Abstract :
A matrix optimization problem of interest is the infimization, for arbitrary F, G set membership, variant Rn × m, of double vertical bar;F − PGdouble vertical bar; with respect to positive semidefinite, symmetric P set membership, variant Rn × n, where double vertical bar; · double vertical bar; denotes the Frobenius norm. One motivation for its study is that solutions may be used as estimates of the inverse Hessian of a nonlinear differentiable function f: Rn → R1, which is to be minimized with respect to a parameter vector x set membership, variant Rn by a quasi-Newton-type algorithm. Another is where the compliance matrix of an elastic structure is to be estimated from experimental measurements of the displacements resulting from some static loading. Necessary and sufficient conditions for the existence of the minimum are given. A sufficient condition is that rank[G] = n, which is also necessary and sufficient for uniqueness of minimizers. Arbitrarily accurate infimization can be achieved via solution of an instance of the minimization problem where rank[G] = n. However, no general solution procedure for the latter is known, which motivates algorithmic solution. An algorithm is given which, for the case rank[G] = n, arbitrarily accurately approximates the unique minimizer. The algorithm is globally convergent, requiring the computation of inner products and solutions of symmetric eigenvalue problems.
