Minmax Programming Problems Research Articles

A Single-Variable Method for Solving the Min–Max Programming Problem with Addition–Overlap Function Composition

Min–max programming problems with addition–min constraints have been studied in the literature to model data transfer in BitTorrent-like peer-to-peer file-sharing systems. It is well known that the class of overlap functions contains various operators, including the “min” operator. The aim of this paper is to generalize the above min–max programming problem with addition–overlap function constraints. We demonstrate that this new optimization problem can be transformed into a simplified single-variable optimization problem, which makes it easier to find an optimal solution. The bisection method will be used to find this optimal solution. In addition, when the overlap function is explicitly specified, an iterative method is given to compute the optimal objective value with a polynomial time complexity. A numerical example is provided to illustrate the procedures.

Finding minimal solution to generalized min-max programming problem with addition-min composition

Finding minimal solution to generalized min-max programming problem with addition-min composition

Solving minimal-optimal solutions for the generalized min-max programming problem with addition-min composition

Several forms of min-max programming problems with addition-min constraints have been introduced in the literature to model data transfer of BitTorrent-like peer-to-peer file-sharing systems. Solution algorithms are also provided to find the optimal solution which has equal values of all its variables. In this paper, we present a generalized min-max programming problem subject to addition-min constraints. A simple iterative approach is proposed to find the minimal-optimal solution for this generalized min-max programming problem. Numerical examples are provided to illustrate how the optimal solution with equal values for all its variables can be improved to become minimal-optimal.

Generalized min-max programming problems subject to addition-min fuzzy relational inequalities

In this paper, we explore a new generalized min-max programming problem with constraints of addition-min fuzzy relational inequalities. This new generalized min-max programming model provides unified settings that enable the system manager to understand the system congestion level or the worst individual cost in the BitTorrent-like peer-to-peer file-sharing system. Theoretical results are presented to illustrate how the optimal value of generalized min-max programming problem can be obtained by solving a single-variable optimization model. Two approaches (an analytic method and an iterative approach) are provided to solve this single-variable optimization model. The complexity analyses of these two approaches are provided. Numerical examples demonstrate our proposed approaches.

An active-set approach to finding a minimal-optimal solution to the min-max programming problem with addition-min fuzzy relational inequalities

In the literature, a BitTorrent-like peer-to-peer (BT-P2P) file-sharing system has been modeled as a system of fuzzy relational inequalities (FRI) with addition-min composition. And a min-max programming problem has been proposed to study its system congestion. From a cost-saving viewpoint, optimal solutions to the min-max programming problem may not be the minimal-optimal solution. The “minimal” solution gives better cost performance while the “optimal” solution gives the least system congestion. Such a drawback has been studied in the literature. In this paper, we propose a simple active-set approach to finding a cost-saving optimal solution, i.e. a minimal-optimal solution to the min-max programming problem. The complexity of our approach is O(m2n), where m is the number of decision variables and n is the number of constraints. Numerical examples are given to illustrate our procedures. Since our active-set approach depends on the given order of decision variables, by using a different sequence of decision variables, it may be able to find other minimal-optimal solutions. Therefore, our method may be able to provide more choices for the manager to use in decision-making.

Some results for the minimal optimal solution of min-max programming problem with addition-min fuzzy relational inequalities

Some results for the minimal optimal solution of min-max programming problem with addition-min fuzzy relational inequalities

Min–max programming problem with constraints of addition-min-product fuzzy relation inequalities

Min–max programming problem with constraints of addition-min-product fuzzy relation inequalities

Dichotomy algorithm for solving weighted min-max programming problem with addition-min fuzzy relation inequalities constraint

Addition-min fuzzy relation inequalities were introduced for describing the quantitative relationship in the peer-to-peer (P2P) file sharing system. In order to reduce the network congress, the system manager has to minimize the variables in the addition-min fuzzy relation inequalities. The corresponding min-max programming has been established in the existing works. However in such model, all the variables are viewed equally. In this paper, considering the different important degrees of the variables, we propose and investigate the weighted min–max programming subject to the addition-min fuzzy relation inequalities. Due to the different objective function, the existing methods are no longer effective for our proposed problem. Dichotomy algorithm has been developed in this paper for searching the approximate optimal solution of the weighted min-max programming. The feasibility and effectiveness of the dichotomy algorithm is illustrated by numerical example.

A single-variable method for solving min–max programming problem with addition-min fuzzy relational inequalities

In this paper, we study the min–max programming problem with n addition-min fuzzy relational inequality constraints. We prove that when the problem is feasible, an optimal solution always exists with all variables being of the same value. Based on this result, the min–max programming problem can be simplified as a single-variable optimization problem with the same optimal objective value. To solve the corresponding single-variable optimization problem, we propose an analytical method and an iterative method by successively approximating the lower bound of the optimal value. Numerical examples are given to illustrate our methods.

Optimal-Vector-Based Algorithm for Solving Min-Max Programming Subject to Addition-Min Fuzzy Relation Inequality

Optimal management model in BitTorrent-like peer-to-peer (P2P) file sharing system can be reduced into a min–max programming problem subject to addition-min fuzzy relation inequality. To deal with such problem, we introduce the concept of optimal vector. Besides, a novel optimal-vector-based (OVB) algorithm is developed to find a minimal optimal solution. Based on practical application consideration, solution obtained by the OVB algorithm is theoretically demonstrated to be better than or equal to that obtained by the existing method. Advantage of our proposed OVB algorithm is illustrated by detailed numerical examples.

MBLDA: A novel multiple between-class linear discriminant analysis

Linear discriminant analysis (LDA) with its extensions is a group of classical methods in dimensionality reduction for supervised learning. However, when some classes are far away from the others, it may be difficult to find the optimal direction by LDA because of the average between-class scatter. Moreover, LDAs are always time consuming for high dimensional problem since the involved generalized eigenvalue problem is needed to be solved. In this paper, a multiple between-class linear discriminant analysis (MBLDA) is proposed for dimensionality reduction. MBLDA finds the transformation directions by approximating the solution to a min-max programming problem, leading to well separability in the reduced space with a fast learning speed on the high-dimensional problem. It is proved theoretically that the proposed method can deal with the special generalized eigenvalue problem by solving a underdetermined homogeneous system of linear equations. Experimental results on the artificial and benchmark datasets show that MBLDA can not only reduce the dimension to a powerful linear discriminant level but also have a fast learning speed.

Min-Max Programming Problem Subject to Addition-Min Fuzzy Relation Inequalities

A BitTorrent-like peer-to-peer file-sharing system can be reduced into a system of addition-min fuzzy relation inequalities. Addition-min is a new composition, and the solution set of addition-min fuzzy relation inequalities differs much from that of general max-t-norm fuzzy relation inequalities or equations. In order to avoid network congestion and improve the stability of data transmission, a min-max programming problem is proposed subject to addition-min fuzzy relation inequalities in this paper. Based on some relevant theorems on resolution, a novel algorithm is developed step by step to find an optimal solution of the proposed problem. Moreover, an application example is given to illustrate the feasibility and efficiency of the algorithm. Finally, some further discussions are made concerning the optimal solution of the proposed problem.

Nonsmooth semi-infinite minmax programming involving generalized (Φ,ρ)-invexity

This paper introduces some new generalizations of the concept of (Φ,ρ)-invexity for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. These functions are used to derive the necessary and sufficient optimality conditions for a class of nonsmooth semi-infinite minmax programming problems, where set of restrictions are indexed in a compact set. Utilizing the sufficient optimality conditions, the authors formulate three types of dual models and establish weak and strong duality results. The results of the paper extend and unify naturally some earlier results from the literature.

Minmax programming problems with (p, r) - ρ - (η, θ)-invexity

In this paper, we establish the sufficient optimality conditions for minmax programming and generalized fractional minmax programming problems. Based on the sufficient conditions, a duality model is constructed and duality results are derived under a new type of generalised invexity assumption, called (p, r) − ρ − (η, θ)-invexity. Our study extends some of the previously known results in the framework of generalised invexity and duality models.Wealso provide some examples to illustrate our work.

Solution properties and error bounds for semi-infinite complementarity problems

In this paper, we deal with the semi-infinite complementarity problems (SICP), in which several important issues are covered, such as solvability, semismoothness of residual functions, and error bounds. In particular, we characterize the solution set by investigating the relationship between SICP and the classical complementarity problem. Furthermore, we show that the SICP can be equivalently reformulated as a typical semi-infinite min-max programming problem by employing NCP functions. Finally, we study the concept of error bounds and introduce its two variants, $\varepsilon$-error bounds and weak error bounds, where the concept of weak error bounds is highly desirable in that the solution set is not restricted to be nonempty.

On semi-infinite minmax programming with generalized invexity

Here, we consider the minmax programming problem with a set of restrictions indexed in a compact. As a novelty, we obtain optimality criteria of the Kuhn--Tucker type involving a limited number of restrictions and prove both necessity and sufficiency under new weaker invexity assumptions. Also some dual problems are introduced and it is proved that the weak and strong duality properties hold within the same environment.

A simple approach to optimality conditions in minmax programming

Considering the minmax programming problem, lower and upper subdifferential optimality conditions, in the sense of Mordukhovich, are derived. The approach here, mainly based on the nonsmooth dual objects of Mordukhovich, is completely different from that of most of the previous works where generalizations of the alternative theorem of Farkas have been applied. The results obtained are close to those known in the literature. However, one of the main achievements of this article is that we could also derive necessary optimality conditions for the minmax program of the usual Karush–Kuhn–Tucker type, which seems to be new in this field of study.

First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets

In this paper, based on some new characterizations of subderivativeand subdifferential of sup-types functions, we develop thefirst-order necessary and sufficient optimality conditions forconvex semi-infinite min-max programming problems in which the indexsets are not necessarily compact.

On optimality and duality theorems of nonlinear disjunctive fractional minmax programs

This paper is concerned with the study of optimality conditions for disjunctive fractional minmax programming problems in which the decision set can be considered as a union of a family of convex sets. Dinkelbach’s global optimization approach for finding the global maximum of the fractional programming problem is discussed. Using the Lagrangian function definition for this type of problem, the Kuhn–Tucker saddle point and stationary-point problems are established. In addition, via the concepts of Mond–Weir type duality and Schaible type duality, a general dual problem is formulated and some weak, strong and converse duality theorems are proven.

Partial vector invexity for n-set functions and minmax programming

Partial vector invexity is introduced for a class of n-set functions and is utilized to establish sufficient optimality conditions and duality results for minmax programming problems involving n-set functions.