisfies all, but one of the properties of the usual gap function. However, the partial gap function is convex and we provide upper and lower estimates of its directional derivative.     

Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure

In this paper, we define two new concepts of efficiency for vector optimization with variable ordering structure, namely the sharp and robust efficiencies, and we study their connections with classical concepts of efficiency in vector optimization. Then, we get necessary optimality conditions for them using Fréchet and Mordukhovich calculus coupled with the Gerstewitz’s (Tammer’s) scalarizing functional and openness results for set-valued maps.     

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.     

Partial order relations on family of sets and scalarizations for set optimization

In this study, some new order relations on family of sets are introduced by using Minkowski difference. The relations between these orders and the ordering cone of the vector space are obtained. It is shown that depending on the corresponding cone, these order relations are partial orders on the family of nonempty bounded sets. Some relationships between these order relations and upper and lower set less order relations are investigated. Also, two scalarizing functions are introduced in order to replace set optimization problems with respect to these partial order relations with scalar optimization problems. Moreover, necessary and sufficient optimality conditions are presented.     

A unified vector optimization problem: complete scalarizations and applications

The aim of this article is to introduce and analyse a general vector optimization problem in a unified framework. Using a well-known nonlinear scalarizing function defined by a solid set, we present complete scalarizations of the solution set to the vector problem without any convexity assumptions. As applications of our results we obtain new optimality conditions for several classical optimization problems by characterizing their solution set.     

Robustness to uncertain optimization using scalarization techniques and relations to multiobjective optimization

In this paper, we propose two kinds of robustness concepts by virtue of the scalarization techniques (Benson’s method and elastic constraint method) in multiobjective optimization, which can be characterized as special cases of a general non-linear scalarizing approach. Moreover, we introduce both constrained and unconstrained multiobjective optimization problems and discuss their relations to scalar robust optimization problems. Particularly, optimal solutions of scalar robust optimization problems are weakly efficient solutions for the unconstrained multiobjective optimization problem, and these solutions are efficient under uniqueness assumptions. Two examples are employed to illustrate those results. Finally, the connections between robustness concepts and risk measures in investment decision problems are also revealed.     

Saddle points and scalarizing sets in multiple objective linear programming

The main purpose of this paper is to study saddle points of the vector Lagrangian function associated with a multiple objective linear programming problem. We introduce three concepts of saddle points and establish their characterizations by solving suitable systems of equalities and inequalities. We deduce dual programs and prove a relationship between saddle points and dual solutions, which enables us to obtain an explicit expression of the scalarizing set of a given saddle point in terms of normal vectors to the value set of the problem. Finally, we present an algorithm to compute saddle points associated with non-degenerate vertices and the corresponding scalarizing sets.     

Properly optimal elements in vector optimization with variable ordering structures

In this paper, proper optimality concepts in vector optimization with variable ordering structures are introduced for the first time and characterization results via scalarizations are given. New type of scalarizing functionals are presented and their properties are discussed. The scalarization approach suggested in the paper does not require convexity and boundedness conditions.     

Continuity and Convexity of a Nonlinear Scalarizing Function in Set Optimization Problems with Applications

Hern\(\acute{\mathrm{a}}\)ndez and Rodríguez-Marín (J Math Anal Appl 325:1–18, 2007) introduced a nonlinear scalarizing function for sets, which is a generalization of the Gerstewitz’s function. This paper aims at investigating some properties concerned with the nonlinear scalarizing function for sets. The continuity and convexity of the nonlinear scalarizing function for sets are showed under some suitable conditions. As applications, the upper semicontinuity and the lower semicontinuity of strongly approximate solution mappings to the parametric set optimization problems are also given.     

Nonconvex scalarization in set optimization with set-valued maps

The aim of this work is to obtain scalar representations of set-valued optimization problems without any convexity assumption. Using a criterion of solution introduced by Kuroiwa [D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria, in: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, World Scientific, River Edge, NJ, 1999, pp. 221-228], which is based on ordered relations between sets, we characterize this type of solutions by means of nonlinear scalarization. The scalarizing function is a generalization of the Gerstewitz's nonconvex separation function. As applications of our results we give two existence theorems for set-valued optimization problems.     

Decision support system for multicriterial R&D and information systems projects selection

Decision support system described here makes it possible to carry out multicriterial selection of hundreds of projects simultaneously, with tens of criterion functions with bivalent variables including polynomial ones and quotients of linear functions, and tens of resources limitations. Stewart's idea [J. Oper. Res. Soc. 42 (1) (1991) 17] of the special scalarizing function based on the modified reference point approach, which has been hitherto applied for linear benefits and for balance criteria functions only, and its optimization by effective gradient method is used here in the situation described by Santhanam and Kyparisis [Comput. Oper. Res. 22 (8) (1995) 807], involving synergistic effects of second- and third-orders in benefit and cost criterion function, and in resource requirements respecting resource sharing and hierarchical contingency relationships among candidate projects. In addition, the system enables us making the dialogue of the solution in a way of adaptive creation of weights of criterion functions and also flexible projects portfolio changes. A test of efficiency is presented.     

Using aspiration levels in an interactive interior multiobjective linear programming algorithm

In this paper, we propose the use of an interior-point linear programming algorithm for multiple objective linear programming (MOLP) problems. At each iteration, a Decision Maker (DM) is asked to specify aspiration levels for the various objectives, and an achievement scalarizing function is applied to project aspiration levels onto the nondominated set. The interior-point algorithm is used to find an interior solution path from a starting solution to a nondominated solution corresponding to the optimum of the achievement scalarizing function. The proposed approach allows the DM to re-specify aspiration levels during the solution process and thus steer the interior solution path toward different areas in objective space. We illustrate the use of the approach with a numerical example.