Sensitivity analysis in vector optimization

For a vector optimization problem that depends on a parameter vector, the sensitivity analysis of perturbation, proper perturbation, and weak perturbation maps is dealth with. Each of the perturbation maps is defined as a set-valued map which associates to each parameter value the set of all minimal, properly minimal, and weakly minimal points of the perturbed feasible set in the objective space with respect to a fixed ordering cone. Using contingent cones in a finite-dimensional Euclidean space, we investigate the relationship between the contingent derivatives of the three types of perturbation maps and three types of minimal point sets for the contingent derivative of the feasible-set map in the objective space. These results provide quantitative informations on the behavior of the perturbation maps.The authors would like to thank the referees for their valuable comments and suggestions.     

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.     

A gap between multiobjective optimization and scalar optimization

We give an example to illustrate a gap between multiobjective optimization and single-objective optimization, which solves a problem proposed in Ref. 1.     

A Hausdorff-type distance,a directional derivative of a set-valued map and applications in set optimization

Abstract

In this paper, we follow Kuroiwa’s set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F.     

Explicit gradient information in multiobjective optimization

This work presents an algorithm that converges to points that satisfy a first-order necessary condition of weakly Pareto solutions of multiobjective optimization problems. Hints on how to include second-order information are given. Preliminary numerical results are encouraging.     

Scalarization and lagrange duality in multiobjective optimization

In this paper, we are concerned with scalarization and the Lagrange duality in multiobjective optimization. After exposing a property of a cone-subconvexlike function, we prove two theorems on scalarization and three theorems of the Lagrange duality.     

向量优化中的二阶灵敏性分析

借助二阶相依导数的概念,研究了向量优化问题中扰动映射的二阶灵敏性.     

Decision making in multiobjective optimization aided by the multicriteria tournament decision method

This paper proposes a new method for multicriteria analysis, named Multicriteria Tournament Decision (MTD). It provides the ranking of alternatives from best to worst, according to the preferences of a human decision-maker (DM). It has some positive aspects such as: it has a simple algorithm with intuitive appeal; it involves few input parameters (just the importance weight of each criterion).The helpfulness of MTD is demonstrated by using it to select the final solution of multiobjective optimization problems in an a posteriori decision making approach. Having at hand a discrete approximation of the Pareto front (provided by a multiobjective evolutionary search algorithm), the choice of the preferred Pareto-optimal solution is performed using MTD.A simple method, named Gain Analysis method (GAM), for verifying the existence of a better solution (a solution associated to higher marginal rates of return) than the one originally chosen by the DM, is also introduced here. The usefulness of MTD and GAM methods is confirmed by the suitable results shown in this paper.     

Approximative solution methods for multiobjective combinatorial optimization

In this paper we present a review of approximative solution methods, that is, heuristics and metaheuristics designed for the solution of multiobjective combinatorial optimization problems (MOCO). First, we discuss questions related to approximation in this context, such as performance ratios, bounds, and quality measures. We give some examples of heuristics proposed for the solution of MOCO problems. The main part of the paper covers metaheuristics and more precisely non-evolutionary methods. The pioneering methods and their derivatives are described in a unified way. We provide an algorithmic presentation of each of the methods together with examples of applications, extensions, and a bibliographic note. Finally, we outline trends in this area. The research of M. Ehrgott has been partially supported by University of Auckland grant 3602178/9275 and grant Ka 477/27-1 of the Deutsche Forschungsgemeinschaft (DFG).     

A survey of recent developments in multiobjective optimization

Multiobjective Optimization (MO) has many applications in such fields as the Internet, finance, biomedicine, management science, game theory and engineering. However, solving MO problems is not an easy task. Searching for all Pareto optimal solutions is expensive and a time consuming process because there are usually exponentially large (or infinite) Pareto optimal solutions. Even for simple problems determining whether a point belongs to the Pareto set is -hard. In this paper, we discuss recent developments in MO. These include optimality conditions, applications, global optimization techniques, the new concept of epsilon Pareto optimal solution, and heuristics.     

On local optima in multiobjective combinatorial optimization problems

In this article, local optimality in multiobjective combinatorial optimization is used as a baseline for the design and analysis of two iterative improvement algorithms. Both algorithms search in a neighborhood that is defined on a collection of sets of feasible solutions and their acceptance criterion is based on outperformance relations. Proofs of the soundness and completeness of these algorithms are given.     

Interactive algorithm for multiobjective optimization

A man-machine interactive algorithm is given for solving multiobjective optimization problems involving one decision maker. The algorithm, a modification of the Frank-Wolfe steepest ascent method, gives at each iteration a significant freedom and ease for the decision-maker's self-expression, and requires a minimal information on his local estimate of the steepest-ascent direction. The convergence of the iterative algorithm is proved under natural assumptions on the convergence and stability of the basic Frank-Wolfe algorithm.     

Interactive algorithm for multiobjective optimization

The definition of a class of matrices in Ref. 1 is modified.     

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.     

Hard uncertainties in multiobjective layout optimization of photovoltaic power plants

This paper describes how to treat hard uncertainties defined by so-called uncertainty maps in multiobjective optimization problems. For the uncertainty map being set-valued, a Taylor formula is shown under appropriate assumptions. The hard uncertainties are modeled using parametric set optimization problems for which a scalarization result is given. The presented new approach for the solution of multiobjective optimization problems with hard uncertainties is then applied to the layout optimization of photovoltaic power plants. Since good weather forecasts are difficult to obtain for future years, weather data are really hard uncertainties arising in the planning process. Numerical results are presented for a real-world problem on the Galapagos island Isabela.     

A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization

We consider a multiobjective program with inequality and equality constraints and a set constraint. The equality constraints are Fréchet differentiable and the objective function and the inequality constraints are locally Lipschitz. Within this context, a Lyusternik type theorem is extended, establishing afterwards both Fritz–John and Kuhn–Tucker necessary conditions for Pareto optimality.     

Multiple subgradient descent bundle method for convex nonsmooth multiobjective optimization

The aim of this paper is to propose a new multiple subgradient descent bundle method for solving unconstrained convex nonsmooth multiobjective optimization problems. Contrary to many existing multiobjective optimization methods, our method treats the objective functions as they are without employing a scalarization in a classical sense. The main idea of this method is to find descent directions for every objective function separately by utilizing the proximal bundle approach, and then trying to form a common descent direction for every objective function. In addition, we prove that the method is convergent and it finds weakly Pareto optimal solutions. Finally, some numerical experiments are considered.     

Nonsmooth multiobjective optimization using limiting subdifferentials

In this study, using the properties of limiting subdifferentials in nonsmooth analysis and regarding a separation theorem, some weak Pareto-optimality (necessary and sufficient) conditions for nonsmooth multiobjective optimization problems are proved.     

Quasi-Newton methods for solving multiobjective optimization

This paper presents a quasi-Newton-type algorithm for nonconvex multiobjective optimization. In this algorithm, the iterations are repeated until termination conditions are met, which is when a suitable descent direction cannot be found anymore. Under suitable assumptions, global convergence is established.     

A novel evolution strategy for multiobjective optimization problem

Recent literatures have suggested that multiobjective evolutionary algorithms (MOEAs) can serve as a more exploratory and effective tool in solving multiobjective optimization problems (MOPs) than traditional optimizers. In order to contain a good approximation of Pareto optimal set with wide diversity associated with the inherent characters and variability of MOPs, this paper proposes a new evolutionary approach—(μ, λ) multiobjective evolution strategy ((μ, λ)-MOES). Following the highlight of how to balance proximity and diversity of individuals in exploration and exploitation stages respectively, some cooperative techniques are devised. Firstly, a novel combinatorial exploration operator that develops strong points from Gaussian mutation of proximity exploration and from Cauchy mutation of diversity preservation is elaborately designed. Additionally, we employ a complete nondominance selection so as to ensure maximal pressure for proximity exploitation while a fitness assignment determined by dominance and population diversity information is simultaneous used to ensure maximal diversity preservation. Moreover, a dynamic external archive is introduced to store elitist individuals as well as relatively better individuals and exchange information with the current population when performing archive increase scheme and archive decrease scheme. By graphical presentation and examination of selected performance metrics on three prominent benchmark test functions, (μ, λ)-MOES is found to outperform SPEA-II to some extent in terms of finding a near-optimal, well-extended and uniformly diversified Pareto optimal front.