Calmness and Exact Penalization in Constrained Scalar Set-Valued Optimization

In this paper, we study a class of constrained scalar set-valued optimization problems, which includes scalar optimization problems with cone constraints as special cases. We introduce (local) calmness of order??? for this class of constrained scalar set-valued optimization problems. We show that the (local) calmness of order??? is equivalent to the existence of a (local) exact set-valued penalty map.     

Duality and Exact Penalization for Vector Optimization via Augmented Lagrangian

In this paper, we introduce an augmented Lagrangian function for a multiobjective optimization problem with an extended vector-valued function. On the basis of this augmented Lagrangian, set-valued dual maps and dual optimization problems are constructed. Weak and strong duality results are obtained. Necessary and sufficient conditions for uniformly exact penalization and exact penalization are established. Finally, comparisons of saddle-point properties are made between a class of augmented Lagrangian functions and nonlinear Lagrangian functions for a constrained multiobjective optimization problem.     

Well-Posedness and Optimization under Perturbations

Estimates of the size of sets of approximate solutions are obtained for well-posed optimization problems in a Banach space, and extended to problems subject to perturbations of a general form. An estimate of the perturbations guaranteeing a prescribed level of suboptimality is presented.     

Some Results about Duality and Exact Penalization

In this paper, we introduce the concept of the valley at 0 augmenting function and apply it to construct a class of valley at 0 augmented Lagrangian functions. We establish the existence of a path of optimal solutions generated by valley at 0 augmented Lagrangian problems and its convergence toward the optimal set of the original problem and obtain the zero duality gap property between the primal problem and the valley at 0 augmented Lagrangian dual problem. Moreover, we establish the exact penalization representation results in the framework of valley at 0 augmented Lagrangian.     

Duality and Exact Penalization for General Augmented Lagrangians

We consider a problem of minimizing an extended real-valued function defined in a Hausdorff topological space. We study the dual problem induced by a general augmented Lagrangian function. Under a simple set of assumptions on this general augmented Lagrangian function, we obtain strong duality and existence of exact penalty parameter via an abstract convexity approach. We show that every cluster point of a sub-optimal path related to the dual problem is a primal solution. Our assumptions are more general than those recently considered in the related literature.     

集值映射向量优化问题的锥弱有效解的镇定性和稳定性(英文)

本文研究了集值映射向量优化问题的锥弱有效解的镇定性和稳定性,我们引进了集值映射向量优化问题的镇定性和稳定性的定义,并证明了集值映射向量优化问题的镇定性和稳定性的一些主要定理.     

Calmness of Linear Constraint Systems under Structured Perturbations with an Application to the Path-Following Scheme

Set-Valued and Variational Analysis - We are concerned with finite linear constraint systems in a parametric framework where the right-hand side is an affine function of the perturbation parameter....     

集值映射向量优化问题的锥弱有效解的镇定性和稳定性

本文研究了集值映射向量优化问题的锥弱有效解的镇定性和稳定性,我们引进了集值映射向量优化问题的镇定性和稳定性的定义,并证明了集值映射问题优化问题的镇定性和稳定性的一些主要定理。     

An Approximate Exact Penalty in Constrained Vector Optimization on Metric Spaces

In this paper, we use the penalty approach in order to study a class of constrained vector minimization problems on complete metric spaces. A penalty function is said to have the generalized exact penalty property iff 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. For our class of problems, we establish the generalized exact penalty property and obtain an estimation of the exact penalty.     

Semi-Infinite Optimization under Convex Function Perturbations: Lipschitz Stability

This paper is devoted to the study of the stability of the solution map for the parametric convex semi-infinite optimization problem under convex function perturbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the problems under consideration reduce to linear semi-infinite optimization. Examples are given to illustrate the obtained results.     

Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this paper, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.     

非线性脉冲扰动下带强迫项的次线性时滞微分系统解的渐近性

讨论了带强迫项的次线性时滞微分系统在非线性脉冲扰动下系统解的渐近性,得到了带强迫项的次线性时滞微分系统在非线性脉冲扰动下系统解渐近吸引的充分性条件.     

Convergence Analysis of a Class of Nonlinear Penalization Methods for Constrained Optimization via First-Order Necessary Optimality Conditions

We propose a scheme to solve constrained optimization problems by combining a nonlinear penalty method and a descent method. A sequence of nonlinear penalty optimization problems is solved to generate a sequence of stationary points, i.e., each point satisfies a first-order necessary optimality condition of a nonlinear penalty problem. Under some conditions, we show that any limit point of the sequence satisfies the first-order necessary condition of the original constrained optimization problem.     

Primal-Dual Solution Perturbations in Convex Optimization

Solutions to optimization problems of convex type are typically characterized by saddle point conditions in which the primal vector is paired with a dual multiplier vector. This paper investigates the behavior of such a primal-dual pair with respect to perturbations in parameters on which the problem depends. A necessary and sufficient condition in terms of certain matrices is developed for the mapping from parameter vectors to saddle points to be single-valued and Lipschitz continuous locally. It is shown that the saddle point mapping is then semi-differentiable, and that its semi-derivative at any point and in any direction can be calculated by determining the unique solutions to an auxiliary problem of extended linear-quadratic programming and its dual. A matrix characterization of calmness of the solution mapping is provided as well.     

On Nonlinear Perturbations of Nonlinear Dynamical Systems, and Applications to Control

The nonlinear variation-of-constants formula is generalizedto infinite-dimensional systems and applied to the stabilityproblem and used to obtain bounds on the states of a nonlinearcontrol system when the controls are bounded. The theory isillustrated by a simple model from nuclear reactor dynamics.     

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.