Liwei Zhang

THE IMPLICIT LX METHOD OF THE ABS CLASS

We describe an algorithm of the ABS class, which solves a general qonsingular linear system in n 3/3 + 0(n 2) multiplications without the assumption that the coefficient matrix be regular. The method can be viewed as a variation of the implicit LU algorithm of the ABS class, whose associated factorization contains a factor which is not triangular (but can be reduced to triangular form after suitable row permutations). We describe king the Abaffan properties of the method, including in particular an efficient way of upd matrix after column interchanges. Such a problem arises in the application to the simplex algorithm, where the implicit LX algorithm provides a faster technique than the standard LU factorization for the pivoting operation if the number of equality constraints m is greater than n/2

A Perturbation approach for an inverse quadratic programming problem

We consider an inverse quadratic programming (QP) problem in which the parameters in both the objective function and the constraint set of a given QP problem need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a linear complementarity constrained minimization problem with a positive semidefinite cone constraint. With the help of duality theory, we reformulate this problem as a linear complementarity constrained semismoothly differentiable (SC1) optimization problem with fewer variables than the original one. We propose a perturbation approach to solve the reformulated problem and demonstrate its global convergence. An inexact Newton method is constructed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. As the objective function of the problem is a SC1 function involving the projection operator onto the cone of positively semi-definite symmetric matrices, the analysis requires an implicit function theorem for semismooth functions as well as properties of the projection operator in the symmetric-matrix space. Since an approximate proximal point is required in the inexact Newton method, we also give a Newton method to obtain it. Finally we report our numerical results showing that the proposed approach is quite effective.

ABS algorithms for linear equations and optimization

In this paper we review basic properties and the main achievements obtained by the class of ABS methods, developed since 1981 to solve linear and nonlinear algebraic equations and optimization problems.

Quantitative Stability Analysis for Distributionally Robust Optimization with Moment Constraints

In this paper we consider a broad class of distributionally robust optimization (DRO) problems where the probability of the underlying random variables depends on the decision variables and the ambiguity set is defined through parametric moment conditions with generic cone constraints. Under some moderate conditions, including Slater-type conditions of a cone constrained moment system and Holder continuity of the underlying random functions in the objective and moment conditions, we show local Holder continuity of the optimal value function of the inner maximization problem with respect to (w.r.t.) the decision vector and other parameters in moment conditions, and local Holder continuity of the optimal value of the whole minimax DRO w.r.t. the parameter. Moreover, under the second order growth condition of the Lagrange dual of the inner maximization problem, we demonstrate and quantify the outer semicontinuity of the set of optimal solutions of the minimax DRO w.r.t. variation of the parameter. Finally, w...

A perturbation approach for a type of inverse linear programming problems

We consider an inverse linear programming (LP) problem in which the parameters in both the objective function and the constraint set of a given LP problem need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a linear complementarity constrained minimization problem. With the help of the smoothed Fischer–Burmeister function, we propose a perturbation approach to solve the inverse problem and demonstrate its global convergence. An inexact Newton method is constructed to solve the perturbed problem and numerical results are reported to show the effectiveness of the approach.

A class of nonlinear Lagrangians for nonconvex second order cone programming

This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function.

Convergence Analysis for Mathematical Programs with Distributionally Robust Chance Constraint

Convergence analysis for optimization problems with chance constraints concerns impact of variation of probability measure in the chance constraints on the optimal value and the optimal solutions and research on this topic has been well documented in the literature of stochastic programming. In this paper, we extend such analysis to optimization problems with distributionally robust chance constraints where the true probability distribution is unknown, but it is possible to construct an ambiguity set of probability distributions and the chance constraint is based on the most conservative selection of probability distribution from the ambiguity set. The convergence analysis focuses on impact of the variation of the ambiguity set on the optimal value and the optimal solutions. We start by deriving general convergence results under abstract conditions such as continuity of the robust probability function and uniform convergence of the robust probability functions and followed with detailed analysis of these ...

Quantitative stability analysis of stochastic quasi-variational inequality problems and applications

We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is defined over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem, we effectively convert stability of the SQVIP into that of a one stage stochastic program with stochastic cone constraints. Under some moderate conditions, we derive Hölder outer semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter. The stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with SQVIP constraints.

Smoothing augmented Lagrangian method for nonsmooth constrained optimization problems

In this paper, we propose a smoothing augmented Lagrangian method for finding a stationary point of a nonsmooth and nonconvex optimization problem. We show that any accumulation point of the iteration sequence generated by the algorithm is a stationary point provided that the penalty parameters are bounded. Furthermore, we show that a weak version of the generalized Mangasarian Fromovitz constraint qualification (GMFCQ) at the accumulation point is a sufficient condition for the boundedness of the penalty parameters. Since the weak GMFCQ may be strictly weaker than the GMFCQ, our algorithm is applicable for an optimization problem for which the GMFCQ does not hold. Numerical experiments show that the algorithm is efficient for finding stationary points of general nonsmooth and nonconvex optimization problems, including the bilevel program which will never satisfy the GMFCQ.

A sample average approximation regularization method for a stochastic mathematical program with general vertical complementarity constraints

Based on the log-exponential function, a sample average approximation (SAA) regularization method is proposed for solving a stochastic mathematical program with general vertical complementarity constraints (SMPVCC) considered by Birbil et al. (2006). Detailed convergence analysis of this method is investigated. It is demonstrated that under some regularity conditions, any accumulation point of the sequence of optimal solutions of SAA regularized problem is almost surely an optimal solution of the SMPVCC as the parameter tends to zero and the sample size tends to infinity. Furthermore, the optimal value sequence of SAA regularized problem converges to the optimal value of SMPVCC with exponential convergence rate with probability one and a sequence of stationary points of regularized SAA problem converges almost surely to a stationary point of SMPVCC. Finally, we show that a stochastic Stackelberg game can be formulated as a SMPVCC problem and an equilibrium solution can be obtained by the method proposed.