Error bound for a perturbed minimization problem related with the sum of smallest eigenvalues

Abstract

Abstract. Let C be a n × n symmetric matrix. For each integer 1 ≤ k < n we consider the minimization problem m(ε) := min X {Tr{C X } + ε f (X )}. Here the variable X is an n × n symmetric matrix, whose eigenvalues satisfythe number ε is a positive (perturbation) parameter and f is a Lipchitz-continuous function (in general nonlinear). It is well known that when ε = 0 the minimum value, m(0), is the sum of the smallest k eigenvalues of C. Assuming that the eigenvalues of C satisfywe establish the following upper and lower bounds for the minimum value m(ε):wheref is the minimum value of f over the solution set of unperturbed problem and L is the Lipschitz-constant of f . The above inequality shows that the error by replacing the upper bound (or the lower bound) by the exact value is at least quadratic in the perturbation parameter. We also treat the case that λ k+1 (C) = λ k (C). We compare the exact solution with the upper and lower bounds for some examples.Mathematical subject classification: 15A42, 15A18, 90C22.