Optimality Conditions in Uncertain Multi-Objective Optimization Problems with a Variable Domination Structure

Background: Nowadays, variable domination structure is instrumental in studying multiobjective decision making problems. We investigate multiobjective location problems with respect to variable domination structure and its applications in supply chain management. Methods: We formulate practical problems in supply chain management as an optimization problem with a variable domination structure. Moreover, we present the mathematical methods to solve such problems. We investigate two kinds of solutions derived from the concept of minimal and nondominated solutions from vector optimization problems with respect to variable domination structure. Furthermore, we explore how these solution concepts are characterized in practical problems. Results: We ex- pose how those solutions are beneficial in practical problems. However, these results hold true for multiobjective decision making problems with a continuous feasible set; we present a practical problem in the case of a finite set of feasible locations. Conclusions: In many multiobjective location problems, each location’s characterizations, preferences, and restrictions are involved in the decision making process. This study investigates the decision making problems, where different preferences of objective functions at each location are assumed. Moreover, we present a numerical experiment for selecting a new hub airport.

In this paper, an iterative algorithm for solving the strong vector equilibrium problem with variable domination structure (VSVEP) is considered. First, an auxiliary problem for the VSVEP is introduced and the relationships between the auxiliary problem and VSVEP are discussed. Then, using the auxiliary principle technique, a projection iterative algorithm to compute the approximate solutions of the VSVEP is proposed and analysed. Furthermore, convergence of the iterative sequences generated by this algorithm is investigated under suitable conditions of continuity and convexity. These results extend and improve some recent works in this field.

In this paper, we investigate an uncertain multiobjective optimization problem involving nonsmooth and nonconvex functions. The notion of a (local/global) robust weak sharp efficient solution is introduced. Then, we establish necessary and sufficient optimality conditions for local and/or the robust weak sharp efficient solutions of the considered problem. These optimality conditions are presented in terms of multipliers and Mordukhovich/limiting subdifferentials of the related functions.

This paper addresses a large class of vector optimization problems in infinite-dimensional spaces with respect to two important binary relations derived from domination structures. Motivated by theoretical challenges as well as by applications to some models in behavioral sciences, we establish new variational principles that can be viewed as far-going extensions of the Ekeland variational principle to cover domination vector settings. Our approach combines advantages of both primal and dual techniques in variational analysis with providing useful sufficient conditions for the existence of variational traps in behavioral science models with variable domination structures.

In this study, we investigate an uncertain multiobjective optimization problem through a setvalued optimization problem, and introduce a Newton method to find robust weakly efficient points of the considered uncertain optimization problem.We assume that the problem under consideration has uncertainty only in the objective function, and the involved uncertainty set is of finite cardinality.Also, for each uncertain scenario, the components of the objective function of the problem are assumed to be twice continuously differentiable and locally strong convex.Utilizing the concept of a partition set from set optimization, we formulate a class of vector optimization problems to solve the formulated set optimization problem pertaining to the considered uncertain multiobjective optimization.We derive a Newton method to solve this class of vector optimization problems that facilitates generating a sequence of points whose any limit point is a weakly robust efficient solution of the considered problem.The proposed method is found to have a local superlinear convergence rate under standard hypotheses with a regularity condition.Additionally, assuming Lipschitz continuity of the Hessian of the objective function for all scenarios, we show local quadratic convergence of the method.Finally, we provide numerical examples to discuss and illustrate the performance of the proposed method.

In this article, a modified quasi-Newton method is developed for the robust counterpart (RC) of an uncertain multiobjective optimization problem (UMOP). Specifically, the RC of a UMOP is the minimum of an objective-wise worst-case (OWWC) type RC. To determine the descent direction for the RC, a subproblem using a common Hessian approximation is constructed. An inexact line search technique based on the Armijo technique is employed to find an appropriate step length. Furthermore, a modified version of the Davidon–Fletcher–Powell (DFP) update formula is developed for the RC. A modified quasi-Newton algorithm is built using descent direction step length, and DFP Hessian update formula. Furthermore, the convergence of the modified quasi-Newton algorithm is also discussed. Finally, the modified quasi-Newton algorithm is illustrated with a set of test problems and compared with the weighted sum method. Moreover, it is demonstrated that the algorithm is effective for both convex and non-convex problems.

In this paper, by means of linear and nonlinear (regular) weak separation functions, we obtain some characterizations of robust optimality conditions for uncertain optimization problems, especially saddle point sufficient optimality conditions. Additionally, the relationships between three approaches used for robustness analysis: image space analysis, vector optimization and set-valued optimization, are discussed. Finally, an application for finding a shortest path is given to verify the validity of the results derived in this paper.

In this paper, the image space analysis is employed to study a generalized Ky Fan quasi-inequality with cone constraints. By virtue of a nonlinear scalarization function and a positive linear operator, a nonlinear (regular) weak separation function and a linear regular weak separation function are introduced. Nonlinear and, in particular, linear separations for the generalized Ky Fan quasi-inequality with cone constraints are characterized. Some necessary and sufficient optimality conditions, especially a saddle-point sufficient optimality condition for the generalized Ky Fan quasi-inequality with cone constraints, are obtained. As applications, some sufficient conditions for (weak) vector equilibrium flows of vector traffic equilibrium problems with capacity arc constraints, are derived.

In the research of optimization problems, optimality conditions play an important role. By using some derivatives, various types of necessary and/or sufficient optimality conditions have been introduced by many researchers. Especially, in convex programming, necessary and sufficient optimality conditions in terms of the subdifferential have been studied extensively. Recently, necessary and sufficient optimality conditions for quasiconvex programming have been investigated by the authors. However, there are not so many results concerned with Karush–Kuhn–Tucker type optimality conditions for non-differentiable quasiconvex programming. In this paper, we study a Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. We show some closedness properties for Greenberg–Pierskalla subdifferential. Under the Slater constraint qualification, we show a necessary and sufficient optimality condition for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. Additionally, we introduce a necessary and sufficient constraint qualification of the optimality condition. As a corollary, we show a necessary and sufficient optimality condition for convex programming in terms of the subdifferential.

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish “no-gap” second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.

We consider optimal control problems that can have singular or sliding modes as solutions or parts of solutions [1; 2, p. 26]; in these modes, the Pontryagin function is independent of some control variables or has a nonunique maximum on the set of control variables. For such problems with specific degeneration properties [3, p. 47], the classical optimality conditions (if they apply at all) are noneffective: the Lagrange–Pontryagin necessary conditions are satisfied and can produce infinitely many spurious solutions, while the sufficient conditions fail to hold. One needs effective conditions, which are lacking in the theory yet. The aim of the present paper is to fill the gap to a certain extent by suggesting a reasonably complete set of necessary and sufficient local optimality conditions obtained as a generalization of the classical second-order conditions in terms of the corresponding Clebsch and Riccati inequalities and the maximum principle. To this end, we use a special construction (known as the method of multiple maxima [3–5]) of the Krotov function in the Krotov sufficient conditions.

Based on a given set of uncertain scenarios, robust optimization aims to determine the best possible choices. The best solution in robust optimization is usually found by identifying the best out of the worst-case analysis or by using the min–max counterpart. The min–max approach is a pessimistic approach and commonly discards the effect of some uncertain scenarios in the process of finding an optimum solution. Thus, in this article, a new robust counterpart is introduced with the help of the generalized ordered weighted aggregation (GOWA) operator to solve uncertain multi-objective optimization problems with an uncertainty set of finite cardinality. After introducing GOWA robustness, the elementary properties of the GOWA robust objective function, such as continuity and local Lipschitz continuity, are analysed. Then, the relationship is discussed between the concept of GOWA robustness and (the pessimistic) min–max and (the optimistic) min–min robustness. It is shown that min–min and min–max robustness are particular cases of GOWA robustness. The GOWA robust counterpart is found to be an optimization problem with a set-valued objective mapping. Two approaches are provided to find robust solutions for GOWA: weighted-sum and epsilon-constraint. Finally, an objective-wise GOWA robust counterpart is provided to solve uncertain multi-objective optimization problems and relate it to min–max robustness. Several geometrical and numerical illustrations support the entire article.

Invited Papers.- Combined Relaxation Methods for Generalized Monotone Variational Inequalities.- Abstract Convexity and the Monge-Kantorovich Duality.- Optimality Conditions and Duality for Multiobjective Programming Involving (C, ?, ?, d) type-I Functions.- Contributed Papers.- Partitionable Variational Inequalities with Multi-valued Mappings.- Almost Convex Functions: Conjugacy and Duality.- Pseudomonotonicity of a Linear Map on the Interior of the Positive Orthant.- An Approach to Discrete Convexity and Its Use in an Optimal Fleet Mix Problem.- A Unifying Approach to Solve a Class of Parametrically-Convexifiable Problems.- Mathematical Programming with (?, ?)-invexity.- Some Classes of Pseudoconvex Fractional Functions via the Charnes-Cooper Transformation.- Equilibrium Problems Via the Palais-Smale Condition.- Points of Efficiency in Vector Optimization with Increasing-along-rays Property and Minty Variational Inequalities.- Higher Order Properly Efficient Points in Vector Optimization.- Higher-order Pseudoconvex Functions.- Sufficient Optimality Conditions and Duality in Nonsmooth Multiobjective Optimization Problems under Generalized Convexity.- Optimality Conditions for Tanaka's Approximate Solutions in Vector Optimization.- On the Work of W. Oettli in Generalized Convexity and Nonconvex Optimization - a Review and Some Perspectives.- Local and Global Consumer Preferences.- Optimality Conditions for Convex Vector Functions by Mollified Derivatives.- On Arcwise Connected Convex Multifunctions.- A Sequential Method for a Class of Bicriteria Problems.- Decomposition of the Measure in the Integral Representation of Piecewise Convex Curves.- Rambling Through Local Versions of Generalized Convex Functions and Generalized Monotone Operators.- Monotonicity and Dualities.- On Variational-like Inequalities with Generalized Monotone Mappings.- Almost Pure Nash Equilibria in Convex Noncooperative Games.- A Spectral Approach to Solve Box-constrained Multi-objective Optimization Problems.

In this article, the second-order optimality conditions for nonlinear programming with multiple interval-valued objective functions (in brief, NPMIOF) are studied. In the first part, the second- and first-order necessary optimality conditions for some types of efficient solutions of NPMIOF are established under of Abadie second- and first-order constraint qualifications. In the next part, we investigate both primal and dual second-order sufficient optimality conditions. Our second-order necessary conditions enhance the previous results. The second-order sufficient optimality conditions are new.

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.