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

• Autor(a) principal: Travaglia,Marcos Vinicio
• Data de Publicação: 2010
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Computational & Applied Mathematics
• Texto Completo: http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022010000200004
• topic: matrix analysis; sum of smallest eigenvalues; minimization problem involving matrices; nonlinear perturbation; semidefinite programming

Let C be a n×n symmetric matrix. For each integer 1 < k < n we consider the minimization problem m(ε): = minX{ Tr{CX} + εƒ(X)}. Here the variable X is an n×n symmetric matrix, whose eigenvalues satisfy <img border=0 src="../../../../img/revistas/cam/v29n2/a04img01.gif"> the number ε is a positive (perturbation) parameter and ƒ 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 satisfy λ1(C) < ... < λk(C) < λk+1(C) < ∙∙∙ < λn(C), we establish the following upper and lower bounds for the minimum value m(ε): <img border=0 src="../../../../img/revistas/cam/v29n2/a04img02.gif"> where <img border=0 src="../../../../img/revistas/cam/v29n2/f4_barra.gif" align=absmiddle>is the minimum value of ƒ over the solution set of unperturbed problem and L is the Lipschitz-constant of ƒ. 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 λk+1(C) = λk(C). We compare the exact solution with the upper and lower bounds for some examples. Mathematical subject classification: 15A42, 15A18, 90C22.