Abstract
The paper describes a class of mathematical problems at an intersection of operator theory and combinatorics, and discusses their application in complex system analysis. The main object of study is duality gap bounds in quadratic programming which deals with problems of maximizing quadratic functionals subject to quadratic constraints Such optimization is known to be universal, in the sense that many computationally hard questions can be reduced to quadratic programming On the other hand, it is conjectured that an efficient algorithm of solving general non-convex quadratic programs exactly does not exist.
A specific technique of ”relaxation”, which essentially replaces deterministic decision parameters by random variables, is known experimentally to yield high quality approximate solutions in some non-convex quadratic programs arising in engineering applications. However, proving good error bounds for a particular relaxation scheme is usually a challenging mathematical problem. In this paper relaxation techniques of dynamical system analysis will be described. It will be shown how operator theoretic methods can be used to give error bounds for these techniques or to provide counterexamples. On the other hand, it will be demonstrated that some difficult problems of operator theory have equivalent formulations in terms of relaxation bounds in quadratic programming, and can be approached using the insights from combinatorics and system theory.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization,SIAM Journal on Optimization, 5 (1995), 13–51.
M.X. Goemans and D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), 1115–1145.
P. Halmos, Ten problems in Hilbert space, Bull. Amer. math. Soc., 76 (1970), 887–933.
J. Hastad, Some optimal inapproximability results, Proc. of the 29th ACM Symp. on Theory Comput., 1997.
J.-L. Krivine, Constantes de Grothendieck et fonctions de type positif sur les spheres, Adv. in Math., 31 (1979), 16–30.
A. Megretski and A. Rantzer, System Analysis via Integral Quadratic Constraints, IEEE Transactions on Automatic Control, volume 47, no. 6, pp. 819–830, June 1997.
Y. Nesterov and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming, SIAM, Philadelphia, PA, 1994.
Y. Nesterov, Quality of semidefinite relaxation for nonconvex quadratic optimization, Manuscript, 1997.
G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, Journal Amer. Math. Soc., 10 (1997), no. 2, 351–369.
O. Toker and H. Ozbay, On the complexity of purely complex μ computation and related problems in multidimensional systems, IEEE Trans. Aut. Contr., 43 (1998), no. 3, 409–414.
S. Treil, The gap between complex structured singular value μ and its upper bound is infinite, To appear in IEEE Trans. Aut. Contr.
Yinyu Ye, Approximating quadratic programming with bound constraints, Manuscript, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Megretski, A. (2001). Relaxations of Quadratic Programs in Operator Theory and System Analysis. In: Borichev, A.A., Nikolski, N.K. (eds) Systems, Approximation, Singular Integral Operators, and Related Topics. Operator Theory: Advances and Applications, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8362-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8362-7_15
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9534-7
Online ISBN: 978-3-0348-8362-7
eBook Packages: Springer Book Archive