Scalar Lagrange Multiplier Rules for Set-Valued Problems in Infinite-Dimensional Spaces

This paper deals with Lagrange multiplier rules for constrained set-valued optimization problems in infinite-dimensional spaces, where the multipliers appear as scalarization functions of the maps instead of the derivatives. These rules provide necessary conditions for weak minimizers under hypotheses of stability, convexity, and directional compactness. Counterexamples show that the hypotheses are minimal.     

β-光滑Banach空间中一类数学规划问题的最优必要条件

在一般Banach空间中，提出数学规划问题的偏静态条件定义，在目标函数和约束函数均为扩张值下半连续函数的情形下，获得了罚问题解的一个结果，该结果改进和推广了已有的相应结果，并由此证明了β-光滑Banach空间中的模糊乘子规则，从而找出该类数学规划问题的最优必要条件。     

Conjugate Duality in Constrained Set-Valued Vector Optimization Problems

In this article, under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are proposed for a constrained set-valued vector optimization problem. Weak duality, strong duality, and stability criteria are investigated. The inclusion relations between the image sets of the dual problems are also discussed.     

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.     

Banach空间中集值测度的表示定理

本文在可分自反Ｒａｎａｃｈ空的的情形下，给出了任何一列两两等比、一致有界的矢值测度可以生成一个有界闭凸值集值测度的所谓表示定理，而这个定现对κ空间首先在［３］中建立。同时，找到了由一列两两等比、一致有界变差矢值测度所生成集值测度与这列矢值测度Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ导数之间的关系。     

Eigenvalues of Set-Valued Operators in Banach Spaces

This work deals with the spectral analysis of set-valued operators from a Banach space X into its dual space X^*. The main goal of the paper is to study semicontinuity properties of the spectrum operator.Mathematics Subject Classifications (2000) 47H04, 47H12, 58C40.UMR 2071 Universidad de Chile-CNRS.     

Constrained Approximation in Banach Spaces

   Abstract. We consider the problem of approximating vectors from a complemented subspace Z ₊ of a Banach space X by the projections onto Z ₊ of vectors from a subspace Y ₊ with a norm constraint on their projections onto the complementary subspace. Sufficient conditions are found for the existence of a unique best approximant and a characterization via a critical point equation is provided, thus extending known results on Hilbert spaces. These results are then applied in the case that X is L ^p (T), where T denotes the unit circle, Z ₊ consists of functions supported on a subset of the circle, and Y ₊ is the corresponding Hardy space.     

Constrained Approximation in Banach Spaces

Abstract. We consider the problem of approximating vectors from a complemented subspace Z ₊ of a Banach space X by the projections onto Z ₊ of vectors from a subspace Y ₊ with a norm constraint on their projections onto the complementary subspace. Sufficient conditions are found for the existence of a unique best approximant and a characterization via a critical point equation is provided, thus extending known results on Hilbert spaces. These results are then applied in the case that X is L ^p (T), where T denotes the unit circle, Z ₊ consists of functions supported on a subset of the circle, and Y ₊ is the corresponding Hardy space.     

Lagrange Multiplier Rules for Weak Approximate Pareto Solutions of Constrained Vector Optimization Problems in Hilbert Spaces

In the Hilbert space case, in terms of proximal normal cone and proximal coderivative, we establish a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems. In this case, our Lagrange multiplier rule improves the main result on vector optimization in Zheng and Ng (SIAM J. Optim. 21: 886–911, 2011). We also introduce a notion of a fuzzy proximal Lagrange point and prove that each Pareto (or weak Pareto) solution is a fuzzy proximal Lagrange point.     

Banach空间中广义集值变分包含问题的迭代解

本文研究Banach空间中一类广义集值变分包含问题 ,建立了广义集值变分包含问题的迭代解的一些算法 ,并统一和推广了一些最新文献中的结果 .     

Strong Convergence Theorem for Multiple Sets Split Feasibility Problems in Banach Spaces

The purpose of this article is to study the iterative approximation of solution to multiple sets split feasibility problems in p-uniformly convex real Banach spaces that are also uniformly smooth. We propose an iterative algorithm for solving multiple sets split feasibility problems and prove a strong convergence theorem of the sequence generated by our algorithm under some appropriate conditions in p-uniformly convex real Banach spaces that are also uniformly smooth.     

On the Existence and Iterative Approximation Problems of Solutions for Set-Valued Variational Inclusions in Banach Spaces

The purpose of this paper is to introduce and study a class of set-valued variational inclusions in Banach spaces. By using Michael's selection theorem and Nadler's theorem, some existence theorems and iterative algorithms for solving this kind of set-valued variational inclusion in Banach spaces are established and suggested.     

Convex Optimization on Banach Spaces

Greedy algorithms which use only function evaluations are applied to convex optimization in a general Banach space \(X\). Along with algorithms that use exact evaluations, algorithms with approximate evaluations are treated. A priori upper bounds for the convergence rate of the proposed algorithms are given. These bounds depend on the smoothness of the objective function and the sparsity or compressibility (with respect to a given dictionary) of a point in \(X\) where the minimum is attained.     

Existence of Exact Penalty for Constrained Optimization Problems in Metric Spaces

In this paper we use the penalty approach in order to study constrained minimization problems in a complete metric space with locally Lipschitzian mixed constraints. 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 sufficient conditions for the exact penalty property.      

Banach空间中广义集值拟变分包含的灵敏性分析

研究了Banach空间中一类广义集值拟变分包含问题的灵敏性分析．利用预解算子的技巧,在对给定条件没有假设可微性和单调性下,建立了这类问题与广义预解方程类的等价性．     

Strong Convergence of an Inexact Proximal Point Algorithm for Equilibrium Problems in Banach Spaces

We develop an inexact proximal point algorithm for solving equilibrium problems in Banach spaces which consists of two principal steps and admits an interesting geometric interpretation. At a certain iterate, first we solve an inexact regularized equilibrium problem with a flexible error criterion to obtain an axillary point. Using this axillary point and the inexact solution of the previous iterate, we construct two appropriate hyperplanes which separate the current iterate from the solution set of the given problem. Then the next iterate is defined as the Bregman projection of the initial point onto the intersection of two halfspaces obtained from the two constructed hyperplanes containing the solution set of the original problem. Assuming standard hypotheses, we present a convergence analysis for our algorithm, establishing that the generated sequence strongly and globally converges to a solution of the problem which is the closest one to the starting point of the algorithm.     

Banach空间中的β扰动优化

通过在Banach空间中引入一种新拓扑来证明其上的β扰动优化定理成立。     

Pareto Solutions of Polyhedral-valued Vector Optimization Problems in Banach Spaces

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra. We establish some results on structure and connectedness of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set and Pareto optimal value set of (SVOP). In particular, we improve and generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in Euclidean spaces.