Optimization Online


The matricial relaxation of a linear matrix inequality

J. William Helton (helton***at***math.ucsd.edu)
Igor Klep (igor.klep***at***fmf.uni-lj.si)
Scott McCullough (sam***at***math.ufl.edu)

Abstract: Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask: (Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X) PsD? (Q2) when do they have the same solution set? Such questions can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables x_j. With this relaxation, the domination questions (Q1) and (Q2) have elegant answers, indeed reduce to constructible semidefinite programs. Assume there is an X such that L_1(X) and L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^* L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are unitarily equivalent. Such algebraic certificates are typically called Positivstellensštze and the above are examples of such for linear polynomials. The paper goes on to derive a cleaner and more powerful Putinar-type Positivstellensatz for polynomials positive on a bounded set of the form {X | L(X) PsD}. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map from a subspace of matrices to a matrix algebra is "completely positive". Complete positivity (CP) is one of the main techniques of modern operator theory. Thus on one hand it provides tools for studying LMIs and in the other direction, since CP is well developed, it gives perspective on the difficulties in solving LMI domination problems.

Keywords: linear matrix inequality (LMI), completely positive, semidefinite programming, Positivstellensatz, Gleichstellensatz, archimedean quadratic module, free positivity

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation: Mathematical Programming, to appear

Download: [PDF]

Entry Submitted: 03/03/2010
Entry Accepted: 03/03/2010
Entry Last Modified: 03/01/2012

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society