Nonessential objectives within network approaches for MCDM

In Gal and Hanne [Eur. J. Oper. Res. 119 (1999) 373] the problem of using several methods to solve a multiple criteria decision making (MCDM) problem with linear objective functions after dropping nonessential objectives is analyzed. It turned out that the solution does not need be the same when using various methods for solving the system containing the nonessential objectives or not. In this paper we consider the application of network approaches for multicriteria decision making such as neural networks and an approach for combining MCDM methods (called MCDM networks). We discuss questions of comparing the results obtained with several methods as applied to the problem with or without nonessential objectives. Especially, we argue for considering redundancies such as nonessential objectives as a native feature in complex information processing. In contrast to previous results on nonessential objectives, the current paper focuses on discrete MCDM problems which are also denoted as multiple attribute decision making (MADM).     

Multiobjective optimization problems with modified objective functions and cone constraints and applications

In this paper, we consider a differentiable multiobjective optimization problem with generalized cone constraints (for short, MOP). We investigate the relationship between weakly efficient solutions for (MOP) and for the multiobjective optimization problem with the modified objective function and cone constraints [for short, (MOP) _η (x)] and saddle points for the Lagrange function of (MOP) _η (x) involving cone invex functions under some suitable assumptions. We also prove the existence of weakly efficient solutions for (MOP) and saddle points for Lagrange function of (MOP) _η (x) by using the Karush-Kuhn-Tucker type optimality conditions under generalized convexity functions. As an application, we investigate a multiobjective fractional programming problem by using the modified objective function method.     

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.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

Risk-control approach for bottleneck transportation problem with randomness and fuzziness

Solving transportation problems is essential in engineering and supply chain management, where profitability depends on optimal traffic flow. This study proposes risk-control approaches for two bottleneck transportation problems with random variables and preference levels to objective functions with risk parameters. Each proposed model is formulated as a multiobjective programming problem using robust-based optimization derived from stochastic chance constraints. Since it is impossible to obtain a transportation pattern that optimizes all objective functions, our proposed models are numerically solved by introducing an aggregation function for the multiobjective problem. An exact algorithm that performs deterministic equivalent transformations and introduces auxiliary problems is also developed.     

Quasi-Newton methods for multiobjective optimization problems

This work is an attempt to develop multiobjective versions of some well-known single objective quasi-Newton methods, including BFGS, self-scaling BFGS (SS-BFGS), and the Huang BFGS (H-BFGS). A comprehensive and comparative study of these methods is presented in this paper. The Armijo line search is used for the implementation of these methods. The numerical results show that the Armijo rule does not work the same way for the multiobjective case as for the single objective case, because, in this case, it imposes a large computational effort and significantly decreases the speed of convergence in contrast to the single objective case. Hence, we consider two cases of all multi-objective versions of quasi-Newton methods: in the presence of the Armijo line search and in the absence of any line search. Moreover, the convergence of these methods without using any line search under some mild conditions is shown. Also, by introducing a multiobjective subproblem for finding the quasi-Newton multiobjective search direction, a simple representation of the Karush–Kuhn–Tucker conditions is derived. The H-BFGS quasi-Newton multiobjective optimization method provides a higher-order accuracy in approximating the second order curvature of the problem functions than the BFGS and SS-BFGS methods. Thus, this method has some benefits compared to the other methods as shown in the numerical results. All mentioned methods proposed in this paper are evaluated and compared with each other in different aspects. To do so, some well-known test problems and performance assessment criteria are employed. Moreover, these methods are compared with each other with regard to the expended CPU time, the number of iterations, and the number of function evaluations.     

Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints

Multiobjective DC optimization problems arise naturally, for example, in data classification and cluster analysis playing a crucial role in data mining. In this paper, we propose a new multiobjective double bundle method designed for nonsmooth multiobjective optimization problems having objective and constraint functions which can be presented as a difference of two convex (DC) functions. The method is of the descent type and it generalizes the ideas of the double bundle method for multiobjective and constrained problems. We utilize the special cutting plane model angled for the DC improvement function such that the convex and the concave behaviour of the function is captured. The method is proved to be finitely convergent to a weakly Pareto stationary point under mild assumptions. Finally, we consider some numerical experiments and compare the solutions produced by our method with the method designed for general nonconvex multiobjective problems. This is done in order to validate the usage of the method aimed specially for DC objectives instead of a general nonconvex method.     

Multiobjective optimal control of a four-body kinematic chain

Optimal control of mechanical systems is an active area of research. However, so far, most contributions are taking only one single objective into account, whereas for many practical problems, one is interested in optimizing several conflicting objectives at the same time. Applying singleobjective optimization to each of them leads to several trajectories each being optimal for one objective, but ignoring all others. In contrast to that, combining all objectives and using multiobjective optimization leads to a variety of trade off solutions taking all objectives into account simultaneously. We use the direct discretization method DMOCC (Discrete Mechanics and Optimal Control for Constrained systems) to approximate trajectories of the underlying optimal control problems, resulting in restricted optimization problems of high dimension. For the multiobjective part, we apply a reference point technique which successively utilizes an auxiliary distance function to gain the trade off solutions. The presented approach is illustrated by the multiobjective optimal control of a constrained multibody system. A four-body kinematic chain is controlled in a rest to rest maneuver, for which minimal control effort and minimal required maneuver time are the conflicting objectives. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tradeoff directions in multiobjective optimization problems

Multiobjective optimization problems (MOP) typically have conflicting objectives wherein a gain in one objective is at the expense of another. Tradeoff directions, which measure the change in some objectives relative to changes in others, provide important information as to the best direction of improvement from the current solution. In this paper we present a general definition of tradeoffs as a cone of directions and provide a general method of calculating tradeoffs at every Pareto optimal point of a convex MOP. This extends current definitions of tradeoffs which assume certain conditions on the feasible set and the objective functions. Two comprehensive numerical examples are provided to illustrate the tradeoff directions and the methods used to calculate them.     

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.     

On the convergence of multiobjective evolutionary algorithms

We consider the usage of evolutionary algorithms for multiobjective programming (MOP), i.e. for decision problems with alternatives taken from a real-valued vector space and evaluated according to a vector-valued objective function. Selection mechanisms, possibilities of temporary fitness deterioration, and problems of unreachable alternatives for such multiobjective evolutionary algorithms (MOEAs) are studied. Theoretical properties of MOEAs such as stochastic convergence with probability 1 are analyzed.     

Robust multiobjective optimization with application to Internet routing

Robust optimization addressing decision making under uncertainty has been very well developed for problems with a single objective function and applied to areas of human activity such as portfolio selection, investment decisions, signal processing, and telecommunication-network planning. As these decision problems typically have several decisions or goals, we extend robust single objective optimization to the multiobjective case. The column-wise uncertainty model can be carried over to the multiobjective case without any additional assumptions. For the row-wise uncertainty model, we show under additional assumptions that robust efficient solutions are efficient to specific instance problems and can be found as the efficient solutions of another deterministic problem. Being motivated by the fact that Internet traffic must be maintained in a reliable yet affordable manner in situations of complex and dynamic usage, we apply the row-wise model to an intradomain multiobjective routing problem with polyhedral traffic uncertainty. We consider traditional objective functions corresponding to link utilizations and implement the biobjective case using the parametric simplex algorithm to compute robust efficient routings. We also present computational results for the Abilene network and analyze their meaning in the context of the application.     

Nonsmooth multiobjective programming: strong Kuhn–Tucker conditions

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond–Weir type dual problems are formulated. We also establish the weak and strong duality theorems.     

On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization

We consider a smooth multiobjective optimization problem with inequality constraints. Weak Kuhn?CTucker (WKT) optimality conditions are said to hold for such problems when not all the multipliers of the objective functions are zero, while strong Kuhn?CTucker (SKT) conditions are said to hold when all the multipliers of the objective functions are positive. We introduce a new regularity condition under which (WKT) hold. Moreover, we prove that for another new regularity condition (SKT) hold at every Geoffrion-properly efficient point. We show with an example that the assumption on proper efficiency cannot be relaxed. Finally, we prove that Geoffrion-proper efficiency is not needed when the constraint set is polyhedral and the objective functions are linear.     

Stochastic MOLP with Incomplete Information: An Interactive Approach with Recourse

Numerous multiobjective linear programming (MOLP) methods have been proposed in the last two decades, but almost all for contexts where the parameters of problems are deterministic. However, in many real situations, parameters of a stochastic nature arise. In this paper, we suppose that the decision-maker is confronted with a situation of partial uncertainty where he possesses incomplete information about the stochastic parameters of the problem, this information allowing him to specify only the limits of variation of these parameters and eventually their central values. For such situations, we propose a multiobjective stochastic linear programming methodology; it implies the transformation of the stochastic objective functions and constraints in order to obtain an equivalent deterministic MOLP problem and the solving of this last problem by an interactive approach derived from the STEM method. Our methodology is illustrated by a didactical example.     

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.     

Solving Ill-posed Bilevel Programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem and a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem.     

Dimension reduction in multiobjective optimization

In multiobjective optimization there is often the problem of the existence of a large number of objectives. For more than two objectives there is a difficulty with the representation and visualization of the solutions in the objective space. Therefore, it is not clear for the decision maker the trade-off between the different alternative solutions. Thus, this creates enormous difficulties when choosing a solution from the Pareto-optimal set and constitutes a central question in the process of decision making. Based on statistical methods as Principle Component Analysis and Cluster Analysis, the problem of reduction of the number of objectives is addressed. Several test examples with different number of objectives have been studied in order to evaluate the process of decision making through these methods. Preliminary results indicate that this statistical approach can be a valuable tool on decision making in multiobjective optimization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Generalization of Mixed Problems with an Application to Multiobjective Optimal Control

This paper generalizes to multiobjective optimization the notion of mixed problems as Philippe Michel calls it for single-objective optimization. This notion is then applied to a multiobjective control problem under constraints in the discrete time framework to obtain strong Pontryagin maximum principles in the finite-horizon case. The infinite-horizon case is also treated with conditions ensuring that the multipliers associated to the objective functions are not all zero.     

Characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method

In this paper, a new approximation method is introduced to characterize a so-called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with (F,ρ)-convex functions of order 2. In this method, an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point. In order to prove the equivalence between the original multiobjective programming problem and its associated F-approximated vector optimization problem, the suitable (F,ρ)-convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem.