Duality of Nonscalarized Multiobjective Linear Programs: Dual Balance,Level Sets,and Dual Clusters of Optimal Vectors

A new concept of duality is proposed for multiobjective linear programs. It is based on a set expansion process for the computation of optimal solutions without scalarization. The duality gap qualifications are investigated; the primal–dual balance set and level set equations are derived. It is demonstrated that the nonscalarized dual problem presents a cluster of optimal dual vectors that corresponds to a unique optimal primal vector. Comparisons are made with linear utility, minmax and minmin scalarizations. Connections to Pareto optimality are studied and relations to sensitivity and parametric programming are discussed. The ideas are illustrated by examples.     

Pareto Analysis vis-à-vis Balance Space Approach in Multiobjective Global Optimization

There is much controversy about the balance space approach, introduced first in Ref. 1, pp. 138–140, with the consideration of the balance number and balance vectors, and then further developed in Ref. 2, with the consideration of balance points and balance sets. There were attempts to identify the balance space approach with some other methods of multiobjective optimization, notably the method proposed in Ref. 3 and most recently Pareto analysis, as presented in Ref. 4. In this paper, we compare Pareto analysis with the balance space approach on several examples to demonstrate the interrelation and the differences of the two methods. As a byproduct, it is shown that, in some cases, the entire Pareto sets, proper and adjoint, can be determined very simply, without any special investigation of the (nonscalarized, nonconvex) multiobjective global optimization problem. The method of parameter introduction is presented in application to determining the Pareto sets and balance set. The use of computer graphics software complemented with the Gauss–Jordan matrix reduction algorithm is proposed for a class of otherwise intractable problems with nonconvex constraint sets.     

Radial Solutions and Orthogonal Trajectories in Multiobjective Global Optimization

This paper presents a new, ray-oriented method for the global solution of nonscalarized vector optimization problems and a framework for the application of the Karush–Kuhn–Tucker theorem to such problems. Properties of nonlinear multiobjective problems implied by the Karush–Kuhn–Tucker necessary conditions are investigated. The regular case specific to nonscalarized MOPs is singled out when a nonlinear MOP with nonlinearities only in the constraints reduces to a nondegenerate linear system. It is shown that the trajectories of the Lagrange multipliers corresponding to the components of the vector cost function are orthogonal to the corresponding trajectories of the vector deviations in the balance space (to the balance set for Pareto solutions). Illustrative examples are presented.     

Bayesian approach to global optimization and application to multiobjective and constrained problems

In this paper, the Bayesian methods of global optimization are considered. They provide the minimal expected deviation from the global minimum. It is shown that, using the Bayesian methods, the asymptotic density of calculations of the objective function is much greater around the point of global minimum. The relation of this density to the parameters of the method and to the function is defined.Algorithms are described which apply the Bayesian methods to problems with linear and nonlinear constraints. The Bayesian approach to global multiobjective optimization is defined. Interactive procedures and reduction of multidimensional data in the case of global optimization are discussed.     

Asymptotic convergence of a simulated annealing algorithm for multiobjective optimization problems

In this paper we consider a simulated annealing algorithm for multiobjective optimization problems. With a suitable choice of the acceptance probabilities, the algorithm is shown to converge asymptotically, that is, the Markov chain that describes the algorithm converges with probability one to the Pareto optimal set.     

Sensitivity analysis in multiobjective optimization

Sensitivity analysis in multiobjective optimization is dealt with in this paper. Given a family of parametrized multiobjective optimization problems, the perturbation map is defined as the set-valued map which associates to each parameter value the set of minimal points of the perturbed feasible set in the objective space with respect to a fixed ordering convex cone. The behavior of the perturbation map is analyzed quantitatively by using the concept of contingent derivatives for set-valued maps. Particularly, it is shown that the sensitivity is closely related to the Lagrange multipliers in multiobjective programming.This research was made while the author stayed at the International Institute for Applied Systems Analysis, Laxenburg, Austria.The author would like to thank an anonymous referee for his helpful suggestions; particularly, he pointed out that Proposition 2.2 and Theorem 2.1 are valid also in infinite-dimensional spaces.     

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.     

Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators

The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function.     

多目标优化正则条件的一个注记

给出带不等式约束的非光滑多目标优化问题正则条件的一个例子.通过该例,指出最近由Burachik和Rizvi利用线性化锥提出的可微多目标优化问题的正则条件不能利用Clarke导数推广到非光滑情形.     

Portfolio construction on the Athens Stock Exchange: a multiobjective optimization approach

In this research article, our purpose is to propose a single-period multiobjective mixed-integer programming model for equity portfolio construction, in order to generate the Pareto optimal portfolios, using a variant of the well-known ε-constraint method. The decision maker's investment policy, i.e. constraints regarding the portfolio structure, is strongly taken into account. An illustrative application in the Athens Stock Exchange market is also presented.     

On duality in multiobjective semi-infinite optimization

Abstract

In this paper, we consider multiobjective semi-infinite optimization problems which are defined in a finite-dimensional space by finitely many objective functions and infinitely many inequality constraints. We present duality results both for the convex and nonconvex case. In particular, we show weak, strong and converse duality with respect to both efficiency and weak efficiency. Moreover, the property of being a locally properly efficient point plays a crucial role in the nonconvex case.     

Scalarizing vector optimization problems

A scalarization of vector optimization problems is proposed, where optimality is defined through convex cones. By varying the parameters of the scalar problem, it is possible to find all vector optima from the scalar ones. Moreover, it is shown that, under mild assumptions, the dependence is differentiable for smooth objective maps defined over reflexive Banach spaces. A sufficiency condition of optimality for a general mathematical programming problem is also given in the Appendix.     

Bucket elimination for multiobjective optimization problems

Multiobjective optimization deals with problems involving multiple measures of performance that should be optimized simultaneously. In this paper we extend bucket elimination (BE), a well known dynamic programming generic algorithm, from mono-objective to multiobjective optimization. We show that the resulting algorithm, MO-BE, can be applied to true multi-objective problems as well as mono-objective problems with knapsack (or related) global constraints. We also extend mini-bucket elimination (MBE), the approximation form of BE, to multiobjective optimization. The new algorithm MO-MBE can be used to obtain good quality multi-objective lower bounds or it can be integrated into multi-objective branch and bound in order to increase its pruning efficiency. Its accuracy is empirically evaluated in real scheduling problems, as well as in Max-SAT-ONE and biobjective weighted minimum vertex cover problems.     

Characterizations of solution sets for parametric multiobjective optimization problems

In this article, we investigate the nonemptiness and compactness of the weak Pareto optimal solution set of a multiobjective optimization problem with functional constraints via asymptotic analysis. We then employ the obtained results to derive the necessary and sufficient conditions of the weak Pareto optimal solution set of a parametric multiobjective optimization problem. Our results improve and generalize some known results.     

多面体集下多目标优化问题近似解的若干性质

研究了多目标优化问题的近似解. 首先证明了多面体集是 co-radiant集,并证明了一些性质. 随后研究了多面体集下多目标优化问题近似解的特殊性质.     

Vector-valued lagrangian and multiobjective fractional programming duality

A class of multiobjective fractional programming problems is considered and duality results are established in terms of properly efficient solutions of the primal and dual programs. Further a vector-valued ratio type Lagrangian is introduced and certain vector saddlepoint results are presented.     

A novel elitist multiobjective optimization algorithm: Multiobjective extremal optimization

Recently, a general-purpose local-search heuristic method called extremal optimization (EO) has been successfully applied to some NP-hard combinatorial optimization problems. This paper presents an investigation on EO with its application in numerical multiobjective optimization and proposes a new novel elitist (1 + λ) multiobjective algorithm, called multiobjective extremal optimization (MOEO). In order to extend EO to solve the multiobjective optimization problems, the Pareto dominance strategy is introduced to the fitness assignment of the proposed approach. We also present a new hybrid mutation operator that enhances the exploratory capabilities of our algorithm. The proposed approach is validated using five popular benchmark functions. The simulation results indicate that the proposed approach is highly competitive with the state-of-the-art multiobjective evolutionary algorithms. Thus MOEO can be considered a good alternative to solve numerical multiobjective optimization problems.     

Sensitivity of Pareto Solutions in Multiobjective Optimization

The paper presents a sensitivity analysis of Pareto solutions on the basis of the Karush-Kuhn-Tucker (KKT) necessary conditions applied to nonlinear multiobjective programs (MOP) continuously depending on a parameter. Since the KKT conditions are of the first order, the sensitivity properties are considered in the first approximation. An analogue of the shadow prices, well known for scalar linear programs, is obtained for nonlinear MOPs. Two types of sensitivity are investigated: sensitivity in the state space (on the Pareto set) and sensitivity in the cost function space (on the balance set) for a vector cost function. The results obtained can be used in applications for sensitivity computation under small variations of parameters. Illustrative examples are presented.Research of this author was partially supported by Grant BEC2003-09067-C04-03.Research of this author was partially supported by NSERC Grant RGPIN-3492-00.Research of this author was partially supported by Grant BEC2003-09067-C04-02.     

Weak sharp minima for piecewise linear multiobjective optimization in normed spaces

In a general normed space, we consider a piecewise linear multiobjective optimization problem. We prove that a cone-convex piecewise linear multiobjective optimization problem always has a global weak sharp minimum property. By a counter example, we show that the weak sharp minimum property does not necessarily hold if the cone-convexity assumption is dropped. Moreover, under the assumption that the ordering cone is polyhedral, we prove that a (not necessarily cone-convex) piecewise linear multiobjective optimization problem always has a bounded weak sharp minimum property.     

Methods of variational analysis in multiobjective optimization

This article studies new applications of advanced methods of variational analysis and generalized differentiation to constrained problems of multiobjective/vector optimization. We pay most attention to general notions of optimal solutions for multiobjective problems that are induced by geometric concepts of extremality in variational analysis, while covering various notions of Pareto and other types of optimality/efficiency conventional in multiobjective optimization. Based on the extremal principles in variational analysis and on appropriate tools of generalized differentiation with well-developed calculus rules, we derive necessary optimality conditions for broad classes of constrained multiobjective problems in the framework of infinite-dimensional spaces. Applications of variational techniques in infinite dimensions require certain ‘normal compactness’ properties of sets and set-valued mappings, which play a crucial role in deriving the main results of this article.