Liansheng Zhang

A filled function method with one parameter for box constrained global optimization

In this paper, a new auxiliary function with one parameter on box constrained for escaping the current local minimizer of global optimization problem is proposed. First, a new definition of the filled function for box constrained minimization problem is given and under mild assumptions, this new auxiliary function is really a filled function. Then a new solution algorithm is proposed according to the theoretical analysis. And some numerical results demonstrate the efficiency of this method for box constrained global optimization.

On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems

In this paper, we study a new exact and smooth penalty function for the nonlinear mixed discrete programming problem by augumenting only one variable no matter how many constraints. Through the smooth and exact penalty function, we can transform the nonlinear mixed discrete programming problem into an unconstrained optimization model. We demonstrate that under mild conditions, when the penalty parameter is sufficiently large, optimizers of this penalty function are precisely the optimizers of the nonlinear mixed discrete programming problem. Alternatively, under some mild assumptions, the local exactness property is also presented. The numerical results demonstrate that the new penalty function is an effective and promising approach. As important applications, we solve an increasingly popular search engine advertising problem via the new proposed penalty function.

Global minimization of non-smooth unconstrained problems with filled function

For smooth or non-smooth unconstrained global optimization problems, an one parameter filled function is derived to identify their global optimizers or approximately global optimizers. The theoretical properties of the proposed function are investigated. Based on the filled function, an algorithm is designed for solving unconstrained global optimization problems. The algorithm consists of two phases: local minimization and filling. The former is intended to minimize the objective function and obtain a local optimizer, the latter aims to find a better initial point for the first phase. Numerical experimentation is also provided. The preliminary computational results confirm that the proposed filled function approach is promising.

Modified T-F Function Method for Finding Global Minimizer on Unconstrained Optimization

This paper indicates that the filled function which appeared in one of the papers by Y. L. Shang et al. (2007) is also a tunneling function; that is, we prove that under some general assumptions this function has the characters of both tunneling function and filled function. A solution algorithm based on this T-F function is given and numerical tests from test functions show that our T-F function method is very effective in finding better minima.

A class of augmented Lagrangians for equality constraints in nonlinear programming problems

In this paper a class of augmented Lagrangians is considered, for solving equality constrained nonlinear optimization problems via unconstrained minimization techniques. This class of augmented Lagrangians is obtained by multiplying the penalty term on the first order necessary optimality condition in a class of augmented Lagrangians of Di Pillo and Grippo by a penalty parameter. Under suitable assumptions, the exactly corresponding relationship is established between the solution of the original constrained problem and the unconstrained minimization of this class of augmented Lagrangians on the product space of problem variables and multipliers for sufficiently large but finite values of penalty parameters. Therefore, a solution of the original constrained problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of an augmented Lagrangian on the product space of problem variables and multipliers. In particular, for quadratic programming problems with equality constraints, the optimizer is obtained by minimizing a quadratic function on the expanded space.

Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming

This paper presents a filled function method for finding a global optimizer of integer programming problem. The method contains two phases: the local minimization phase and the filling phase. The goal of the former phase is to identify a local minimizer of the objective function, while the filling phase aims to search for a better initial point for the first phase with the aid of the filled function. A two-parameter filled function is proposed, and its properties are investigated. A corresponding filled function algorithm is established. Numerical experiments on several test problems are performed, and preliminary computational results are reported.

A new algorithm for constrained global optimization based on filled function

In this paper, we propose a filled function method for constrained global optimization. This filled function contains only one parameter which is easily to be chosen. Then we investigate the properties of this function and design a new algorithm based on this function. Last, we make a numerical test. The numerical results show the efficiency of this global optimization method.

A novel filled function for unconstrained global optimization

In this paper, we present a novel filled function approach for finding better minimizer of an unconstrained global optimization. The proposed new filled function contains no parameters, it has more advantages over those with parameters and have wide applications in real world life. Moreover, the iterations process in the proposed filled function algorithm can be easily actualized. we also make a numerical test to demonstrate the efficiency of the proposed approach. The numerical results show that our filled function approach is efficient and reliable.

A mixed approach for nonlinear equations based on simple smooth penalty function and filled function

In the paper, we proposed a novel mixed approach for system of nonlinear equations based on simple smooth penalty function method and filled function method. First, we transform the problem of nonlinear equations into a global optimization problem. Then, we use the filled function method to minimize the constructed simple smooth penalty function. Under some mild assumptions, we can get a global optimizer of the penalty problem if the penalty parameter is sufficiently large. By this way, a root of the system of nonlinear equations can be obtained.