An elementary proof of the Lagrange multiplier theorem in normed linear spaces

We present an elementary proof of the Lagrange multiplier theorem for optimization problems with equality constraints in normed linear spaces. Most proofs in the literature rely on advanced concepts and results, such as the implicit function theorem and the Lyusternik theorem. By contrast, the proof given in this article employs only basic results from linear algebra, the critical-point condition for unconstrained minima and the fact that a continuous function attains its minimum over a closed ball in the finite-dimensional space.     

Condition Number Theorems in Linear-Quadratic Optimization

The condition number of a given mathematical problem is often related to the reciprocal of its distance from ill-conditioning. Such a property is proved here in the infinite-dimensional setting for linear-quadratic convex optimization of two types: linearly constrained convex quadratic problems, and minimum norm least squares solutions. A uniform version of such theorem is obtained in both cases for suitably equi-bounded classes of optimization problems. An application to the conditioning of a Ritz method is presented. For least squares problems it is shown that the semi-Fredholm property of the operators involved determines the validity of a condition number theorem.     

About duality and alternative in multiobjective optimization

McLinden's result on the equivalence between a theorem of the alternative and a duality theorem for constrained optimization problems is extended to the multiobjective case. We then discuss some existing results on that topic and present an alternative approach to duality relations in conditionally complete lattices.The author is indebted to the referee for his useful remarks on a previous version of the paper.     

一个一般的Motzkin定理及其应用

本文考虑局部凸拓扑向量空间中包含多值映射的不等式系统，在很一般的条件下建立了一个Motzkin型择一定理，并给出了该定理在向量最优化问题中的应用，本文结果涵盖并推广了许多已知择一定理     

Farkas' theorem of nonconvex type and its application to a min-max problem

This note is concerned with the generalization of Farkas' theorem and its application to derive optimality conditions for a mix-max problem. Farkas' theorem is generalized to a system of inequalities described by sup-min type positively homogeneous functions. This generalization allows us to deal with optimization problems consisting of objective and constraint functions whose directional derivatives are not necessarily convex with respect to the directions. As an example of such problems, we formulate a min-max problem and derive its optimality conditions.The author would like to express his sincere thanks to Professors S. Suzuki and T. Asano of Sophia University and Professor K. Shimizu of Keio University for encouragement and suggestions.     

向量优化问题中的弱S-有效解

在局部凸拓扑线性空间中, 提出了集值向量优化问题的弱S-有效解和S-次似凸性概念. 在S-次似凸性假设下建立了择一性定理, 并利用择一性定理建立了弱S-有效解的标量化定理. 此外, 通过几个具体例子解释了主要结果.     

大博弈中Nash均衡的存在性

研究了具有任意多个局中人的非合作博弈(大博弈)中Nash均衡的存在性.将1969年Ma的截口定理推广得到新的截口定理.用这个新的截口定理进一步证明了:1)大博弈中Nash均衡的存在性;2)纯策略集为紧度量空间而且支付函数为连续函数时,连续大博弈中混合策略Nash均衡的存在性.并且存在性定理推出了2010年Salonen的结果,即此研究结果较Salonen的结论更具普遍意义.     

建立在L-凸空间上的截口定理及其应用

In this paper,the author gives a new section theorem in L-convex spaces.And as its applications,the author proves a coincident theorem and a two-functional minimax theorem established in L-convex spaces.     

Separation theorems for nonconvex sets and application in optimization

The aim of this paper is to present separation theorems for two disjoint closed sets, without convexity condition. First, a separation theorem for a given closed cone and a point outside from this cone, is proved and then it is used to prove a separation theorem for two disjoint sets. Illustrative examples are provided to highlight the important aspects of these theorems. An application to optimization is also presented to prove optimality condition for a nonconvex optimization problem.     

Abstract convergence theorem for quasi-convex optimization problems with applications

Abstract

Quasi-convex optimization is fundamental to the modelling of many practical problems in various fields such as economics, finance and industrial organization. Subgradient methods are practical iterative algorithms for solving large-scale quasi-convex optimization problems. In the present paper, focusing on quasi-convex optimization, we develop an abstract convergence theorem for a class of sequences, which satisfy a general basic inequality, under some suitable assumptions on parameters. The convergence properties in both function values and distances of iterates from the optimal solution set are discussed. The abstract convergence theorem covers relevant results of many types of subgradient methods studied in the literature, for either convex or quasi-convex optimization. Furthermore, we propose a new subgradient method, in which a perturbation of the successive direction is employed at each iteration. As an application of the abstract convergence theorem, we obtain the convergence results of the proposed subgradient method under the assumption of the Hölder condition of order p and by using the constant, diminishing or dynamic stepsize rules, respectively. A preliminary numerical study shows that the proposed method outperforms the standard, stochastic and primal-dual subgradient methods in solving the Cobb–Douglas production efficiency problem.     

On Schur’s theorem and its converses for n-Lie algebras

An n-Lie algebra analogue of Schur’s theorem and its converse as well as a Lie algebra analogue of Baer’s theorem and its converse are presented. Also, it is shown that, an n-Lie algebra with finite dimensional derived subalgebra and finitely generated central factor is isoclinic to some finite dimensional n-Lie algebra.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty

The celebrated von Neumann minimax theorem is a fundamental theorem in two-person zero-sum games. In this paper, we present a generalization of the von Neumann minimax theorem, called robust von Neumann minimax theorem, in the face of data uncertainty in the payoff matrix via robust optimization approach. We establish that the robust von Neumann minimax theorem is guaranteed for various classes of bounded uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the columnwise affine parameter uncertainty.     

On Ha＇s Version of Set-valued Ekeland＇s Variational Principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.     

Floquet theorem for linear implicit nonautonomous difference systems

The aim of this paper is to develop the Floquet theory for linear implicit difference systems (LIDS). It is proved that any index-1 LIDS can be transformed into its Kronecker normal form. Then the Floquet theorem on the representation of the fundamental matrix of index-1 periodic LIDS has been established. As an immediate consequence, the Lyapunov reduction theorem is proved. Some applications of the obtained results are discussed.     

On the Pila–Wilkie theorem

This expository paper gives an account of the Pila–Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We also include complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin–Gromov theorem that are used in this proof. For the latter we follow Binyamini and Novikov.     

常微分方程解的存在唯一性定理教学研究

探讨了常微分方程初值问题解的存在唯一性定理教学策略.为便于教学和有利于学生理解并掌握其思想方法,对定理证明过程的表述作了命题化处理,给出了Picard逐步逼近法的应用实例,提出了教学讨论与知识拓展的一些有益内容.     

Henig Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions

In this paper,we study Henig efficiency in vector optimization with nearly cone-subconvexlikeset-valued function.The existence of Henig efficient point is proved and characterization of Henig efficiencyis established using the method of Lagrangian multiplier.As an interesting application of the results in thispaper,we establish a Lagrange multiplier theorem for super efficiency in vector optimization with nearly cone-subconvexlike set-valued function.     

An (α ,β)-Optimal solution concept in Fuzzy Optimization problems

We propose an (α,β)-optimal solution concept of fuzzy optimization problem based on the possibility and necessity measures. It is well known that the set of all fuzzy numbers can be embedded into a Banach space isometrically and isomorphically. Inspired by this embedding theorem, we can transform the fuzzy optimization problem into a biobjective programming problem by applying the embedding function to the original fuzzy optimization problem. Then the (α,β)-optimal solutions of fuzzy optimization problem can be obtained by solving its corresponding biobjective programming problem. We also consider the fuzzy optimization problem with fuzzy coefficients (i.e., the coefficients are assumed as fuzzy numbers). Under a setting of core value of fuzzy numbers, we provide the Karush–Kuhn–Tucker optimality conditions and show that the optimal solution of its corresponding crisp optimization problem (the usual optimization problem) is also a (1,1)-optimal solution of the original fuzzy optimization problem.