An LMI approach to constrained optimization with homogeneous forms

This paper considers the problem of determining the minimum Euclidean distance of a point from a polynomial surface in R n . It is well known that this problem is in general non-convex. The main purpose of the paper is to investigate to what extent linear matrix inequality (LMI) techniques can be ex...

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Bibliographic Details
Published in:Systems & control letters 2001, Vol.42 (1), p.11-19
Main Authors: Chesi, G., Tesi, A., Vicino, A., Genesio, R.
Format: Article
Language:English
Subjects:
Applied sciences
Computer science
control theory
systems
Control system synthesis
Control theory. Systems
Exact sciences and technology
Homogeneous form
Linear matrix inequalities (LMIs)
Optimization
Robustness
Stability
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Summary:This paper considers the problem of determining the minimum Euclidean distance of a point from a polynomial surface in R n . It is well known that this problem is in general non-convex. The main purpose of the paper is to investigate to what extent linear matrix inequality (LMI) techniques can be exploited for solving this problem. The first result of the paper shows that a lower bound to the global minimum can be achieved via the solution of a one-parameter family of linear matrix inequalities (LMIs). It is also pointed out that for some classes of problems the solution of a single LMI problem provides the lower bound. The second result concerns the tightness of the bound. It is shown that optimality of the lower bound amounts to solving a system of linear equations. An application example is finally presented to show the features of the approach.
ISSN:0167-6911
1872-7956
DOI:10.1016/S0167-6911(00)00072-4