Minimax Programming Problems Research Articles

Optimality and duality in nondifferentiable minimax programming problems involving $$\epsilon $$-quasi solutions

Optimality and duality in nondifferentiable minimax programming problems involving $$\epsilon $$-quasi solutions

A Minimax-Program-Based Approach for Robust Fractional Multi-Objective Optimization

In this paper, by making use of some advanced tools from variational analysis and generalized differentiation, we establish necessary optimality conditions for a class of robust fractional minimax programming problems. Sufficient optimality conditions for the considered problem are also obtained by means of generalized convex functions. Additionally, we formulate a dual problem to the primal one and examine duality relations between them. In our results, by using the obtained results, we obtain necessary and sufficient optimality conditions for a class of robust fractional multi-objective optimization problems.

Optimality conditions and sensitivity analysis in parametric convex minimax programming

ABSTRACT In this paper, we study optimality conditions as well as sensitivity analysis of parametric convex minimax programming problems. The main tools used here are the formulas for computing the subdifferentials of maximum functions under some suitable regularity conditions. More precisely, we use these tools to study optimality conditions for the problem under consideration. These results are then applied to obtain optimality conditions for multiobjective optimization problems, which constitute the first part of the paper. In the second part, we derive formulas for computing the subdifferential in the sense of convex analysis and singular subdifferential of the optimal value function for the problem in question.

Higher-order Mond-Weir duality of set-valued fractional minimax programming problems

In this paper, we consider a set-valued fractional minimax programming problem (abbreviated as SVFMPP) (MFP), in which both the objective and constraint maps are set-valued. We use the concept of higher-order ?-cone arcwisely connectivity, introduced by Das [1], as a generalization of higher-order cone arcwisely connected setvalued maps. We explore the higher-order Mond-Weir (MWD) form of duality based on the supposition of higher-order ?-cone arcwisely connectivity and prove the associated higher-order converse, strong, and weak theorems of duality between the primary (MFP) and the analogous dual problem (MWD).

A specific minimax programming with the max-plus inequalities constraints

Recently, the max-plus inequalities have been introduced for describing the time constraints of the workpiece transmission in a factory workshop. Each solution of the max-plus inequalities system represents actually a feasible time scheme in the factory workshop. For searching all the feasible time schemes, we attempt to solve the corresponding max-plus inequalities system. An effective approach, based on the so-called row-nonnegative matrix, is proposed to find out all the solutions to the max-plus inequalities system. Besides, considering the practical application background, the system manager usually hopes to postpone the processing finishing times. To coincide with such an objective, we establish a relevant minimax programming problem, in which the objective function is to minimize the maximum of all variables, while the constraint is the above-mention system of max-plus inequalities. A detailed minimum-discriminant-matrix-based algorithm is provided for searching an optimal solution of our studied minimax programming. Moreover, we further develop a novel approach to compute all its optimal solution(s), i.e., the whole optimal solution set. Our proposed resolution approaches are illustrated by some numerical examples.

Set-valued minimax programming problems under $ \sigma $-arcwisely connectivity

<abstract><p>A set-valued minimax programming problem (in short, SVMP) is taken into consideration in this study. We present the idea of $ \sigma $-arcwisely connectivity of set-valued maps (in short, SVM) in the broader sense of arcwisely connected SVMs. The sufficient criteria for Karush-Kuhn-Tucker (KKT) optimality are constituted for the problem (MP) under contingent epidifferentiation and $ \sigma $-arcwisely connectivity suppositions. In addition, we develop the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) kinds models of duality and verify the associated strong, weak, and converse theorems of duality among the primal (MP) and the associated figures of duals under $ \sigma $-arcwisely connectivity supposition.</p></abstract>

Design of Prediction-Based Controller for Networked Control Systems with Packet Dropouts and Time-Delay

A novel prediction-based controller design is proposed for networked control systems (NCSs) with stochastic packet dropouts and time-delay in their control channel. The sequence of packet dropouts, which are modelled as a Bernoulli process, is compensated by a zero-order holder (ZOH)-based module, whereas a state predictor is utilized for obtaining the predicted states at the time delayed. In view of dropout compensator and state predictor, a novel modified model predictive controller (MPC) is designed and proposed in the following procedures. Compared to cost function of a general model predictive controller, variables of states are substituted by the predicted ones as obtained from state predictor preliminarily. Then, a logical programming approach is applied to include all the possible circumstances in the prediction horizon. Consequently, the cost function is reformed as simultaneous minimax linear matrix inequalities (LMI) with constraints. As a result, toolbox YALMIP is employed in order to solve such minimax programming problem eventually. Simulation results are presented to show the feasibility and performance of proposed method.

E-differentiable minimax programming under E-convexity

In this paper, a new class of minimax programming problems is considered in which the functions involved are E-differentiable. The so-called parametric and nonparametric necessary E-optimality conditions are derived for the considered E-differentiable minimax programming problem. Further, sufficient optimality conditions are established for such nondifferentiable extremum problems under E-convexity hypotheses. Moreover, the example of a nonsmooth minimax programming problem with E-differentiable functions is given to illustrate the aforesaid results. Furthermore, the so-called Mond-Weir E-dual problem and Wolfe E-dual problem are defined for the considered E-differentiable minimax programming problem and several E-duality theorems are established also under appropriate E-convexity hypotheses.

Fuzzy relation weighted minimax programming with addition-min composition

In the existing works, system of addition-min fuzzy relation inequalities (FRIs) was introduced to model a peer-to-peer (P2P) file sharing system. For decreasing the network congestion, a minimax programming has been studied. However, in such programming, the different weights of the terminals (or peers) were not considered. In order to distinguish the different importance degrees of the terminals, we propose and investigate the weighted minimax programming problem in this work. A novel algorithm based on some discrete function-values is developed to obtain its optimal solution. Effectiveness of our proposed algorithms are further illustrated by some numerical examples.

Optimality Conditions for Set-Valued Minimax Programming Problems via Second-Order Contingent Epiderivative

Optimality Conditions for Set-Valued Minimax Programming Problems via Second-Order Contingent Epiderivative

Necessary optimality conditions for minimax programming problems with mathematical constraints

In this paper, necessary optimality conditions in terms of upper and/or lower subdifferentials of both cost and constraint functions are derived for minimax optimization problems with inequality, equality and geometric constraints in the setting of non-differentiatiable and non-Lipschitz functions in Asplund spaces. Necessary optimality conditions in the fuzzy form are also presented. An application of the fuzzy necessary optimality condition is shown by considering minimax fractional programming problem.

Second-order sequence-based necessary optimality conditions in constrained nonsmooth vector optimization and applications

Several notions of sequential directional derivatives and sequential local approximations are introduced. Under (first-order) Hadamard differentiability assumptions of the data at the point of study, these concepts are utilized to analyze second-order necessary optimality conditions, which rely on given sequences, for local weak solutions in nonsmooth vector optimization problems with constraints. Some applications to minimax programming problems are also derived.

Optimality and duality for nonsmooth minimax programming problems using convexifactor

The aim of this work is to study optimality conditions for nonsmooth minimax programming problems involving locally Lipschitz functions by means of the idea of convexifactors that has been used in [J. Dutta, S. Chandra, Convexifactors, generalized convexity and vector optimization, Optimization, 53 (2004) 77-94]. Further, using the concept of optimality conditions, Mond-Weir and Wolfe type duality theory has been developed for such a minimax programming problem. The results in this paper extend the corresponding results obtained using the generalized Clarke subdifferential in the literature.

Set-valued minimax programming problems under generalized cone convexity

In this paper, a set-valued minimax programming problem (MP) is considered. The Karush–Kuhn–Tucker (KKT) sufficient optimality conditions are established for the problem (MP) via contingent epiderivative under generalized cone convexity assumptions. Mond–Weir (MWD), Wolfe (WD) and mixed (MD) types duality are constructed for the problem (MP) and desired duality theorems between them are established.

Optimality conditions and duality for nonsmooth minimax programming problems under generalized invexity

In this paper, we consider a class of nonsmooth minimax programming problems in which functions are locally Lipschitz. Sucient optimality conditions are discussed under locally Lipschitz generalized (?,?)-invex functions. Moreover, usual duality results are proved under the said assumptions.

A Projection Neural Network for Constrained Quadratic Minimax Optimization.

This paper presents a projection neural network described by a dynamic system for solving constrained quadratic minimax programming problems. Sufficient conditions based on a linear matrix inequality are provided for global convergence of the proposed neural network. Compared with some of the existing neural networks for quadratic minimax optimization, the proposed neural network in this paper is capable of solving more general constrained quadratic minimax optimization problems, and the designed neural network does not include any parameter. Moreover, the neural network has lower model complexities, the number of state variables of which is equal to that of the dimension of the optimization problems. The simulation results on numerical examples are discussed to demonstrate the effectiveness and characteristics of the proposed neural network.

Nondifferentiable minimax programming problems with applications

This paper is devoted to the study of optimality conditions and duality in nondifferentiable minimax programming problems and applications. Employing some advanced tools of variational analysis and generalized differentiation, we establish new necessary conditions for optimal solutions of a minimax programming problem involving inequality and equality constraints. Sufficient conditions for the existence of such solutions to the considered problem are also obtained by way of \(L\)-invex-infine functions. We state a dual problem to the primal one and explore weak, strong and converse duality relations between them. In addition, some of these results are applied to a nondifferentiable multiobjective optimization problem.

On minimax programming problems involving right upper-Dini-derivative functions

Abstract In this paper, we derive necessary and sufficient optimality conditions for a general minimax programming problem involving some classes of generalized convexities with the tool-right upper-Dini-derivative. Moreover, using the concept of optimality conditions, Mond-Weir type duality theory has been developed for such a minimax programming problem. MSC:26A51, 49J35, 90C32.

A New Objective Penalty Function Approach for Solving Constrained Minimax Problems

In this paper, a new objective penalty function approach is proposed for solving minimax programming problems with equality and inequality constraints. This new objective penalty function combines the objective penalty and constraint penalty. By the new objective penalty function, a constrained minimax problem is converted to minimizations of a sequence of continuously differentiable functions with a simple box constraint. One can thus apply any efficient gradient minimization methods to solve the minimizations with box constraint at each step of the sequence. Some relationships between the original constrained minimax problem and the corresponding minimization problems with box constraint are established. Based on these results, an algorithm for finding a global solution of the constrained minimax problems is proposed by integrating the particular structure of minimax problems and its global convergence is proved under some conditions. Furthermore, an algorithm is developed for finding a local solution of the constrained minimax problems, with its convergence proved under certain conditions. Preliminary results

On Nonsmooth Semi-Infinite Minimax Programming Problem with(Φ,ρ)-Invexity

We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz(Φ,ρ)-invexity assumptions, we derive the sufficiency of the necessary optimality conditions for the same problem. We also formulate dual and establish weak, strong, and strict converse duality theorems for Problem (SIP) and its dual. These results extend several known results to a wider class of problems.