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

Published online by Cambridge University Press:  17 February 2009

N. Q. Huy ,
V. Jeyakumar  and
G. M. Lee
N. Q. Huy
Affiliation:
Department of Mathematics, Hanoi Pedagogical UniversityNo. 2, Vinh Phuc, Vietnam.
V. Jeyakumar
Affiliation:
Department of Applied Mathematics, University of New South Wales, Sydney NSW 2052, Australia; e-mail: jeya@maths.unsw.edu.au.
G. M. Lee
Affiliation:
Department of Applied Mathematics, Pukyong National University, Pusan 608–737, Korea; e-mail: gmlee@pknu.ac.kr.
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Abstract

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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 optinusation problems with discrete constraints.

Keywords

smooth nonconvex minimisationglobal optimality conditionsbox constraintsdiscrete constraintslinear matrix inequalitiesmulti-extremal problems.
Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 4 , April 2006 , pp. 439 - 450
Copyright
Copyright © Australian Mathematical Society 2006

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