This paper presents a global optimization method for solving general nonlinear programming problems subjected to box constraints. Regardless of convexity or nonconvexity, by introducing a differential flow on the dual feasible space, a set of complete solutions to the original problem is obtained, and criteria for global optimality and existence of solutions are given. Our theorems improve and generalize recent known results in the canonical duality theory. Applications to a class of constrained optimal control problems are discussed. Particularly, an analytical form of the optimal control is expressed. Some examples are included to illustrate this new approach.

In this paper, we consider the following general box constrained nonlinear programming problem (the primal problem

Problem (

Inspired and motivated by these facts, a differential flow for constructing the canonical dual function is introduced and a new approach to solve the general (especially nonconvex) nonlinear programming problem

The paper is organized as follows. In Section

In the beginning of this paper, we have mentioned that our primal goal is to find the global minimizers to a general (mainly nonconvex) box-constrained optimization problem

The main idea of constructing the differential flow and the canonical dual problem is as follows. For simplicity without generality, we assume that

Let

The dual feasible space

Notice that

Suppose that

Let

Let

Since

By Lemma

The canonical dual problem

By introducing the Lagrange multiplier vector

In addition, we have

Theorem

Due to introduceing a differential flow

Suppose that

Since

Notice that the function

Actually, Lemma

Moreover, for the proper parameter

Suppose that

If

For any given parameter

Thus,

Theorem

In order to study the existence conditions of the canonical dual solutions, we let

Suppose that

We first show that for any given

By Lemma

Suppose that

Since

Clearly, when

Let

If

To prove Theorem

Before beginning of applications to optimal control problems, we present two examples to find global minimizers by differential flows.

As a particular example of

In this example, we see that a differential flow is useful in solving a nonconvex optimization problem. For the global optimization problem, people usually compute the global minimizer numerically. Even in using canonical duality method, one has to solve a canonical dual problem numerically. Nevertheless, the differential flow directs us to a new way for finding a global minimizer. Particularly, one may expect an exact solution of the problem provided that the corresponding differential equation has an analytic solution.

Given a symmetric matrix

If we choose

In this section, we consider the following constrained linear-quadratic optimal control problem:

It is well known that the central result in the optimal control theory is the

Define the Hamilton-Jacobi-Bellman function

Unfortunately, above conditions are not, in general, sufficient for optimality. In such a case, we need to go through the process of comparing all the candidates for optimality that the necessary conditions produce, and picking out an optimal solution to the problem. Nevertheless, Lemma

Let

For any given

Lemma

Suppose that

The proof of Theorem

Substituting

Next, we give an example to illustrate our results.

We consider

Following the idea of Lemma

The optimal control

The dual variable

The authors would like to thank the referees for their helpful comments on the early version of this paper. This work was supported by the National Natural Science Foundation of China (no. 10971053).