Convergence of an Interior Point Algorithm for Continuous Minimax

We propose an algorithm for the constrained continuous minimax problem. The algorithm uses a quasi-Newton search direction, based on subgradient information, conditional on maximizers. The initial problem is transformed to an equivalent equality constrained problem, where the logarithmic barrier function is used to ensure feasibility. In the case of multiple maximizers, the algorithm adopts semi-infinite programming iterations toward epiconvergence. Satisfaction of the equality constraints is ensured by an adaptive quadratic penalty function. The algorithm is augmented by a discrete minimax procedure to compute the semi-infinite programming steps and ensure overall progress when required by the adaptive penalty procedure. Progress toward the solution is maintained using merit functions.     

A nonmonotonic trust region algorithm for a class of semi-infinite minimax programming

Based on the discretization methods for solving semi-infinite programming problems, this paper presents a new nonmonotonic trust region algorithm for a class of semi-infinite minimax programming problem. Under some mild assumptions, the global convergence of the proposed algorithm is given. Numerical tests are reported that show the efficiency of the proposed method.     

An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems

We propose an algorithm for the global optimization of continuous minimax problems involving polynomials. The method can be described as a discretization approach to the well known semi-infinite formulation of the problem. We proceed by approximating the infinite number of constraints using tools and techniques from semidefinite programming. We then show that, under appropriate conditions, the SDP approximation converges to the globally optimal solution of the problem. We also discuss the numerical performance of the method on some test problems. Financial support of EPSRC Grant GR/T02560/01 gratefully acknowledged.     

一类半无限规划的一种渐近替代约束方法和收敛性

A class of constrained semi infinite minimax problem is transformed into a simpleconstrained problem, by means of discretization decoraposirion and maximum entropy method,making use of surrogate constraint, The paper deals with the convergence of this asymptotic aI-proach method.     

Error bounds of two smoothing approximations for semi-infinite minimax problems

In the paper we investigate smoothing method for solving semi-infinite minimax problems. Not like most of the literature in semi-infinite minimax problems which are concerned with the continuous time version（i.e., the one dimensional semi-infinite minimax problems）, the primary focus of this paper is on multi- dimensional semi-infinite minimax problems. The global error bounds of two smoothing approximations for the objective function are given and compared. It is proved that the smoothing approximation given in this paper can provide a better error bound than the existing one in literature.     

A cutting plane method for solving minimax problems in the complex plane

It is shown how the combined discretization and cutting plane method for general convex semi-infinite programming problems recently presented in [40] can be effectively implemented for the solution of minimax problems in the complex plane. In contrast to other approaches, the minimax problem does not have to be linearized. The performance of the algorithm is demonstrated by a number of highly accurate numerical examples.     

Semi-Infinite Programming and Applications to Minimax Problems

A minimisation problem with infinitely many constraints – semi-infinite programming problem (SIP) is considered. The problem is solved using a two stage procedure that searches for global maximum violation of the constraints. A version of the algorithm that searches for any violation of constraints is also considered, and the performance of the two algorithm is compared. An application to solving minimax problem (with and without coupled constraints) is given and a comparison with the algorithm for continuous minimax of Rustem and Howe (2001) is included. Finally, we consider an application to chemical engineering problems.     

A globally most violated cutting plane method for complex minimax problems with application to digital filter design

In [4,6], the authors have presented a numerical method for the solution of complex minimax problems, which implicitly solves discretized versions of the equivalent semi-infinite programming problem on increasingly finer grids. While this method only requires the most violated constraint at the current iterate on a finite subset of the infinitely many constraints of the problem, we consider here a related and more direct approach (applicable to general convex semi-infinite programming problems) which makes use of the globally most violated constraint. Numerical examples with up to 500 unknowns, which partially originate from digital filter design problems, are discussed.     

Chebyshev Approximation of the Analytical Solution of Dirichlet Problem

In this paper linear programming method for minimax approximation is used to obtain an approximation to the analytical solution of a Dirichlet problem using the logarithmic potential function as an approximating function. This approach has the advantage of producing a better approximation than that using other solution of the potential equation as an approximating or basis function for a problem in $n=2$ dimensions.     

Extending the mixed algebraic-analysis Fourier–Motzkin elimination method for classifying linear semi-infinite programmes

Motivated by a recent Basu–Martin–Ryan paper, we obtain a reduced primal-dual pair of a linear semi-infinite programming problem by applying an amended Fourier–Motzkin elimination method to the linear semi-infinite inequality system. The reduced primal-dual pair is equivalent to the original one in terms of consistency, optimal values and asymptotic consistency. Working with this reduced pair and reformulating a linear semi-infinite programme as a linear programme over a convex cone, we reproduce all the theorems that lead to the full eleven possible duality state classification theory. Establishing classification results with the Fourier–Motzkin method means that the two classification theorems for linear semi-infinite programming, 1969 and 1974, have been proved by new and exciting methods. We also show in this paper that the approach to study linear semi-infinite programming using Fourier–Motzkin elimination is not purely algebraic, it is mixed algebraic-analysis.     

An interior point algorithm for semi-infinite linear programming

We consider the generalization of a variant of Karmarkar's algorithm to semi-infinite programming. The extension of interior point methods to infinite-dimensional linear programming is discussed and an algorithm is derived. An implementation of the algorithm for a class of semi-infinite linear programs is described and the results of a number of test problems are given. We pay particular attention to the problem of Chebyshev approximation. Some further results are given for an implementation of the algorithm applied to a discretization of the semi-infinite linear program, and a convergence proof is given in this case.     

Properties of ε-value functions for a two person differential game

We consider the problem of approximate minimax for the Bolza problem of optimal control. Starting from the method of dynamic programming (Bellman) we define the ?-value function to be the approximation for the value function being a solution to the Hamilton–Jacobi equation.     

一类广义半无限规划问题的一阶最优性条件

本文利用一个精确增广Lagrange函数研究了一类广义半无限极小极大规划问题。在一定的条件下将其转化为标准的半无限极小极大规划问题。研究了这两类问题的最优解和最优值之间的关系,利用这种关系和标准半无限极小极大规划问题的一阶最优性条件给出了这类广义半无限极小极大规划问题的一个新的一阶最优性条件。     

Interior penalty functions and duality in linear programming

Logarithmic additive terms of barrier type with a penalty parameter are included in the Lagrange function of a linear programming problem. As a result, the problem of searching for saddle points of the modified Lagrangian becomes unconstrained (the saddle point is sought with respect to the whole space of primal and dual variables). Theorems on the asymptotic convergence to the desired solution and analogs of the duality theorems for the arising optimization minimax and maximin problems are formulated.     

Optimality conditions for nondifferentiable minimax fractional programming with complex variables

We show that a minimax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. We establish the necessary and sufficient optimality conditions of nondifferentiable minimax fractional programming problem with complex variables under generalized convexities.     

极大极小随机规划逼近问题最优解集和最优值的稳定性

研究了特殊的二层极大极小随机规划逼近收敛问题. 首先将下层初始随机规划最优解集拓展到非单点集情形, 且可行集正则的条件下, 讨论了下层随机规划逼近问题最优解集关于上层决策变量参数的上半收敛性和最优值函数的连续性. 然后把下层随机规划的epsilon-最优解向量函数反馈到上层随机规划的目标函数中, 得到了上层随机规划逼近问题的最优解集关于最小信息概率度量收敛的上半收敛性和最优值的连续性.     

Convex Semi-Infinite Parametric Programming: Uniform Convergence of the Optimal Value Functions of Discretized Problems

The continuity of the optimal value function of a parametric convex semi-infinite program is secured by a weak regularity condition that also implies the convergence of certain discretization methods for semi-infinite problems. Since each discretization level yields a parametric program, a sequence of optimal value functions occurs. The regularity condition implies that, with increasing refinement of the discretization, this sequence converges uniformly with respect to the parameter to the optimal value function corresponding to the original semi-infinite problem. Our result is applicable to the convergence analysis of numerical algorithms based on parametric programming, for example, rational approximation and computation of the eigenvalues of the Laplacian.     

Robust multi-market newsvendor models with interval demand data

We present a robust model for determining the optimal order quantity and market selection for short-life-cycle products in a single period, newsvendor setting. Due to limited information about demand distribution in particular for short-life-cycle products, stochastic modeling approaches may not be suitable. We propose the minimax regret multi-market newsvendor model, where the demands are only known to be bounded within some given interval. In the basic version of the problem, a linear time solution method is developed. For the capacitated case, we establish some structural results to reduce the problem size, and then propose an approximation solution algorithm based on integer programming. Finally, we compare the performance of the proposed minimax regret model against the typical average-case and worst-case models. Our test results demonstrate that the proposed minimax regret model outperformed the average-case and worst-case models in terms of risk-related criteria and mean profit, respectively.     

Perturbation analysis of the heat transfer in porous media with small thermal conductivity

An approximate analytical solution for the one-dimensional problem of heat transfer between an inert gas and a porous semi-infinite medium is presented. Perturbation methods based on Laplace transforms have been applied using the solid thermal conductivity as small parameter. The leading order approximation is the solution of Nusselt (or Schumann) problem. Such solution is corrected by means of an outer approximation. The boundary condition at the origin has been taking into account using an inner approximation for a boundary layer. The gas temperature presents a discontinuous front (due to the incompatibility between initial and boundary conditions) which propagates at constant velocity. The solid temperature at the front has been smoothed out using an internal layer asymptotic approximation. The good accuracy of the resulting asymptotic expansion shows its usefulness in several engineering problems such as heat transfer in porous media, in exhausted chemical reactions, mass transfer in packed beds, or in the analysis of capillary electrochromatography techniques.     

Duality for Semi-Infinite Minimax Fractional Programming Problem Involving Higher-Order (Φ, ρ)-V-Invexity

In this paper, we introduce the concept of higher-order (Φ, ρ)-V-invexity and present two types of higher-order dual models for a semi-infinite minimax fractional programming problem. Weak, strong, and strict converse duality theorems are discussed under the assumptions of higher-order (Φ, ρ)-V-invexity to establish a relation between the primal and dual problems.