Li-Ping Pang

An algorithm for solving optimization problems with fuzzy relational inequality constraints

An algorithm for solving a kind of optimization problems with fuzzy relational inequalities was proposed by Guo and Xia [6]. However, it is too expensive to verify the optimal condition. In this paper, some rules for reducing these problems are proposed and the relationship between minimal solutions and FRI paths is also given. These lead to a new algorithm for solving this kind of problems. Numerical experiments are presented for illustrating the efficiency of the proposed algorithm.

A method for solving the system of linear equations and linear inequalities

A method, called the (I.) ABS-MPVT algorithm, for solving a system comprising linear equations and linear inequalities is presented. This method is characterized by solving the system of linear equations first via the ABS algorithms and then solving an unconstrained minimization obtained by substituting the ABS general form of solutions into the system of linear inequalities. For the unconstrained minimization problem it can be solved by a (modified) parallel algorithm. The convergence of this method is also given.

Stochastic methods based on Newton method to the stochastic variational inequality problem with constraint conditions

Abstract Two stochastic methods for solving a class of stochastic variational inequality problems (SVIPs) are presented, using the stochastic approximation (SA) method and the sample average approximation (SAA) method. They are constructed by SA and SAA methods based on the Newton method where the underlying functions are the expected values of stochastic functions. Local convergences are given under Lipschitzian conditions. Numerical experiments show that the proposed methods are efficient.

An approximate bundle-type auxiliary problem method for solving generalized variational inequalities

By combining the bundle idea with the auxiliary problem method we present an algorithm for solving a generalized variational inequality problem (GVIP). The proposed algorithm only requires the approximate function values and the approximate subgradients of the involved function instead of the exact ones at some points, which makes the algorithm easier to implement. In addition to that, the conditions imposed on the auxiliary functions are weaker than the preexisting results: convex is enough, not necessarily strongly convex. We also prove the weak convergence of the proposed algorithm under some conditions.

A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information

We propose an inexact proximal bundle method for constrained nonsmooth nonconvex optimization problems whose objective and constraint functions are known through oracles which provide inexact information. The errors in function and subgradient evaluations might be unknown, but are merely bounded. To handle the nonconvexity, we first use the redistributed idea, and consider even more difficulties by introducing inexactness in the available information. We further examine the modified improvement function for a series of difficulties caused by the constrained functions. The numerical results show the good performance of our inexact method for a large class of nonconvex optimization problems. The approach is also assessed on semi-infinite programming problems, and some encouraging numerical experiences are provided.

An adaptive fixed-point proximity algorithm for solving total variation denoising models

Abstract We study an adaptive fixed-point proximity algorithm to solve the total variation denoising model. The objective function is a sum of two convex functions and one of them is composed by an affine transformation, which is usually a regularization term. By decoupling and splitting, the problem is changed into two subproblems. Due to the nonsmooth and nondifferentiability of the subproblem, we solve its proximity minimization problem instead of the original one. To overcome the “staircase” effect during the process of denoising, an adaptive criterion on proximity parameters is put forward. At last we apply the improved algorithm to solve the isotropic total variation denoising model. The numerical results are given to illustrate the efficiency of the algorithm.

Sensitivity Analysis of Multiobjective Optimization Problems with Parameterized Quasi-Variational Inequalities

ABSTRACT This paper mainly establishes the sensitivity analysis of a multiobjective optimization problem with parameterized quasi-variational inequalities (QVIs). Using the (regular) coderivative of the associated epigraphical multifunction, the (regular) subdifferentials of the efficient frontier maps are estimated, which involve the (regular) coderivatives of the solution mapping to the parameterized QVIs. Under the linear independent constraint qualification, the defined auxiliary set-valued mappings in the parameterized QVIs are clam. The detailed formulae of subdifferentials of the efficient frontier maps are obtained and examples are simultaneously provided for analyzing and illustrating the obtained results.

An infeasible bundle method for nonconvex constrained optimization with application to semi-infinite programming problems

The main difficulty for solving semi-infinite programming (SIP) problem is precisely that it has infinitely many constraints. By using a maximum function, the SIP problem can be rewritten as a nonconvex nonsmooth constrained optimization (NNCO) problem. Global convergence in most of constrained optimization algorithms has traditionally been enforced by the use of a penalty function or filter strategy. In this paper, we propose an infeasible bundle method for NNCO problem based on the so-called improvement functions, without a penalty function and filter strategy. The method appears to be more direct and easier to implement, in the sense that it is closer in spirit and structure to the well-developed unconstrained bundle methods. Under a special constraint qualification, the sequence generated by this algorithm converges to the KKT point of the NNCO problem as well as the SIP problems. Preliminary numerical results show that this algorithm is robust and efficient for NNCO problems and SIP problems.

Inexact SA Method for Constrained Stochastic Convex SDP and Application in Chinese Stock Market

We propose stochastic convex semidefinite programs (SCSDPs) to handle uncertain data in applications. For these models, we design an efficient inexact stochastic approximation (SA) method and prove the convergence, complexity, and robust treatment of the algorithm. We apply the inexact method for solving SCSDPs where the subproblem in each iteration is only solved approximately and show that it enjoys the similar iteration complexity as the exact counterpart if the subproblems are progressively solved to sufficient accuracy. Numerical experiments show that the method we proposed was effective for uncertain problem.

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.