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On the quality of semidefinite approximations of uncertain semidefinite
programs affected by box uncertainty

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Aharon Ben-Tal and Arkadi Nemirovski

Let P(z) = A_0 + z_1 A_1 + ... + z_L A_L be an affine mapping taking values
in the space of m x m real symmetric matrices such that A_0 is positive
semidefinite. Consider the following question: what is the largest R such
that the set P(\{z: \|z\|_\infty \leq R\}) is contained in the positive
semidefinite cone? In general, it is NP-hard to compute a ``tight enough''
approximation of R. One can, however, easily build a simple semidefinite
program such that its optimal value r is a lower bound on R. We demonstrate
that the ratio R/r does not exceed {\pi\sqrt{k}\over 2}, where k is the
maximum of the ranks of A_1, A_2,...,A_L. We present 3 applications of the
result: we demonstrate that one can build efficiently lower bounds, exact
within the factor {\pi\over 2}, for the following two quantities:
(1) the largest R such that all instances of an interval symmetric matrix
\{A=A^T: |A_{ij}-A^*_{ij}| \leq R D_{ij} \forall i,j\} are positive
semidefinite;
(2) the largest R such that all instances of an interval square matrix
\{A: |A_{ij}-A^*_{ij} \leq R D_{ij} \forall i,j\} admit a common quadratic
Lyapunov stability certificate.
Besides this, we present an alternative proof (which does not use the
Goemans-Williamson construction) of the fact, established by Yu. Nesterov,
that the standard semidefinite upper bound on the maximum of a positive
semidefinite quadratic form over the unit cube is at most {\pi\over 2} times
larger than the true value of the maximum.
Research report #2/00, April 2000, MINERVA Optimization Center, Technion -
Israel Institute of Technology, Technion City, Haifa 32000, Israel

Contact: nemirovs@ie.technion.ac.il

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