Scalarization Technique Research Articles

Using the technique of scalarization to solve the multiobjective programming problems with fuzzy coefficients

Scalarization of the multiobjective programming problems with fuzzy coefficients using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Since the set of all fuzzy numbers can be embedded into a normed space, this motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to evaluate the multiobjective programming problems with fuzzy coefficients. Two solution concepts are proposed in this paper by considering different convex cones.

Scalarization for pointwise well-posed vectorial problems

The aim of this paper is to develop a method of study of Tykhonov well-posedness notions for vector valued problems using a class of scalar problems. Having a vectorial problem, the scalarization technique we use allows us to construct a class of scalar problems whose well-posedness properties are equivalent with the most known well-posedness properties of the original problem. Then a well-posedness property of a quasiconvex level-closed problem is derived.

Scalarization of the fuzzy optimization problems

Scalarization of the fuzzy optimization problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Two solution concepts are proposed by considering two convex cones. The set of all fuzzy numbers can be embedded into a normed space. This motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to solve the fuzzy optimization problems. By applying scalarization to the optimization problem with fuzzy coefficients, we obtain its corresponding scalar optimization problem. Finally, we show that the optimal solution of its corresponding scalar optimization problem is the optimal solution of the original fuzzy optimization problem.

A discussion of scalarization techniques for multiple objective integer programming

In this paper we consider solution methods for multiobjective integer programming (MOIP) problems based on scalarization. We define the MOIP, discuss some common scalarizations, and provide a general formulation that encompasses most scalarizations that have been applied in the MOIP context as special cases. We show that these methods suffer some drawbacks by either only being able to find supported efficient solutions or introducing constraints that can make the computational effort to solve the scalarization prohibitive. We show that Lagrangian duality applied to the general scalarization does not remedy the situation. We also introduce a new scalarization technique, the method of elastic constraints, which is shown to be able to find all efficient solutions and overcome the computational burden of the scalarizations that use constraints on objective values. Finally, we present some results from an application in airline crew scheduling as evidence.

A scalarization technique for computing the power and exponential moments of Gaussian random matrices

We consider the problems of computing the power and exponential moments EXs and EetX of square Gaussian random matrices X=A+BWC for positive integer s and real t, where W is a standard normal random vector and A, B, C are appropriately dimensioned constant matrices. We solve the problems by a matrix product scalarization technique and interpret the solutions in system-theoretic terms. The results of the paper are applicable to Bayesian prediction in multivariate autoregressive time series and mean-reverting diffusion processes.

Scalarization techniques or relationship between a social welfare function and a Pareto optimality concept

This paper discusses some scalarization techniques and one application of the multi-objective optimization problem into a mathematical economics. A function of social welfare and a concept of Pareto optimality in a finite pure exchange economy are considered. Here it is proved that a set of Pareto optimality allocations is path-connected and uncountable, if the utility functions of economical agents are monotone, concave and strictly quasi-concave. In the end, it is proved that there exist non-negative weights and Pareto optimality allocations such that the social welfare function has maximum not proving continuity of this function.

Necessary Optimality Conditions for Lipschitz Multiobjective Optimization Problems

Abstract Optimality conditions are established in terms of Lagrange–Kuhn–Tucker multipliers for multiobjective optimization problems by a scalarization technique. Throughout this note, the data are assumed to be locally Lipschitz.

On generalized vector pre-variational and pre-quasivariational inequalities

In this paper, we consider generalized vector pre-variational inequality problems as well as vector pre-quasivariational inequality problems in abstract space setting. We establish several existence results for these two classes of problems by employing the Ky Fan lemma and a scalarization technique, respectively.

Electromagnetic vector potentials and the scalarization of sources in a nonhomogeneous medium.

Electromagnetic source equivalence is considered for the case of an isotropic nonhomogeneous medium. Equivalent transformations of the transversally oriented (with respect to a chosen axis) current sources into longitudinally oriented sources are derived. They allow the reduction of any given distribution of arbitrarily oriented sources to an equivalent distribution of single-component parallel electric and magnetic sources. The technique is referred to as source scalarization; and, together with a recently developed vector potential field representation in an isotropic, nonhomogeneous, lossy medium, which may contain sources of arbitrary orientation, it is applied to produce a complete description of the field in terms of two scalar wave potentials. The proposed source scalarization technique is illustrated by a simple numerical example: the radiation of an electromagnetic pulse by an asymmetrical loop of magnetic currents.

Controlled Markov Set‐chains with Set‐valued Rewards—The Average Case

In this paper, the average case of finite state controlled Markov set‐chains with Rp‐set‐valued rewards are considered. The optimization is done by a pseudo‐order relation on the set of all convex and compact subsets of Rp induced by a closed convex cone. We introduce a v‐step contractive property (minorization condition) for the average case and by use of this method the average expected reward set from a periodic policy is characterized. And, applying the scalarization technique, a Pareto optimal policy is obtained. Also, a numerical example is given to comprehend our results.

Stopped Markov decision processes with multiple constraints

In this paper, a optimization problem for stopped Markov decision processes with vector-valued terminal reward and multiple running cost constraints is considered. Applying the idea of occupation measures and using the scalarization technique for vector maximization problems we obtain the equivalent Mathematical Programming problem and show the existence of a Pareto optimal pair of stationary policy and stopping time requiring randomization in at most k states, where k is the number of constraints. Moreover Lagrange multiplier approaches are considered. The saddle-point statements are given, whose results are applied to obtain a related parametric Mathematical Programming, by which the problem is solved. Numerical examples are given.

Vector equilibrium problem and vector optimization

This paper examines the vector equilibrium model based on a vector cost consideration. This is a generalization of the well-known Wardrop traffic equilibrium principle where road users choose paths based on just a single cost. The concept of parametric equilibria is introduced and used to establish relations with parametric complementarity and variational inequality problems. Relations with some vector optimization problems via scalarization techniques are given under appropriate conditions. Some solution methods for solving vector equilibrium problems are also discussed.

The generalized vector quasi-variational-like inequalities

In this paper, we introduce and study a class of Generalized Vector Quasi-Variational-Like Inequality Problem (GVQVLIP) involving set-valued mappings with certain monotonicity. By employing the scalarization technique, several existence results for solutions of the (GVQVLIP) are established under noncompact setting in topological vector spaces. These new existence results improve, unify, and generalize many known results for scalar and vector variational inequalities in recent literature.

On Lagrance-Kuhn-Tucker Mulitipliers for Muliobjective Optimization Problems

Optimallty conditions are established in terms of Lagrange-Fritz-John and Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems by scalarization technique

Upper robust mappings and vector minimization: an integral approach

A study of upper robust mapping from a topological space to R n and development of optimality conditions for vector minimization of upper robust mappings are presented in the framework of integral based optimization theory. Under some general assumptions, optimality conditions are established for several well developed scalarization techniques such as weighting, ϵ-constraint and reference point. These optimality conditions are applied to design integral algorithms for finding the set of efficient solutions of a vector optimization problem. A numerical example is presented to illustrate the effectiveness of the algorithm.

Vector optimization and generalized Lagrangian duality

In this paper, foundations of a new approach for solving vector optimization problems are introduced. Generalized Lagrangian duality, related for the first time with vector optimization, provides new scalarization techniques and allows for the generation of efficient solutions for problems which are not required to satisfy any convexity assumptions.

Optimal stationary policies in the vector-valued Markov decision process

In this paper we are concerned with the vector-valued Markov decision process and consider the characterization of optimal stationary policies among the set of all (randomized, history-dependent) policies. Using the scalarization technique developed for the vector maximizing problem in the nonlinear programming, we present a necessary condition and a (different) sufficient condition for a stationary policy to be optimal among the set of all policies.

A Note on the Inversion of the Dyadic Operator

Two mathematical techniques - scalarization and symbolic technique - are promoted as useful techniques for the solution of dyadic differential equations. Both are employed to find a complete, analytic inversion of the dyadic differential operator × x I - a2 I - b × I. An operator of this structure arises in the electromagnetic theory of isotropic chiral media. While describing the identical physical process, different mathematical formulations of chiral media have been put forward in the literature. The operator (and therefore the dyadic Green's function) analyzed here is the most general one that contains the other descriptions as special cases of the parameters.

Multiple objective optimal control of linear systems: the quadratic norm case

Control of a linear system to meet requirements embodied in multiple performance criteria is recognized as the typical design problem. The authors study the design of control when every criterion of interest takes the form of an H/sub 2/ norm defined over certain exogenous inputs and controlled outputs of the system. The objective is to find all stabilizing controllers that are noninferior with respect to the defined objectives, that is, to find the set of Pareto-optimal controllers. The novel feature is the use of the Youla parametrization to represent the admissible controllers. This representation enables the authors to show that each criterion, and hence their weighted sum, is convex over the class of admissible controllers. Since the set of admissible controllers is itself convex, the Pareto-optimal controllers can be determined by the standard scalarization technique. It is concluded that, in general, the controller order is higher than the order of the plant.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Generalized hyperplane methods for generating Λ‐extreme points and trade‐off rates for multiobjective optimization problem

AbstractThe hyperplane method is a generalized scalarization technique for the traditional typical scalarization problem to determine the Pareto‐optimal solution for the multi‐objective optimization problem. This paper presents a generalized hyperplane method to determine the A‐extreme point, which is a generalization of the Pareto‐optimal solution. A situation is considered in which the decisionmaker specifies the A‐extreme points, and his or her preference is to be estimated.‐ It is shown that the trade‐off rate, which is considered as useful information, can be represented by a Lagrange multiplier obtained by solving the generalized hyperplane problem.