Exact penalty property for a class of inequality-constrained minimization problems

We use the penalty approach in order to study constrained minimization problems. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish the exact penalty property for a large class of inequality-constrained minimization problems.     

A sufficient condition for exact penalty functions

In this paper, we use the penalty approach for constrained minimization problems in infinite dimensional Banach spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. We establish a simple sufficient condition for exact penalty property for two large classes of constrained minimization problems.     

化不等式约束问题为无约束的广义可微限域精确罚函数方法

本文给出了广义可微精确罚函数的概念及一类所谓广义限域可微精确罚函数.本文预先选定罚因子,将不等式约束问题化为单一的无约束问题,并给出了具全局收敛性的算法.本文的罚函数构造简单,假设条件少而且算法的构造与收敛性结果是独特的.     

一种新的求解带约束的有限极大极小问题的精确罚函数

提出了一种新的精确光滑罚函数求解带约束的极大极小问题．仅仅添加一个额外的变量,利用这个精确光滑罚函数,将带约束的极大极小问题转化为无约束优化问题． 证明了在合理的假设条件下,当罚参数充分大,罚问题的极小值点就是原问题的极小值点．进一步,研究了局部精确性质．数值结果表明这种罚函数算法是求解带约束有限极大极小问题的一种有效算法．     

Second-Order Smoothing Objective Penalty Function for Constrained Optimization Problems

In this article, a novel objective penalty function as well as its second-order smoothing is introduced for constrained optimization problems (COP). It is shown that an optimal solution to the second-order smoothing objective penalty optimization problem is an optimal solution to the original optimization problem under some mild conditions. Based on the second-order smoothing objective penalty function, an algorithm that has better convergence is introduced. Numerical examples illustrate that this algorithm is efficient in solving COP.     

On the Smoothing of the Square-Root Exact Penalty Function for Inequality Constrained Optimization

In this paper we propose two methods for smoothing a nonsmooth square-root exact penalty function for inequality constrained optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem and of the original optimization problem. We develop an algorithm for solving the optimization problem based on the smoothed penalty function and prove the convergence of the algorithm. The efficiency of the smoothed penalty function is illustrated with some numerical examples, which show that the algorithm seems efficient.     

Existence of Exact Penalty and Its Stability for Nonconvex Constrained Optimization Problems in Banach Spaces

In this paper we use the penalty approach in order to study two constrained minimization problems. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property is stable under perturbations of cost functions, constraint functions and the right-hand side of constraints.     

On Smoothing l1 Exact Penalty Function for Constrained Optimization Problems

This article introduces a smoothing technique to the l₁ exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm.     

Proximal Point Algorithm for Quasi-Convex Minimization Problems in Metric Spaces

In this paper, the proximal point algorithm for quasi-convex minimization problem in nonpositive curvature metric spaces is studied. We prove Δ-convergence of the generated sequence to a critical point (which is defined in the text) of an objective quasi-convex, proper and lower semicontinuous function with at least a minimum point as well as some strong convergence results to a minimum point with some additional conditions. The results extend the recent results of the proximal point algorithm in Hadamard manifolds and CAT(0) spaces.     

A Smoothing Objective Penalty Function Algorithm for Inequality Constrained Optimization Problems

In this article, a smoothing objective penalty function for inequality constrained optimization problems is presented. The article proves that this type of the smoothing objective penalty functions has good properties in helping to solve inequality constrained optimization problems. Moreover, based on the penalty function, an algorithm is presented to solve the inequality constrained optimization problems, with its convergence under some conditions proved. Two numerical experiments show that a satisfactory approximate optimal solution can be obtained by the proposed algorithm.     

Existence of exact penalty for optimization problems with mixed constraints in Banach spaces

In this paper we use the penalty approach in order to study constrained minimization problems in a Banach space with nonsmooth nonconvex mixed constraints. A penalty function is said to have the exact penalty property [J.-B. Hiriart-Urruty, C. Lemarechal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993] if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish sufficient conditions for the exact penalty property.     

TVS-锥度量空间中的统计收敛

本文研究TVS-锥度量空间中的统计收敛以及TVS-锥度量空间的统计完备性.令（X,E,P,d）表示一个TVS-锥度量空间.利用定义在有序Hausdorff拓扑向量空间E上的Minkowski函数ρ,证明了在X上存在一个通常意义下的度量d_ρ,使得X中的序列（x_n）在锥度量d意义下统计收敛到x ∈ X,当且仅当（x_n）在度量d_ρ意义下统计收敛到x.基于此,我们证明了任意一个TVS-锥统计Cauchy序列是几乎处处TVS-锥Cauchy序列,还证明了任意一个TVS-锥统计收敛的序列是几乎处处TVS-锥收敛的.从而,TVS-锥度量空间（X,d）是d-完备的,当且仅当它是d-统计完备的.基于以上结论,通常度量空间中统计收敛的许多性质都可以平行地推广到锥度量空间中统计收敛的情形.     

Existence theorems for variational inclusion problems and the set-valued vector Ekeland variational principle in a complete metric space

In this paper, we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in Ekeland’s sense and the existence results for some variants of set-valued vector Ekeland variational principles in a complete metric space. Our results contain Ekeland’s variational principle as a special case and our approaches are different to those for any existence theorems for such problems.     

Ekeland变分原理的一种推广

研究完备度量空间中一类拟均衡问题的可解性,由此导出名的Ekeland变分原理。     

Algorithms and Convergence Theorems for Mixed Equilibrium Problems in Hilbert Spaces

In this article, we study mixed equilibrium problems, and present algorithms and convergence theorems for the proposed algorithms, like proximal gradient method, Tikhonov regularization method, Mann’s type method, conjugate gradient method. As application, we study minimization problems.     

一类约束不可微优化问题的区间极大熵方法

本文研究求解不等式约束离散ｍｉｎｉｍａｘ问题的区间算法,其中目标函数和约束函数是 Ｃ~１类函数．利用罚函数法和极大熵函数思想将问题转化为无约束可微优化问题,讨论了极大熵函数的区间扩张,证明了收敛性等性质,提出了无解区域删除原则,建立了区间极大熵算法,并给出了数值算例．该算法是收敛、可靠和有效的．     

Fixed Point Theorems in Partially Ordered Metric Spaces and Existence Results for Integral Equations

We derive some new coincidence and common fixed point theorems for self-mappings satisfying a generalized contractive condition in partially ordered metric spaces. As applications of the presented theorems, we obtain fixed point results for generalized contraction of integral type and we prove an existence theorem for solutions of a system of integral equations.     

非紧L-凸度量空间中的一般拟平衡问题组(英文)

本文建立了非紧完备L-凸度量空间中新的不动点定理.作为应用,获得了非紧完备L-凸度量空间中的一般拟平衡问题组和拟平衡问题组的平衡存在定理.     

The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.     

一类Ekeland变分原理的推广

本文给出一个向量值形式的Ekeland变分原理,其目标函数是从完备的度量空间映到锥序拓扑线性空间.此结果是对钟承奎建立的Ekeland变分原理的推广.