ANZIAM  J.  47 (2006), 439-450

Sufficient global optimality conditions for multi-extremal smooth minimisation problems with bounds and linear matrix inequality constraints

N. Q. Huy   Department of Mathematics   Hanoi Pedagogical University No. 2   Vinh Phuc   Vietnam   V. Jeyakumar   Department of Applied Mathematics   University of New South Wales   Sydney NSW 2052   Australia     jeya@maths.unsw.edu.au and G. M. Lee   Department of Applied Mathematics   Pukyong National University   Pusan 608--737   Korea   gmlee@pknu.ac.kr

Abstract

In this paper, we present sufficient conditions for global optimality of a general nonconvex smooth minimisation model problem involving linear matrix inequality constraints with bounds on the variables. The linear matrix inequality constraints are also known as ``semidefinite'' constraints which arise in many applications, especially in control system analysis and design. Due to the presence of nonconvex objective functions, such minimisation problems generally have many local minimisers which are not global minimisers. We develop conditions for identifying global minimisers of the model problem by first constructing a (weighted sum of squares) quadratic underestimator for the twice continuously differentiable objective function of the minimisation problem and then by characterising global minimisers of the easily tractable underestimator over the same feasible region of the original problem. We apply the results to obtain global optimality conditions for optimisation problems with discrete constraints.

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