Minimax Programming Problems Research Articles

Minimax Programming Under Generalized (p, r)-Invexity

Minimax programming problems involving generalized (p, r)-invex functions are considered. Parametric sufficient optimality conditions and duality results are established under the aforesaid assumptions on the objective and constraint functions.

An equitable solution for multicriteria bargaining games

In this paper we study bargaining models where the agents consider several criteria to evaluate the results of the negotiation process. We propose a new solution concept for multicriteria bargaining games based on the distance to a utopian minimum level vector. This solution is a particular case of the class of the generalized leximin solutions and can be characterized as the solution of a finite sequence of minimax programming problems.

H2 ROBUST FILTERING WITH OPTIMALITY GAP CERTIFICATION

In this paper H2 robust filtering is considered. Particular attention is devoted to what we call optimality gap certification of robust filters for both continuous and discrete time invariant systems subject to polytopic parameter uncertainty. After a brief discussion on some of the most expressive methods available for H2 robust filter design, a new one based on a performance certificate calculation is presented. The performance certificate is given in terms of the gap produced by the robust filter between lower and upper bounds of a minimax programming problem where the H2 norm of the estimation error is maximized with respect to the feasible uncertainties and minimized with respect to all linear, rational, stable and causal filters. The calculations are performed through convex programming methods developed to deal with linear matrix inequality (LMI). Many examples borrowed from the literature are solved and it is shown that the proposed method outperforms all other designs.

H2 Optimal Robust Filtering

In this paper, a design procedure for H 2 optimal robust filtering is presented. The robust filter is determined from the equilibrium solution of a minimax programming problem where the H 2 norm of the estimation error is maximized with respect to the feasible uncertainties and minimized with respect to all full order, linear, rational and causal filters. As the main contribution, it is shown that for the class of parameter uncertainty considered, the equilibrium solution of the aforementioned minimax problem can be exactly determined. In contrast to the design methods available in the literature to date, dealing with norm bounded or convex bounded parameter uncertainty, which naturally provide suboptimal solutions to the robust filtering problem, the one presented in this paper does not present any degree of conservatism due to the particular parameter uncertainty model taken into consideration. However, the price to be payed to reach robust optimality is that the order of the optimal robust filter is, in general, very high. The classical static linear approximation problem as well as the filter design problem corresponding to continuous and discrete time linear systems are considered. In the last section, an illustrative example is presented to make clear the main features of the reported results.

Variable Programming: A Generalized Minimax Problem. Part II: Algorithms

In this part of the two-part series of papers, algorithms for solving some variable programming (VP) problems proposed in Part I are investigated. It is demonstrated that the non-differentiability and the discontinuity of the maximum objective function, as well as the summation objective function in the VP problems constitute difficulty in finding their solutions. Based on the principle of statistical mechanics, we derive smooth functions to approximate these non-smooth objective functions with specific activated feasible sets. By transforming the minimax problem and the corresponding variable programming problems into their smooth versions we can solve the resulting problems by some efficient algorithms for smooth functions. Relevant theoretical underpinnings about the smoothing techniques are established. The algorithms, in which the minimization of the smooth functions is carried out by the standard quasi-Newton method with BFGS formula, are tested on some standard minimax and variable programming problems. The numerical results show that the smoothing techniques yield accurate optimal solutions and that the algorithms proposed are feasible and efficient.

Performance Analysis of Capacity of MIMO Systems under Multiuser Interference Based on Worst-Case Noise Behavior

The capacity of a cellular multiuser MIMO system depends on various parameters, for example, the system structure, the transmit and receive strategies, the channel state information at the transmitter and the receiver, and the channel properties. Recently, the main focus of research was on single-user MIMO systems, their channel capacity, and their error performance with space-time coding. In general, the capacity of a cellular multiuser MIMO system is limited by additive white Gaussian noise, intracell interference from other users within the cell, and intercell interference from users outside the considered cell. We study one point-to-point link, on which interference acts. The interference models the different system scenarios and various parameters. Therefore, we consider three scenarios in which the noise is subject to different constraints. A general trace constraint is used in the first scenario. The noise covariance matrix eigenvalues are kept fixed in the second scenario, and in the third scenario the entries on the diagonal of the noise covariance matrix are kept fixed. We assume that the receiver as well as the transmitter have perfect channel state information. We solve the corresponding minimax programming problems and characterize the worst-case noise and the optimal transmit strategy. In all scenarios, the achievable capacity of the MIMO system with worst-case noise is equal to the capacity of some MIMO system in which either the channels are orthogonal or the transmit antennas are not allowed to cooperate or in which no channel state information is available at the transmitter. Furthermore, the minimax expressions fulfill a saddle point property. All theoretical results are illustrated by examples and numerical simulations.

Minimax programming under ( p, r)-invexity

Minimax programming problems involving ( p, r)-invex functions with respect to η are considered. Parametric and non-parametric necessary and sufficient optimality conditions and duality results for a class of minimax programming problems are obtained under ( p, r)-invexity assumption on objective and constraint functions. Some different types of dual problems are introduced and duality theorems is established for each dual problem.

OPTIMALITY CONDITIONS FOR MINIMAX PROGRAMMING OF ANALYTIC FUNCTIONS

In this paper, we investigate a minimax complex programming problem. Several sufficient Optimality conditions are established under the framework of generalized convexity for analytic functions. Employing the sufficient optimality conditions, we have proved the weak, strong and strict converse duality theorems for the complex minimax programming problem.

On nonsmooth minimax programming problems containing V-ρ-invexity

We consider a class of nonsmooth minimax programming problems in which functions are locally Lipschitz. Necessary and sufficient optimality conditions and duality theorems for nonsmooth minimax programming problems are obtained under Vρ-invexity assumptions on objective and constraint functions

On Minimax Fractional Programming of Generalized Convex Set Functions

We derive necessary and sufficient optimality conditions for the discrete minimax programming problem with subdifferentiable (I,ρ,θ)-convex set functions. Then we apply these optimality criteria to construct two models of parameter-free dual problems and a third model of another dual problem. We also establish weak, strong, and strictly converse duality theorems.

Duality for a Minimax Programming Problem Containingn-Set Functions

Parameter necessary and sufficient optimality conditions for the discrete minimax programming problem with differentiablen-set functions are established in Zalmai [Optimization20(1989) 377–395]. In this paper, we establish nonparameter necessary and sufficient optimality conditions. The nonparameter necessary optimality conditions suggest the establishment of two parameter-free duality models for the discrete minimax programming problem with differentiablen-set functions, and the weak, strong, and strict converse duality theorems have been derived.

Duality in fractional minimax programming

AbstractCertain omissions in the recently introduced dual for fractional minimax programming problem ‘minimize max y ∈ Y f(x, y) / h(x, y), subject to g(x) ≤ 0’, are indicated and two modified duals for this problem are presented. Various fractional programming and generalized fractional programming duals are shown to be special cases of this study.

Non-differentiable pseudo-convex functions and duality for minimax programming problems

Optimality results are derived for a general minimax programming problem under non-differentiable pseudo-convexity assumptions. A dual in terms of Dini derivatives is introduced and duality results are established. Finally, two duals again in terms of Dini derivatives are introduced for a generalized fractional minimax programming problem and corresponding results are studied.

Duality for fractional minimax programming problems

AbstractDuality theory is discussed for fractional minimax programming problems. Two dual problems are proposed for the minimax fractional problem: minimize maxy∈Υf(x, y)/h(x, y), subject to g(x) ≤ 0. For each dual problem a duality theorm is established. Mainly these are generalisations of the results of Tanimoto [14] for minimax fractional programming problems. It is noteworthy here that these problems are intimately related to a class of nondifferentiable fractional programming problems.

A multiobjective minimax programming method and its application of optimal design.

A practical approach to nonlinear multiobjective programming problems using penalty functions and the complex search (a derivative free method) was recently proposed by some of the authors. Here, this approach is applied to solving multiobjective minimax programming problems which are modified to multiobjective programming ones by the finite points approximation procedure. Then, a cam design problem is formulated and solved.

Non-linear programming using least pth optimization with extrapolation†

We present a general approach for solving minimax and non-linear programming problems through a sequence of least pth approximations with extrapolation. Several numerical examples illustrate the effectiveness of the present approach. A comparison with the woll-known SUMT method of Fiaeco and McCormick is made under the same conditions and employing Fletcher's quasi.Newton programme.