Mathematical Programming

, Volume 138, Issue 1–2, pp 401–445

# The matricial relaxation of a linear matrix inequality

Full Length Paper Series A

## 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 $${L_1(X) \succeq 0}$$ imply $${L_2(X) \succeq 0}$$?

• (Q2) when are they mutually dominant, that is, when do they have the same solution set?

The matrix cube problem of Ben-Tal and Nemirovski (SIAM J Optim 12:811–833, 2002) is an example of LMI domination. Hence such problems 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. As an example, to test the strength of this relaxation we specialize it to the matrix cube problem and obtain essentially the relaxation given in Ben-Tal and Nemirovski (SIAM J Optim 12:811–833, 2002). Thus our relaxation could be viewed as generalizing it. Assume there is an X such that L 1(X) and L 2(X) are both positive definite, 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
$$\begin{array}{ll}L_2(x) = V_1^{*} L_1(x) V_1 + \cdots + V_\mu^{*} L_1(x) V_{\mu}. \quad \quad \quad ({\rm A}_1)\end{array}$$
As for (Q2) we show that L 1 and L 2 are mutually dominant if and only if, up to certain redundancies described in the paper, L 1 and L 2 are unitarily equivalent. Algebraic certificates for positivity, such as (A1) for linear polynomials, are typically called Positivstellensätze. The paper goes on to derive a Putinar-type Positivstellensatz for polynomials with a cleaner and more powerful conclusion under the stronger hypothesis of positivity on an underlying bounded domain of the form $${ \{X \mid L(X)\succeq0\}. }$$ 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 τ from a subspace of matrices to a matrix algebra is “completely positive”. Complete positivity is one of the main techniques of modern operator theory and the theory of operator algebras. On one hand it provides tools for studying LMIs and on the other hand, since completely positive maps are not so far from representations and generally are more tractable than their merely positive counterparts, the theory of completely positive maps provides perspective on the difficulties in solving LMI domination problems.

## Keywords

Linear matrix inequality (LMI) Completely positive Semidefinite programming Positivstellensatz Gleichstellensatz Archimedean quadratic module Real algebraic geometry Free positivity

## Mathematics Subject Classification

Primary 46L07 14P10 90C22 Secondary 11E25 46L89 13J30

## References

1. 1.
Arveson W.B.: Subalgebras of C*-algebras. Acta Math. 123, 141–224 (1969)
2. 2.
Arveson W.B.: The noncommutative Choquet boundary. J. Am. Math. Soc. 21, 1065–1084 (2008)
3. 3.
Arveson W.B.: The noncommutative Choquet boundary III. Math. Scand. 106, 196–210 (2010)
4. 4.
Arveson, W.B.: Notes on the Unique Extension Property, snippet. http://math.berkeley.edu/~arveson/texfiles.html
5. 5.
Ben-Tal A., Nemirovski A.: On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM J. Optim. 12, 811–833 (2002)
6. 6.
Dritschel M.A., McCullough S.: Boundary representations for families of representations of operator algebras and spaces. J. Oper. Theory 53, 159–167 (2005)
7. 7.
Eberly, W., Giesbrecht, M.: Efficient decomposition of associative algebras. In: Lakshman, Y.N. (ed.) Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, ISSAC ’96, pp. 170–178. ACM Press, Zürich (1996)Google Scholar
8. 8.
Friedl, K., Rońyal, L.: Polynomial time solutions of some problems in computational algebra. In: ACM Symposium on Theory of Computing, vol. 17, pp. 153–162 (1985)Google Scholar
9. 9.
Helton J.W., McCullough S.: A Positivstellensatz for non-commutative polynomials. Trans. Am. Math. Soc. 356, 3721–3737 (2004)
10. 10.
Klep I., Schweighofer M.: A Nichtnegativstellensatz for polynomials in noncommuting variables. Israel J. Math. 161, 17–27 (2007)
11. 11.
Klep I., Schweighofer M.: Pure states, positive matrix polynomials and sums of hermitian squares. Indiana Univ. Math. J. 59, 857–874 (2010)
12. 12.
Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The book of involutions. With a preface in French by J. Tits, vol 44. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI (1998)Google Scholar
13. 13.
Lasserre J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)
14. 14.
Murota K., Kanno Y., Kojima M., Kojima S.: A numerical algorithm for block-diagonal decomposition of matrix *-algebras with application to semidefinite programming. Jpn. J. Ind. Appl. Math. 27, 125–160 (2010)
15. 15.
Parrilo P.A.: Semidefinite programming relaxations for semialgebraic problems. Algebraic and geometric methods in discrete optimization. Math. Program. 96B, 293–320 (2003)
16. 16.
Paulsen V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)
17. 17.
Putinar M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 969–984 (1993)
18. 18.
Vidav I.: On some *-regular rings. Acad. Serbe Sci. Publ. Inst. Math. 13, 73–80 (1959)
19. 19.
Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds): Handbook of Semidefinite Programming. Theory, Algorithms, and Applications. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar

© Springer and Mathematical Optimization Society 2012

## Authors and Affiliations

• J. William Helton
• 1
Email author
• Igor Klep
• 2
• 3
• Scott McCullough
• 4
1. 1.Department of MathematicsUniversity of CaliforniaSan DiegoUSA
2. 2.Fakulteta za matematiko in fizikoUniverza v LjubljaniLjubljanaSlovenia
3. 3.Fakulteta za naravoslovje in matematikoUniverza v MariboruMariborSlovenia
4. 4.Department of MathematicsUniversity of FloridaGainesvilleUSA