Appl Math Optim 40:191–210 (1999)
1999 Springer-Verlag New York Inc.
A Dynamical Systems Analysis of Semideﬁnite Programming
with Application to Quadratic Optimization with
Pure Quadratic Equality Constraints
R. J. Orsi,
R. E. Mahony,
and J. B. Moore
Department of Electrical and Electronic Engineering, University of Melbourne,
Parkville, VIC 3052, Australia
Heudiasyc - UTC UMR 6599, Centre de Recherche de Royallieu,
BP 20529, 60205 Compiegne Cedex, France
Department of Systems Engineering, RSISE, Australian National University,
Canberra, ACT 0200, Australia
Abstract. This paper considers the problem of minimizing a quadratic cost sub-
ject to purely quadratic equality constraints. This problem is tackled by ﬁrst relating
it to a standard semideﬁnite programming problem. The approach taken leads to a
dynamical systems analysis of semideﬁnite programming and the formulation of
a gradient descent ﬂow which can be used to solve semideﬁnite programming
problems. Though the reformulation of the initial problem as a semideﬁnite pro-
gramming problem does not in general lead directly to a solution of the original
problem, the initial problem is solved by using a modiﬁed ﬂow incorporating a
KeyWords. Dynamicalsystems, Semideﬁnite programming, Quadratic optimiza-
tion, Quadratic equality constraints.
AMS Classiﬁcation. 90C26, 90C30, 58F40.
The authors wish to acknowledge the funding of the activities of the Cooperative Research Centre
for Robust and Adaptive Systems by the Australian Commonwealth Government under the Cooperative Re-
search Centre Program. The ﬁrst author would also like to acknowledge the support of Telstra under the TRL
Postgraduate Fellowship scheme.