Zun-Quan Xia

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.

Ellipsoidal Collapse and Previrialization

We study the non-linear evolution of a dust ellipsoid,embedded in a Friedmann flat background universe, in order to determine the evolution of the density of the ellipsoid as the perturbation to it related detaches from general expansion and begins to collapse. We show that while the growth rate of the density contrast of a mass element is enhanced by the shear in agreement with Hoffman (1986a), the angular momentum acquired by the ellipsoid has the right magnitude to counterbalance the effect of the shear. The result confirms the previrialization conjecture (Peebles & Groth 1976; Davis & Peebles 1977; Peebles 1990) by showing that initial asphericities and tidal interactions begin to slow down the collapse after the system has broken away from the general expansion.

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

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.

Finding general solutions of the Quasi-Newton equation via the ABS approach

In this paper we use the ABS algorithm to derive general and special solutions of single or multiple secant equations in Quasi-Newton methods for nonlinear equations or nonlinear optimization. The case where sparsity conditions are present is also considered.

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 decomposition algorithm for convex minimization

For nonsmooth convex optimization, Robert Mifflin and Claudia Sagastizabal introduce a VU-space decomposition algorithm in Mifflin and Sagastizabal (2005) [11]. An attractive property of this algorithm is that if a primal-dual track exists, this algorithm uses a bundle subroutine. With the inclusion of a simple line search, it is proved to be globally and superlinearly convergent. However, a drawback is that it needs the exact subgradients of the objective function, which is expensive to compute. In this paper an approximate decomposition algorithm based on proximal bundle-type method is introduced that is capable to deal with approximate subgradients. It is shown that the sequence of iterates generated by the resulting algorithm converges to the optimal solutions of the problem. Numerical tests emphasize the theoretical findings.

A superlinear space decomposition algorithm for constrained nonsmooth convex program

A class of constrained nonsmooth convex optimization problems, that is, piecewise C^2 convex objectives with smooth convex inequality constraints are transformed into unconstrained nonsmooth convex programs with the help of exact penalty function. The objective functions of these unconstrained programs are particular cases of functions with primal-dual gradient structure which has connection with VU space decomposition. Then a VU space decomposition method for solving this unconstrained program is presented. This method is proved to converge with local superlinear rate under certain assumptions. An illustrative example is given to show how this method works.

Existence of Solutions and Algorithm for a System of Variational Inequalities

We obtain some existence results for a system of variational inequalities (for short, denoted by SVI) by Brouwer fixed point theorem. We also establish the existence and uniqueness theorem using the projection technique for the SVI and suggest an iterative algorithm and analysis convergence of the algorithm.