Scalarization Technique Research Articles

Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

This paper has two objectives. The first one is to propose a new vector quasiequilibrium problem where the ordering relation is defined via an improvement set $ D $, and its weak version, also their Minty-type dual problems and the corresponding set-valued cases. These models provide unified frameworks to deal with well-known exact and approximate vector quasiequilibrium problems with vector-valued or set-valued mappings. The second one is to study solution stability in the sense of Hölder continuity of the unique solution to parametric unified (resp. weak) vector quasiequilibrium problems, by employing the Gerstewitz scalarization techniques. In particular, we deduce a new stability result for the typical vector optimization problem related with (resp. weak) $ D $-optimality, by considering perturbations of both the objective function and the feasible set.

Stability and Scalarization for a Unified Vector Optimization Problem

This paper aims at investigating the Painleve–Kuratowski convergence of solution sets of a sequence of perturbed vector problems, obtained by perturbing the feasible set and the objective function of a unified vector optimization problem, in real normed linear spaces. We establish convergence results, both in the image and given spaces, under the assumptions of domination and strict domination properties. Moreover, scalarization techniques are employed to establish the Painleve–Kuratowski convergence in terms of the solution sets of a sequence of scalarized problems.

Finding efficient solutions in robust multiple objective optimization with SOS-convex polynomial data

In this article, a mathematical programming problem under affinely parameterized uncertain data with multiple objective functions given by SOS-convex polynomials, denoting by (UMP), is considered; moreover, its robust counterpart, denoting by (RMP), is proposed by following the robust optimization approach (worst-case approach). Then, by employing the well-known $$\epsilon $$ -constraint method (a scalarization technique), we substitute (RMP) by a class of scalar problems. Under some suitable conditions, a zero duality gap result, between each scalar problem and its relaxation problems, is established; moreover, the relationship of their solutions is also discussed. As a consequence, we observe that finding robust efficient solutions to (UMP) is tractable by such a scalarization method. Finally, a nontrivial numerical example is designed to show how to find robust efficient solutions to (UMP) by applying our results.

Optimality conditions via a unified direction approach for (approximate) efficiency in multiobjective optimization

The presented paper addresses scalarization techniques for multiobjective optimization problems. A new scalarization technique for solving a general multiobjective optimization problem is proposed. This scalarization method is called the unified direction method, because it generalizes the Pascoletti–Serafini approach (direction approach). Some theoretical results related to the suggested scalarization technique are established, especially the ability of the method for generating (properly, weakly) efficient solutions is shown. Moreover, the relation between approximate optimal solutions of the proposed single objective problem and approximate (weakly, properly) efficient solutions of the multiobjective optimization problem is investigated. In fact, easy-to-check, necessary and sufficient conditions for ϵ-(weak, proper) efficiency are obtained. Moreover, the effectiveness of the proposed approach is illustrated with some illustrative computational examples.

Necessary optimality conditions for a bilevel multiobjective programming problem via a Ψ-reformulation

ABSTRACTIn this paper, we are concerned with a bilevel multiobjective optimization problem . First, using Ψ, a function introduced by Gadhi and Dempe [Necessary optimality conditions and a new approach tomulti-objective bilevel optimization problems. J Optim Theory Appl. 2012;155:100–114], we transform into a one level optimization problem . Second, on terms of convexificators, using a scalarization technique, we derive a Karash-Kuhn-Tucker (KKT)-type necessary optimality conditions to the initial problem under a generalized Abadie constraint qualification without the assumption that the lower-level problem satisfies the Mangasarian Fromovitz constraint qualification. Some examples have been introduced to illustrate our results.

A Revised Pascoletti–Serafini Scalarization Method for Multiobjective Optimization Problems

The presented study deals with the scalarization techniques for solving multiobjective optimization problems. The Pascoletti–Serafini scalarization technique is considered, and it is attempted to sidestep two weaknesses of this method, namely the inflexibility of the constraints and the difficulties of checking proper efficiency. To this end, two modifications for the Pascoletti–Serafini scalarization technique are proposed. First, by including surplus variables in the constraints and penalizing the violations in the objective function, the inflexibility of the constraints is resolved. Moreover, by including slack variables in the constraints, easy-to-check statements on proper efficiency are obtained. Thereafter, the two proposed modifications are combined to obtain the revised Pascoletti–Serafini scalarization method. Theorems are provided on the relation of (weakly, properly) efficient solutions of the multiobjective optimization problem and optimal solutions of the proposed scalarized problems. All the provided results are established with no convexity assumption. Moreover, the capability of the proposed approaches is demonstrated through numerical examples.

Optimality, scalarization and duality in linear vector semi-infinite programming

In this paper, we introduce some qualification conditions for a linear vector semi-infinite programming problem. The new qualification conditions are used for development of necessary conditions for weak efficient, isolated efficient and -efficient solutions of such a problem. Sufficient conditions for the existence of such solutions are also provided. In addition, a new general scalarization technique for solving the reference problem is presented. Finally, we propose a type of Wolf dual problem and examine weakstrong duality relations.

Second-Order KKT Optimality Conditions for MultiObjective Optimal Control Problems

In this paper, we study second-order necessary and sufficient optimality conditions of Karush--Kuhn--Tucker-type for locally optimal solutions in the sense of Pareto to a class of multi-objective optimal control problems with mixed pointwise constraint. To deal with the problems, we first derive second-order optimality conditions for abstract multi-objective optimal control problems which satisfy the Robinson constraint qualification. We then apply the obtained results to our concrete problems. The proofs of obtained results are direct, self-contained without using scalarization techniques.

Robustness to uncertain optimization using scalarization techniques and relations to multiobjective optimization

ABSTRACTIn 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.

Multi‐criteria Optimization of an Industrial World‐Scale Process

AbstractCommercially available flow sheet simulators cannot perform optimization runs with more than one criterion. To overcome this problem, the flow sheet simulator CHEMCAD is combined with an external optimization solver, and the Excel VBA client is used to arrange inter‐process communications. The resulting tool for multi‐criteria optimization in chemical process engineering is applied to two separation problems. First, a dividing‐wall column is used as a theoretical example to test different starting points and different scalarization techniques. Second, a significant decrease in energy demand is achieved for a real continuous world‐scale facility.

Some new solution concepts in generalized fuzzy multiobjective optimization problems

Some new solution concepts to a general fuzzy multiobjective nonlinear programming problem are introduced in this research, and four scalarization techniques are proposed to obtain them. Then, the relation between the set of defined optimal solutions and the set of optimal solutions of the scalarized problems are studied. Moreover, a general scalarized problem is given and shown that these four techniques can be drawn from this problem. Adequate number of numerical examples have been solved to illustrate the techniques.

A ranking algorithm for bi-objective quadratic fractional integer programming problems

An algorithm to solve bi-objective quadratic fractional integer programming problems is presented in this paper. The algorithm uses -scalarization technique and a ranking approach of the integer feasible solution to find all nondominated points. In order to avoid solving non-linear integer programming problems during this ranking scheme, the existence of a linear or a linear fractional function is established, which acts as a lower bound on the values of first objective function of the bi-objective problem over the entire feasible set. Numerical examples are also presented in support of the theory.

Solving Fully Fuzzy Multi-objective Linear Programming Problem Using Nearest Interval Approximation of Fuzzy Number and Interval Programming

This paper focuses on Fully Fuzzy Multi-Objective Linear Programming (FFMOLP) problem in which all the coefficients and decision variables are LR flat fuzzy numbers, the more generalized version of fuzzy numbers and all the constraints are fuzzy inequalities. A new algorithm is proposed for solving FFMOLP problem which first converts it into the Multi-Objective Interval Linear Programming (MOILP) problem. Further, taking the help of fuzzy slack variable, fuzzy surplus variables, nearest interval approximation of fuzzy numbers and scalarization technique, MOILP is then converted into the Crisp Linear Programming (CLP) problem. It is shown that the optimal solution of CLP problem is the fuzzy Pareto optimal solution of FFMOLP problem. The main advantage of the proposed algorithm is that it transforms FFMOLP problem into Crisp Linear Programming problem. Moreover, to apply algorithm, only the knowledge of arithmetic operations of LR flat fuzzy numbers, centre and width of the closed intervals are required. At the end, to illustrate the proposed method and its effectiveness over the existing method, numerical examples are solved and compared.

Error Bounds for Generalized Mixed Vector Equilibrium Problems via a Minimax Strategy

In this paper, by using scalarization techniques and a minimax strategy, error bound results in terms of gap functions for a generalized mixed vector equilibrium problem are established, where the solutions for vector problems may be general sets under natural assumptions, but are not limited to singletons. The other essentially equivalent approach via a separation principle is analyzed. Special cases to the classical vector equilibrium problem and vector variational inequality are also discussed.

The well-posedness for generalized fuzzy games

This paper establishes the stable results for generalized fuzzy games by using a nonlinear scalarization technique. The authors introduce some concepts of well-posedness for generalized fuzzy games. Moreover, the authors identify a class of generalized fuzzy games such that every element of the collection is generalized well-posed, and there exists a dense residual subset of the collection, where every generalized fuzzy game is robust well-posed.

A unified approach to uncertain optimization

In this paper we consider uncertain scalar optimization problems with infinite scenario sets. We apply methods from vector optimization in general spaces, set-valued optimization and scalarization techniques to develop a unified characterization of different concepts of robust optimization and stochastic programming. These methods provide new insights on the interrelation between different concepts for handling uncertainties and naturally lead to new concepts of robustness.

A New Scalarization Technique and New Algorithms to Generate Pareto Fronts

We propose a new scalarization technique for nonconvex multiobjective optimization problems and establish its theoretical properties. By combining our new scalarization approach with existing grid ...

Optimality conditions via scalarization for approximate quasi efficiency in multiobjective optimization

Approximate problems that scalarize and approximate a given multiobjective optimization problem (MOP) became an important and interesting area of research, given that, in general, are simpler and have weaker existence requirements than the original problem. Recently, necessary conditions for approximation of several types of efficiency for MOPs have been obtained through the use of an alternative theorem. In this paper, we use these results in order to extend them to sufficient conditions for approximate quasi (weak, proper) efficiency. For this, we use two scalarization techniques of Tchebycheff type. All the provided results are established without convexity assumptions.

Continuity of approximate solution maps to vector equilibrium problems

This paper considers the parametric primal and dual vector equilibrium problems in locally convex Hausdorff topological vector spaces. Based on linear scalarization technique, we establish sufficient conditions for the continuity of approximate solution maps to these problems. As applications, some new results for vector optimization problem and vector variational inequality are derived. Our results are new and improve the existing ones in the literature.

Economic growth and abatement activities in a stochastic environment: a multi-objective approach

We analyze the effect of uncertainty on economic growth and environmental quality in a concave stochastic bicriteria dynamic problem, solved by means of scalarization techniques and the Hamilton–Jacobi–Bellman equation. Specifically, we focus on an endogenous growth model with purposive abatement activities, where environmental quality, subject to random shocks, is merely a source of utility and does not play any productive role. We show that even in the absence of direct (environmental) productivity effects, along the optimal path production results enhanced by an indirect productivity effect, which transforms environmental quality into the engine of growth. However, this does not prevent uncertainty to dampen economic and environmental performance, leading to a lower social welfare than in a deterministic framework.