Computation of Matrix Norms with Applications to Robust Optimization

The Thesis is devoted to investigating the problem of computing the norm ‖A‖E,F = max x∈E:‖x‖E≤1 ‖Ax‖F of a linear mapping x 7→ Ax acting from a finite-dimensional normed space (E, ‖ · ‖E) to a finite-dimensional normed space (F, ‖ · ‖F ). This problem is important and interesting by its own right and especially due to its role in Robust Optimization. We mainly focus on the case where (E, ‖ · ‖E) = (Rn, ‖ · ‖p) and (F, ‖ · ‖F ) = (Rm, ‖ · ‖r), so that A can be identified with an m × n matrix; the associated norm ‖A‖E,F is denoted by ‖A‖p,r. There are three simple cases ((p = 1, 1 ≤ r ≤ ∞), (r = ∞, 1 ≤ p ≤ ∞), p = r = 2) where ‖ · ‖p,r is easy to compute. We conjecture that these are the only 3 cases where Pp,r is not NP-hard, and prove that Pp,r is NP-hard in the case when 1 ≤ r < p ≤ ∞. We further focus on building efficiently computable upper bounds on ‖·‖p,r. Our first result in this direction is a refinement of Nesterov’s theorems (see [21] and Chapter 13.2 in [22]) stating that in the case of 1 ≤ r ≤ 2 ≤ p ≤ ∞ a natural semidefinite relaxation upper bound Ψp,r(A) on ‖A‖p,r is tight within the absolute constant factor 1 2 √ 3 π − 2 3 ≈ 2.29 (which can be reduced to π/2 ≈ 1.25 when p = 2 or when r = 2). We develop a novel technique for quantifying the quality of the bound Ψp,r and demonstrate that this bound in a wide range of values of p, r, n,m is essentially less conservative than it is suggested by Nesterov’s results. We prove also that the bound Ψp,r coincides with ‖A‖p,r in the case when A has nonnegative entries. Next, we develop a simple interpolation technique allowing to extend the efficiently computable upper bound Ψp,r(A) on ‖A‖p,r from its original domain 1 ≤ r ≤ 2 ≤ p ≤ ∞ to the entire range 1 ≤ p, r ≤ ∞ of values of p, r, and show that the extended bound is tight within a factor depending on p, n, r,m and never exceeding O(1) (max(m,n)) 25 128 . Our analysis demonstrates that this factor does not exceed 9.48 for all p, r, provided that m, n ≤ 100, 000. Finally, we apply our interpolation technique to bound from above the norm of a linear mapping A acting from (Rn, ‖ · ‖p) to the space Sn of symmetric matrices equipped with the standard matrix norm – a situation which is of significant interest for Robust Semidefinite Programming. We demonstrate that for “well-structured”, in certain precise sense, mappings A the norm in question admits efficiently computable upper bound tight within the factor O(1)n1/4.