Parametric Method for Global Optimization

This paper considers constrained and unconstrained parametric global optimization problems in a real Hilbert space. We assume that the gradient of the cost functional is Lipschitz continuous but not smooth. A suitable choice of parameters implies the linear or superlinear (supergeometric) convergence of the iterative method. From the numerical experiments, we conclude that our algorithm is faster than other existing algorithms for continuous but nonsmooth problems, when applied to unconstrained global optimization problems. However, because we solve 2n + 1 subproblems for a large number n of independent variables, our algorithm is somewhat slower than other algorithms, when applied to constrained global optimization.This work was partially supported by the NATO Outreach Fellowship - Mathematics 219.33.We thank Professor Hans D. Mittelmann, Arizona State University, for cooperation and support.     

A Branch-and-Bound Based Method for Solving Monotone Optimization Problems

Monotone optimization problems are an important class of global optimization problems with various applications. In this paper, we propose a new exact method for monotone optimization problems. The method is of branch-and-bound framework that combines three basic strategies: partition, convexification and local search. The partition scheme is used to construct a union of subboxes that covers the boundary of the feasible region. The convexification outer approximation is then applied to each subbox to obtain an upper bound of the objective function on the subbox. The performance of the method can be further improved by incorporating the method with local search procedure. Illustrative examples describe how the method works. Computational results for small randomly generated problems are reported. Dedicated to Professor Alex Rubinov on the occasion of his 65th birthday. The authors appreciate very much the discussions with Professor Alex Rubinov and his suggestion of using local search. Research supported by the National Natural Science Foundation of China under Grants 10571116 and 10261001, and Guangxi University Scientific Research Foundation (No. X051022).     

Optimality Conditions for Vector Optimization Problems

In this paper, some necessary and sufficient optimality conditions for the weakly efficient solutions of vector optimization problems (VOP) with finite equality and inequality constraints are shown by using two kinds of constraints qualifications in terms of the MP subdifferential due to Ye. A partial calmness and a penalized problem for the (VOP) are introduced and then the equivalence between the weakly efficient solution of the (VOP) and the local minimum solution of its penalized problem is proved under the assumption of partial calmness. This work was supported by the National Natural Science Foundation of China (10671135), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the National Natural Science Foundation of Sichuan Province (07ZA123). The authors thank Professor P.M. Pardalos and the referees for comments and suggestions.     

Mixed Type Duality for Set-valued Optimization Problems via Higher-order Radial Epiderivatives

In this article, we introduce a notion of higher-order radial epiderivative for set-valued maps and study its properties. A generalized concept of higher-order strict minimizers in set-valued optimization is proposed as well. By virtue of the radial epiderivative, we establish a mixed dual problem, and then weak, strong, and converse duality theorems are obtained in dealing with generalized strict minimizers.     

Target-Oriented Branch and Bound Method for Global Optimization

We introduce a very simple but efficient idea for branch and bound (&) algorithms in global optimization (GO). As input for our generic algorithm, we need an upper bound algorithm for the GO maximization problem and a branching rule. The latter reduces the problem into several smaller subproblems of the same type. The new & approach delivers one global optimizer or, if stopped before finished, improved upper and lower bounds for the problem. Its main difference to commonly used & techniques is its ability to approximate the problem from above and from below while traversing the problem tree. It needs no supplementary information about the system optimized and does not consume more time than classical & techniques. Experimental results with the maximum clique problem illustrate the benefit of this new method.     

Canonical Dual Transformation Method and Generalized Triality Theory in Nonsmooth Global Optimization*

This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in ⁿ can be reformulated into certain smooth/convex unconstrained dual problems in ^m with m n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.     

Semideterministic Global Optimization Method: Application to a Control Problem of the Burgers Equation

This paper has two objectives. We introduce a new global optimization algorithm reformulating optimization problems in terms of boundary-value problems. Then, we apply this algorithm to a pointwise control problem of the viscous Burgers equation, where the control weight coefficient is progressively decreased. The results are compared with those obtained with a genetic algorithm and an LM-BFGS algorithm in order to check the efficiency of our method and the necessity of using global optimization techniques.