ANZIAM J.
47 (2006), 439450

Sufficient global optimality conditions for multiextremal 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



G. M. Lee
Department of Applied Mathematics
Pukyong National University
Pusan 608737
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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