Consequences of dropping nonessential objectives for the application of MCDM methods

In this article we consider the problem of nonessential objectives for multiobjective optimization problems (MOP) with linear objective functions. In 1977 an approach based on the reduction of size of the matrix of objective functions has been worked out by one of the present authors (Gal, T., Leberling, H., 1977. European Journal of Operations Research 1, 176–184). Although this method for dropping nonessential objectives leads to a mathematically equivalent MOP, problems concerning the application of MOP methods may arise. For instance, dropping some (or all) of the nonessential objectives the question is, how to ensure obtaining the same solution as with all objectives involved. We consider the problem of adapting the parameters of multiobjective optimization methods. For the case of weighting methods a simple procedure for adapting the weights is analyzed. For other methods, e.g. reference point approaches, such a simple possibility for adapting the parameters is not given.     

A path following method for box-constrained multiobjective optimization with applications to goal programming problems

We propose a path following method to find the Pareto optimal solutions of a box-constrained multiobjective optimization problem. Under the assumption that the objective functions are Lipschitz continuously differentiable we prove some necessary conditions for Pareto optimal points and we give a necessary condition for the existence of a feasible point that minimizes all given objective functions at once. We develop a method that looks for the Pareto optimal points as limit points of the trajectories solutions of suitable initial value problems for a system of ordinary differential equations. These trajectories belong to the feasible region and their computation is well suited for a parallel implementation. Moreover the method does not use any scalarization of the multiobjective optimization problem and does not require any ordering information for the components of the vector objective function. We show a numerical experience on some test problems and we apply the method to solve a goal programming problem.     

Abort landing in windshear: Optimal control problem with third-order state constraint and varied switching structure

Optimal abort landing trajectories of an aircraft under different windshear-downburst situations are computed and discussed. In order to avoid an airplane crash due to severe winds encountered by the aircraft during the landing approach, the minimum altitude obtained during the abort landing maneuver is to be maximized. This maneuver is mathematically described by a Chebyshev optimal control problem. By a transformation to an optimal control problem of Mayer type, an additional state variable inequality constraint for the altitude has to be taken into account; here, its order is three. Due to this altitude constraint, the optimal trajectories exhibit, depending on the windshear parameters, up to four touch points and also up to one boundary arc at the minimum altitude level. The control variable is the angle of attack time rate which enters the equations of motion linearly; therefore, the Hamiltonian of the problem is nonregular. The switching structures also includes up to three singular subarcs and up to two boundary subarcs of an angle of attack constraint of first order. This structure can be obtained by applying some advanced necessary conditions of optimal control theory in combination with the multiple-shooting method. The optimal solutions exhibit an oscillatory behavior, reaching the minimum altitude level several times. By the optimization, the maximum survival capability can also be determined; this is the maximum wind velocity difference for which recovery from windshear is just possible. The computed optimal trajectories may serve as benchmark trajectories, both for guidance laws that are desirable to approach in actual flight and for optimal trajectories may then serve as benchmark trajectories both for guidance schemes and also for numerical methods for problems of optimal control.This paper is dedicated to Professor George Leitmann on the occasion of his seventieth birthday.     

Necessary Optimality Conditions and a New Approach to Multiobjective Bilevel Optimization Problems

Multiobjective optimization problems typically have conflicting objectives, and a gain in one objective very often is an expense in another. Using the concept of Pareto optimality, we investigate a multiobjective bilevel optimization problem (say, P). Our approach consists of proving that P is locally equivalent to a single level optimization problem, where the nonsmooth Mangasarian–Fromovitz constraint qualification may hold at any feasible solution. With the help of a special scalarization function introduced in optimization by Hiriart–Urruty, we convert our single level optimization problem into another problem and give necessary optimality conditions for the initial multiobjective bilevel optimization problem P.     

Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization

The use of surrogate based optimization (SBO) is widely spread in engineering design to reduce the number of computational expensive simulations. However, “real-world” problems often consist of multiple, conflicting objectives leading to a set of competitive solutions (the Pareto front). The objectives are often aggregated into a single cost function to reduce the computational cost, though a better approach is to use multiobjective optimization methods to directly identify a set of Pareto-optimal solutions, which can be used by the designer to make more efficient design decisions (instead of weighting and aggregating the costs upfront). Most of the work in multiobjective optimization is focused on multiobjective evolutionary algorithms (MOEAs). While MOEAs are well-suited to handle large, intractable design spaces, they typically require thousands of expensive simulations, which is prohibitively expensive for the problems under study. Therefore, the use of surrogate models in multiobjective optimization, denoted as multiobjective surrogate-based optimization, may prove to be even more worthwhile than SBO methods to expedite the optimization of computational expensive systems. In this paper, the authors propose the efficient multiobjective optimization (EMO) algorithm which uses Kriging models and multiobjective versions of the probability of improvement and expected improvement criteria to identify the Pareto front with a minimal number of expensive simulations. The EMO algorithm is applied on multiple standard benchmark problems and compared against the well-known NSGA-II, SPEA2 and SMS-EMOA multiobjective optimization methods.     

A new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems

Most real-life decision-making activities require more than one objective to be considered. Therefore, several studies have been presented in the literature that use multiple objectives in decision models. In a mathematical programming context, the majority of these studies deal with two objective functions known as bicriteria optimization, while few of them consider more than two objective functions. In this study, a new algorithm is proposed to generate all nondominated solutions for multiobjective discrete optimization problems with any number of objective functions. In this algorithm, the search is managed over (p − 1)-dimensional rectangles where p represents the number of objectives in the problem and for each rectangle two-stage optimization problems are solved. The algorithm is motivated by the well-known ε-constraint scalarization and its contribution lies in the way rectangles are defined and tracked. The algorithm is compared with former studies on multiobjective knapsack and multiobjective assignment problem instances. The method is highly competitive in terms of solution time and the number of optimization models solved.     

A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving nonsmooth convex objective functions. Based on the Yosida regularization of the subdifferential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions, a key property for applications in cooperative games, inverse problems, and numerical multiobjective optimization.     

First and Second-Order Optimality Conditions for Convex Composite Multiobjective Optimization

Multiobjective optimization is a useful mathematical model in order to investigate real-world problems with conflicting objectives, arising from economics, engineering, and human decision making. In this paper, a convex composite multiobjective optimization problem, subject to a closed convex constraint set, is studied. New first-order optimality conditions for a weakly efficient solution of the convex composite multiobjective optimization problem are established via scalarization. These conditions are then extended to derive second-order optimality conditions.     

Optimization of dynamic regional models

In this paper we consider several problems concerning the optimization of Dynamic Regional Models. In particular we describe the optimization of the model of a region. Western Andalusia, in Southern Spain. Fuzzy set theory is applied because of the inherent imprecisions in the judgements about goals and its tradeoffs. Multicriteria optimization is used due to the fact that, in most cases, the search for the optimal policies must be performed taking into account non conmensurable and conflicting objectives. We present a program package which incorporates these methodologies together with simulation, optimization and sensitivity analysis routines.     

Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method

The convexification of a noninferior frontier can be achieved in an appropriate equivalent objective space for general nonconvex multiobjective optimization problems. Specifically, this paper proves that taking the exponentials of the objective functions can act as a convexification scheme. This convexification scheme further leads to the exponential generating method that guarantees the identification of the entire set of noninferior solutions.     

On generalized trade-off directions in nonconvex multiobjective optimization

Trade-off information related to Pareto optimal solutions is important in multiobjective optimization problems with conflicting objectives. Recently, the concept of trade-off directions has been introduced for convex problems. These trade-offs are characterized with the help of tangent cones. Generalized trade-off directions for nonconvex problems can be defined by replacing convex tangent cones with nonconvex contingent cones. Here we study how the convex concepts and results can be generalized into a nonconvex case. Giving up convexity naturally means that we need local instead of global analysis. Received: December 2000 / Accepted: October 2001?Published online February 14, 2002     

Multiobjective Stochastic Control in Fluid Dynamics via Game Theory Approach: Application to the Periodic Burgers Equation

The purpose of the present work is to implement well-known statistical decision and game theory strategies into multiobjective stochastic control problems of fluid dynamics. Such goal is first justified by the fact that deterministic (either singleobjective or multiobjective) control problems that are obtained without taking into account the uncertainty of the model are usually unreliable. Second, in most real-world problems, several goals must be satisfied simultaneously to obtain an optimal solution and, as a consequence, a multiobjective control approach is more appropriate. Therefore, we develop a multiobjective stochastic control algorithm for general fluid dynamics applications, based on the Bayes decision, adjoint formulation and the Nash equilibrium strategies. The algorithm is exemplified by the multiobjective stochastic control of a periodic Burgers equation.     

An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming

An alternative optimization technique via multiobjective programming for constrained optimization problems with interval-valued objectives has been proposed. Reduction of interval objective functions to those of noninterval (crisp) one is the main ingredient of the proposed technique. At first, the significance of interval-valued objective functions along with the meaning of interval-valued solutions of the proposed problem has been explained graphically. Generally, the proposed problems have infinitely many compromise solutions. The objective is to obtain one of such solutions with higher accuracy and lower computational effort. Adequate number of numerical examples has been solved in support of this technique.     

A multiobjective framework for heavily constrained examination timetabling problems

University examination timetabling is a challenging set partitioning problem that comes in many variations, and real world applications usually carry multiple constraints and require the simultaneous optimization of several (often conflicting) objectives. This paper presents a multiobjective framework capable of solving heavily constrained timetabling problems. In this prototype study, we focus on the two objectives: minimizing timetable length while simultaneously optimizing the spread of examinations for individual students. Candidate solutions are presented to a multiobjective memetic algorithm as orderings of examinations, and a greedy algorithm is used to construct violation free timetables from permutation sequences of exams. The role of the multiobjective algorithm is to iteratively improve a population of orderings, with respect to the given objectives, using various mutation and reordering heuristics.     

Singular Arcs in the Generalized Goddard’s Problem

We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control. Support from the French Space Agency CNES (Centre National d’Etudes Spatial) is gratefully acknowledged.     

Using box indices in supporting comparison in multiobjective optimization

Because of the conflicting nature of criteria or objectives, solving a multiobjective optimization problem typically requires interaction with a decision maker who can specify preference information related to the objectives in the problem in question. Due to the difficulties of dealing with multiple objectives, the way information is presented plays a very important role. Questions posed to the decision maker must be simple enough and information shown must be easy to understand. For this purpose, visualization and graphical representations can be useful and constitute one of the main tools used in the literature. In this paper, we propose to use box indices to represent information related to different solution alternatives of multiobjective optimization problems involving at least three objectives. Box indices are an intelligible and easy to handle way to represent data. They are based on evaluating the solutions in a natural and rough enough scale in order to let the decision maker easily recognize the main characteristics of a solution at a glance and to facilitate comparison of two or more solutions in an easily understandable way.     

Experiments with classification-based scalarizing functions in interactive multiobjective optimization

In multiobjective optimization methods, the multiple conflicting objectives are typically converted into a single objective optimization problem with the help of scalarizing functions and such functions may be constructed in many ways. We compare both theoretically and numerically the performance of three classification-based scalarizing functions and pay attention to how well they obey the classification information. In particular, we devote special interest to the differences the scalarizing functions have in the computational cost of guaranteeing Pareto optimality. It turns out that scalarizing functions with or without so-called augmentation terms have significant differences in this respect. We also collect a set of mostly nonlinear benchmark test problems that we use in the numerical comparisons.     

Portfolio selection in the Spanish stock market by interactive multiobjective programming

This study examines new versions of two interactive methods to address multiobjective problems, the aim of which is to enable the decision maker to reach a solution within the range of those considered efficient in a portfolio selection model, in which several objectives are pursued concerning risk and return and given that these are clearly conflicting objectives, the profile of the model proposed is multicriteria. Normally the range of efficient portfolios is fairly extensive thus making the selection of a single one an onerous task. In order to facilitate this process, interactive methods are used aimed at guiding the decision maker towards the optimal solution based on his preferences. Several adaptations were carried out on the original methods in order to facilitate the interactive process, improving the quality of the obtained portfolios, and these were applied to data obtained from the Madrid Stock Market, interaction taking place with two decision makers, one of whom was more aggressive than the other in their selections made.     

A subgradient method for multiobjective optimization

A method for solving quasiconvex nondifferentiable unconstrained multiobjective optimization problems is proposed in this paper. This method extends to the multiobjective case of the classical subgradient method for real-valued minimization. Assuming the basically componentwise quasiconvexity of the objective components, full convergence (to Pareto optimal points) of all the sequences produced by the method is established.     

DMulti-MADS: mesh adaptive direct multisearch for bound-constrained blackbox multiobjective optimization

Computational Optimization and Applications - The context of this research is multiobjective optimization where conflicting objectives are present. In this work, these objectives are only available...