A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS:I.ALGORITHM AND GLOBAL CONVERGENCEXIU NAIHUA

A class of trust region methods for solving linear inequality constrained problems is proposed in this paper. It is shown that the algorithm is of global convergence.The algorithm uses a version of the two-sided projection and the strategy of the unconstrained trust region methods. It keeps the good convergence properties of the unconstrained case and has the merits of the projection method. In some sense, our algorithm can be regarded as an extension and improvement of the projected type algorithm.

低秩矩阵优化若干新进展

低秩矩阵优化是一类含有秩极小或秩约束的矩阵优化问题,在统计与机器学习、信号与图像处理、通信与量子计算、系统识别与控制、经济与金融等众多学科领域有着广泛应用,是当前最优化及其相关领域的一个重点研究方向.然而,低秩矩阵优化是一个NP-难的非凸非光滑优化问题,其研究成果并非十分丰富,亟待进一步深入研究.主要从理论和算法两个方面总结和评述若干新结果,同时列出相关的重要文献,奉献给读者.

A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS Ⅱ．LOCAL CONVERGENCE RATE AND NUMERICAL TESTS

In this paper we prove that a class of trust region methods presented in part I is superlinearly convergent. Numerical tests are reported thereafter. Results by solving a set of typical problems selected from literatures have demonstrated that our algorithm is effective.

Optimality Conditions for Double-sparsity Constrained Optimization

Sparse optimization has witnessed advancements in recent decades,and the step function finds extensive applications across various machine learning and signal processing domains.This paper integrates zero norm and the step function to formulate a doublesparsity constrained optimization problem,wherein a linear equality constraint is also taken into consideration.By defining aτ-Lagrangian stationary point and a KKT point,we establish the first-order and second-order necessary and sufficient optimality conditions for the problem.Furthermore,we thoroughly elucidate their relationships to local and global optimal solutions.Finally,special cases and examples are presented to illustrate the obtained theorems.

Optimality conditions for sparse nonlinear programming

The sparse nonlinear programming （SNP） is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Frechet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn- Tucker （KKT） conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.

Path-following interior point algorithms for the Cartesian P_*(κ)-LCP over symmetric cones

In this paper, we establish a theoretical framework of path-following interior point al- gorithms for the linear complementarity problems over symmetric cones (SCLCP) with the Cartesian P*(κ)-property, a weaker condition than the monotonicity. Based on the Nesterov-Todd, xy and yx directions employed as commutative search directions for semidefinite programming, we extend the variants of the short-, semilong-, and long-step path-following algorithms for symmetric conic linear programming proposed by Schmieta and Alizadeh to the Cartesian P*(κ)-SCLCP, and particularly show the global convergence and the iteration complexities of the proposed algorithms.

Half thresholding eigenvalue algorithm for semidefinite matrix completion

The semidefinite matrix completion(SMC) problem is to recover a low-rank positive semidefinite matrix from a small subset of its entries. It is well known but NP-hard in general. We first show that under some cases, SMC problem and S1/2relaxation model share a unique solution. Then we prove that the global optimal solutions of S1/2regularization model are fixed points of a symmetric matrix half thresholding operator. We give an iterative scheme for solving S1/2regularization model and state convergence analysis of the iterative sequence.Through the optimal regularization parameter setting together with truncation techniques, we develop an HTE algorithm for S1/2regularization model, and numerical experiments confirm the efficiency and robustness of the proposed algorithm.

An approximation algorithm for the k-median warehouse-retailer network design problem

We study the generalizedk-median version of the warehouse-retailer network design problem(kWRND).We formulate the k-WRND as a binary integer program and propose a 6-approximation randomized algorithm based on Lagrangian relaxation.