Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds

Dini derivatives in Riemannian manifold settings are studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given.     

On a pair of nonlinear mixed integer programming problems

We consider maximin and minimax nonlinear mixed integer programming problems which are nonsymmetric in duality sense. Under weaker (pseudo-convex/pseudo-concave) assumptions, we show that the supremum infimum of the maximin problem is greater than or equal to the infimum supremum of the minimax problem. As a particular case, this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems. Further, this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming.     

Reducibility of minimax to minisum 0–1 programming problems

We consider 0–1 programming problems with a minimax objective function and any set of constraints. Upon appropriate transformations of its cost coefficients, such a minimax problem can be reduced to a linear minisum problem with the same set of feasible solutions such that an optimal solution to the latter will also solve the original minimax problem.Although this reducibility applies for any 0–1 programming problem, it is of particular interest for certain locational decision models. Among the obvious implications are that an algorithm for solving a p-median (minisum) problem in a network will also solve a corresponding p-center (minimax) problem.It should be emphasized that the results presented will in general only hold for 0–1 problems due to intrinsic properties of the minimax criterion.     

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.     

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.     

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.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

Symmetric Duality and Self-duality for Nonsmooth Mathematical Programming

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Second order duality for nondifferentiable minimax programming problems with generalized convexity

In this paper, we are concerned with a class of nondifferentiable minimax programming problem and its two types of second order dual models. Weak, strong and strict converse duality theorems from a view point of generalized convexity are established. Our study naturally unifies and extends some previously known results on minimax programming.     

Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs

In this paper, we introduce a new dual program, which is representable as a semidefinite linear programming problem, for a primal convex minimax programming problem, and we show that there is no duality gap between the primal and the dual whenever the functions involved are sum-of-squares convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust sum-of-squares convex programming problems under data uncertainty and to minimax fractional programming problems with sum-of-squares convex polynomials. We obtain these results by first establishing sum-of-squares polynomial representations of non-negativity of a convex max function over a system of sum-of-squares convex constraints. The new class of sum-of-squares convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The sum-of-squares convexity of polynomials can numerically be checked by solving semidefinite programming problems whereas numerically verifying convexity of polynomials is generally very hard.     

Generalized Envelope Theorems: Applications to Dynamic Programming

We show in this paper that the class of Lipschitz functions provides a suitable framework for the generalization of classical envelope theorems for a broad class of constrained programs relevant to economic models, in which nonconvexities play a key role, and where the primitives may not be continuously differentiable. We give sufficient conditions for the value function of a Lipschitz program to inherit the Lipschitz property and obtain bounds for its upper and lower directional Dini derivatives. With strengthened assumptions we derive sufficient conditions for the directional differentiability, Clarke regularity, and differentiability of the value function, thus obtaining a collection of generalized envelope theorems encompassing many existing results in the literature. Some of our findings are then applied to decision models with discrete choices, to dynamic programming with and without concavity, to the problem of existence and characterization of Markov equilibrium in dynamic economies with nonconvexities, and to show the existence of monotone controls in constrained lattice programming problems.     

A quadratic approximation method for minimizing a class of quasidifferentiable functions

Summary An unconstrained nonlinear programming problem with nondifferentiabilities is considered. The nondifferentiabilities arise from terms of the form max [f ₁(x), ...,f _n (x)], which may enter nonlinearly in the objective function. Local convex polyhedral upper approximations to the objective function are introduced. These approximations are used in an iterative method for solving the problem. The algorithm proceeds by solving quadratic programming subproblems to generate search directions. Approximate line searches ensure global convergence of the method to stationary points. The algorithm is conceptually simple and easy to implement. It generalizes efficient variable metric methods for minimax calculations.     

ASYMPTOTIC APPROXIMATION METHOD AND ITS CONVERGENCE ON SEMI-INFINITE PROGRAMMING

The aim of this article is to discuss an asymptotic approximation model and its convergence for the minimax semi-infinite programming problem. An asymptotic surrogate constraints method for the minimax semi-infinite programming problem is presented by making use of two general discrete approximation methods. Simultaneously, the consistence and the epi-convergence of the asymptotic approximation problem are discussed.     

Weak and strong convergence of proximal penalization and proximal splitting algorithms for two-level hierarchical Ky Fan minimax inequalities

The theory of Ky Fan minimax inequalities provides a powerful general framework for the study of convex programming, variational inequalities and economic equilibrium problems. One of the fundamental methods for finding a solution of Ky Fan minimax inequalities is the proximal point algorithm, where a lot of papers have been dedicated to this subject. In this paper, a general class of two-level hierarchical Ky Fan minimax inequalities is introduced in real Hilbert spaces. For a wide class of Bregman functions, an association of inexact implicit Bregman-penalization proximal and Bregman-splitting proximal algorithms are suggested and analysed. Weak and strong convergences are proved under essentially weaker conditions. We conclude this paper with a hierarchical minimization problem and a numerical example.     

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.     

精确罚函数和极小极大问题

在这篇文章中我们研究了对于不等式约束的非线性规划问题如何根据极小极大问题的鞍点来找精确罚问题的解。对于一个具有不等式约束的非线性规划问题,通过罚函数,我们构造出一个极小极大问题,应用交换“极小”或“极大”次序的策略,证明了罚问题的鞍点定理。研究结果显示极小极大问题的鞍点是精确罚问题的解。     

On the accurate identification of active set for constrained minimax problems

In this paper, the problem of identifying the active constraints for constrained nonlinear programming and minimax problems at an isolated local solution is discussed. The correct identification of active constraints can improve the local convergence behavior of algorithms and considerably simplify algorithms for inequality constrained problems, so it is a useful adjunct to nonlinear optimization algorithms. Facchinei et al. [F. Facchinei, A. Fischer, C. Kanzow, On the accurate identification of active constraints, SIAM J. Optim. 9 (1998) 14-32] introduced an effective technique which can identify the active set in a neighborhood of a solution for nonlinear programming. In this paper, we first improve this conclusion to be more suitable for infeasible algorithms such as the strongly sub-feasible direction method and the penalty function method. Then, we present the identification technique of active constraints for constrained minimax problems without strict complementarity and linear independence. Some numerical results illustrating the identification technique are reported.     

ε-OPTIMAL SOLUTION FOR CONTINUOUS MINIMAX PROBLEMS WITH MAXIMUM-ENTROPY FUNCTION

This paper describes and explores a maximum-entropy approach to continuous minimax problem, which is applicable in many fields, such as transportation planning and game theory. It illustrates that the maximum entropy approcach has easy framework and proves that every accumulation of {x_k} generated by maximum-entropy programming is -optimal solution of initial continuous minimax problem. The paper also explains BFGS or TR method for it. Two numerical exam.ples for continuous minimax problem are given     

Nonlinear programming using minimax techniques

A minimax approach to nonlinear programming is presented. The original nonlinear programming problem is formulated as an unconstrained minimax problem. Under reasonable restrictions, it is shown that a point satisfying the necessary conditions for a minimax optimum also satisfies the Kuhn-Tucker necessary conditions for the original problem. A leastpth type of objective function for minimization with extremely large values ofp is proposed to solve the problem. Several numerical examples compare the present approach with the well-known SUMT method of Fiacco and McCormick. In both cases, a recent minimization algorithm by Fletcher is used.This paper is based on work presented at the 5th Hawaii International Conference on System Sciences, Honolulu, Hawaii, 1972. The authors are greatly indebted to V. K. Jha for his programming assistance and J. H. K. Chen who obtained some of the numerical results. This work was supported in part by the National Research Council of Canada under Grant No. A7239, by a Frederick Gardner Cottrell Grant from the Research Corporation, and through facilities and support from the Communications Research Laboratory of McMaster University.