Fu-Sheng Bai

An exact lower order penalty function and its smoothing in nonlinear programming

In this article, we consider a lower order penalty function and its ε-smoothing for an inequality constrained nonlinear programming problem. It is shown that any strict local minimum satisfying the second-order sufficiency condition for the original problem is a strict local minimum of the lower order penalty function with any positive penalty parameter. By using an ε-smoothing approximation to the lower order penalty function, we get a modified smooth global exact penalty function under mild assumptions.

Global optimality conditions for mixed nonconvex quadratic programs

In this article, we present some global optimality conditions for mixed quadratic programming problems. Our approach is based on a L-subdifferential and an associated L-normal cone. Unlike most subdifferentials, the L-subdifferential is formed by functions that are not necessarily linear functions. We derive some sufficient and necessary global optimality conditions for mixed quadratic programs with box constraints and binary constraints. †Dedicated in the memory of Alexander Rubinov.

A new auxiliary function method for general constrained global optimization

In this article, we first propose a method to obtain an approximate feasible point for general constrained global optimization problems (with both inequality and equality constraints). Then we propose an auxiliary function method to obtain a global minimizer or an approximate global minimizer with a required precision for general global optimization problems by locally solving some unconstrained programming problems. Some numerical examples are reported to demonstrate the efficiency of the present optimization method.

A filled function method for quadratic programs with binary constraints

In this article, we first introduce a definition of a local minimizer for the quadratic programs with {−1, 1} constraints and show that the existing necessary condition of global optimality in some recent literature is in fact the necessary and sufficient condition of local optimality under the present definition. A filled function and the corresponding filled function algorithm then are proposed to get the global minimizer of the concerned minimization problem. The existing sufficient condition of global optimality is employed to form one of the termination rules of the algorithm. Several numerical examples are reported to show the efficiency of the present method for solving quadratic programs with binary constraints. †This work is dedicated to the memory of Alexander Rubinov with whom we discussd the first draft of the article.

MONOTONIZATION IN GLOBAL OPTIMIZATION

A general monotonization method is proposed for converting a constrained programming problem with non-monotone objective function and monotone constraint functions into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavefication methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.

A Filled Function Method for Box-constrained System of Nonlinear Equations

In this paper, we present a global optimization method based on the filled function method to solve systems of nonlinear equations. Formulating a system of nonlinear equation into an equivalent global optimization problem, we manage to find a solution or an appropriate solution of the system of nonlinear equations by solving the formulated global optimization problem. A novel filled function method is proposed to solve the global optimization problem. Two numerical examples are presented to illustrate the efficiency of this method