An Efficient Algorithm for Min-Max Convex Semi-Infinite Programming Problems

In this article, we consider the convex min-max problem with infinite constraints. We propose an exchange method to solve the problem by using efficient inactive constraint dropping rules. There is no need to solve the maximization problem over the metric space, as the algorithm has merely to find some points in the metric space such that a certain criterion is satisfied at each iteration. Under some mild assumptions, the proposed algorithm is shown to terminate in a finite number of iterations and to provide an approximate solution to the original problem. Preliminary numerical results with the algorithm are promising. To our knowledge, this article is the first one conceived to apply explicit exchange methods for solving nonlinear semi-infinite convex min-max problems.     

一个求解箱约束二次规划的非内点预估校正算法

利用Chen-Harker-Kanzow-Smale光滑技术,给出了一个求解箱约束二次规划的预估校正的算法,它是Xu‘s方程的进一步研究,它的思想是将问题的K-T条件转化成一组光滑的等式,再用预估校正方法求解.同现存的算法相比,该算法具有较快的收敛速度,且所需的条件相对较弱.本文改进了该领域内的一些最新结果.     

Decomposition Method with a Variable Parameter for a Class of Monotone Variational Inequality Problems

In this paper, we focus on a useful modification of the decomposition method by He et al. (Ref. 1). Experience on applications has shown that the number of iterations of the original method depends significantly on the penalty parameter. The main contribution of our method is that we allow the penalty parameter to vary automatically according to some self-adaptive rules. As our numerical simulations indicate, the modified method is more flexible and efficient in practice. A detailed convergence analysis of our method is also included.     

A Global Optimization Method for Solving Convex Quadratic Bilevel Programming Problems

We use the merit function technique to formulate a linearly constrained bilevel convex quadratic problem as a convex program with an additional convex-d.c. constraint. To solve the latter problem we approximate it by convex programs with an additional convex-concave constraint using an adaptive simplicial subdivision. This approximation leads to a branch-and-bound algorithm for finding a global optimal solution to the bilevel convex quadratic problem. We illustrate our approach with an optimization problem over the equilibrium points of an n-person parametric noncooperative game.     

A Non-Interior Path Following Method for Convex Quadratic Programming Problems with Bound Constraints

We propose a non-interior path following algorithm for convex quadratic programming problems with bound constraints based on Chen-Harker-Kanzow-Smale smoothing technique. Conditions are given under which the algorithm is globally convergent or globally linearly convergent. Preliminary numerical experiments indicate that the method is promising.     

Penalty and Barrier Methods for Convex Semidefinite Programming

In this paper we present penalty and barrier methods for solving general convex semidefinite programming problems. More precisely, the constraint set is described by a convex operator that takes its values in the cone of negative semidefinite symmetric matrices. This class of methods is an extension of penalty and barrier methods for convex optimization to this setting. We provide implementable stopping rules and prove the convergence of the primal and dual paths obtained by these methods under minimal assumptions. The two parameters approach for penalty methods is also extended. As for usual convex programming, we prove that after a finite number of steps all iterates will be feasible.     

An Exterior Newton Method for Strictly Convex Quadratic Programming

We propose an exterior Newton method for strictly convex quadratic programming (QP) problems. This method is based on a dual formulation: a sequence of points is generated which monotonically decreases the dual objective function. We show that the generated sequence converges globally and quadratically to the solution (if the QP is feasible and certain nondegeneracy assumptions are satisfied). Measures for detecting infeasibility are provided. The major computation in each iteration is to solve a KKT-like system. Therefore, given an effective symmetric sparse linear solver, the proposed method is suitable for large sparse problems. Preliminary numerical results are reported.     

Generalized Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems

This article deals with a generalized semi-infinite programming problem (S). Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a min-max problem constrained with compact convex linked constraints.     

Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming

In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ℝ₊₊. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538–543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the proposed algorithm is bounded and that every accumulation point is a solution to the considered problem. Research of Shaohua Pan was partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. Jein-Shan Chen is a member of the Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work was partially supported by National Science Council of Taiwan.     

A Dual Parametrization Method for Convex Semi-Infinite Programming

We formulate convex semi-infinite programming problems in a functional analytic setting and derive optimality conditions and several duality results, based on which we develop a computational framework for solving convex semi-infinite programs.     

Modified Combined Relaxation Method for General Monotone Equilibrium Problems in Hilbert Spaces

In this paper, we study a class of general monotone equilibrium problems in a real Hilbert space which involves a monotone differentiable bifunction. For such a bifunction, a skew-symmetric type property with respect to the partial gradients is established. We suggest to solve this class of equilibrium problems with the modified combined relaxation method involving an auxiliary procedure. We prove the existence and uniqueness of the solution to the auxiliary variational inequality in the auxiliary procedure. Further, we prove also the weak convergence of the modified combined relaxation method by virtue of the monotonicity and the skew-symmetric type property.Communicated by F. GiannessiHis research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation in Shanghai.His research was partially supported by a grant from the National Science Council of Taiwan.     

Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems

For a convex program in a normed vector space with the objective function admitting the Gateaux derivative at an optimal solution, we show that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the Gateaux derivative. For a general continuous convex program, a feasible point is an optimal solution iff it lies in a hyperplane with a normal vector belonging to the subdifferential of the objective function at this point. In several cases, the solution set of a variational inequality problem is shown to coincide with the solution set of a convex program with its dual gap function as objective function, while the mapping involved can be used to express the above normal vectors.The research was supported by the National Science Council of the Republic of China. The authors are grateful to the referees for valuable comments and constructive suggestions.     

Convergence of stationary sequences for variational inequalities with maximal monotone operators

LetT be a maximal monotone operator defined on ^N . In this paper we consider the associated variational inequality 0 T(x ^*) and stationary sequences {x _k ^* for this operator, i.e., satisfyingT(x _k ^* 0. The aim of this paper is to give sufficient conditions ensuring that these sequences converge to the solution setT ^–1(0) especially when they are unbounded. For this we generalize and improve the directionally local boundedness theorem of Rockafellar to maximal monotone operatorsT defined on ^N .     

A Primal-Dual Decomposition Algorithm for Multistage Stochastic Convex Programming

This paper presents a new and high performance solution method for multistage stochastic convex programming. Stochastic programming is a quantitative tool developed in the field of optimization to cope with the problem of decision-making under uncertainty. Among others, stochastic programming has found many applications in finance, such as asset-liability and bond-portfolio management. However, many stochastic programming applications still remain computationally intractable because of their overwhelming dimensionality. In this paper we propose a new decomposition algorithm for multistage stochastic programming with a convex objective and stochastic recourse matrices, based on the path-following interior point method combined with the homogeneous self-dual embedding technique. Our preliminary numerical experiments show that this approach is very promising in many ways for solving generic multistage stochastic programming, including its superiority in terms of numerical efficiency, as well as the flexibility in testing and analyzing the model.Research supported by Hong Kong RGC Earmarked Grant CUHK4233/01E.     

Decomposition Method for a Class of Monotone Variational Inequality Problems

In the solution of the monotone variational inequality problem VI(, F), with

A Class of Infeasible Interior Point Algorithms for Convex Quadratic Programming

This article proposes a class of infeasible interior point algorithms for convex quadratic programming, and analyzes its complexity. It is shown that this algorithm has the polynomial complexity. Its best complexity is O(nL).     

凸函数的若干新性质及应用

本文证明了凸函数的若干新性质 ,讨论了这些性质在求解线性与非线性不等式组和线性规划中的应用 ,为线性与非线性不等式组、线性规划的求解提供了一种新方法 .     

Reflection-Projection Method for Convex Feasibility Problems with an Obtuse Cone

The convex feasibility problem asks to find a point in the intersection of finitely many closed convex sets in Euclidean space. This problem is of fundamental importance in the mathematical and physical sciences, and it can be solved algorithmically by the classical method of cyclic projections.In this paper, the case where one of the constraints is an obtuse cone is considered. Because the nonnegative orthant as well as the set of positive-semidefinite symmetric matrices form obtuse cones, we cover a large and substantial class of feasibility problems. Motivated by numerical experiments, the method of reflection-projection is proposed: it modifies the method of cyclic projections in that it replaces the projection onto the obtuse cone by the corresponding reflection.This new method is not covered by the standard frameworks of projection algorithms because of the reflection. The main result states that the method does converge to a solution whenever the underlying convex feasibility problem is consistent. As prototypical applications, we discuss in detail the implementation of two-set feasibility problems aiming to find a nonnegative [resp. positive semidefinite] solution to linear constraints in ⁿ [resp. in , the space of symmetric n×n matrices] and we report on numerical experiments. The behavior of the method for two inconsistent constraints is analyzed as well.     

A first-order continuous method for the Antipin regularization of monotone variational inequalities in a Banach space

The concept of a generalized projection operator onto a convex closed subset of a Banach space is modified. This operator is used to construct a first-order continuous method for the Antipin regularization of monotone variational inequalities in a Banach space. Sufficient conditions for the convergence of the method are found.     

A Simple Primal-Dual Feasible Interior-Point Method for Nonlinear Programming with Monotone Descent

We propose and analyze a primal-dual interior point method of the feasible type, with the additional property that the objective function decreases at each iteration. A distinctive feature of the method is the use of different barrier parameter values for each constraint, with the purpose of better steering the constructed sequence away from non-KKT stationary points. Assets of the proposed scheme include relative simplicity of the algorithm and of the convergence analysis, strong global and local convergence properties, and good performance in preliminary tests. In addition, the initial point is allowed to lie on the boundary of the feasible set.