Paula Amaral

In practice one has often to deal with the problem of inconsistency between constraints, as the result, among others, of the comple\-xi\-ty of real models. To overcome these conflicts we can outline two major \mbox{actions}: removal of constraints or changes in the coefficients of the model. This last approach, that can be generically described as ``model corre\-ction" is the problem we address in this paper. The correction of the right hand side alone was one of the first approaches. The correction of both the matrix of coefficients and the right hand side introduces non linearity in the constraints. The degree of difficulty in solving the problem of the optimal correction depends on the objective function, whose purpose is to measure the closeness between the original and corrected model. Contrary to other norms, the optimization of the important Frobenius was still an open problem. We have analyzed the problem using the KKT conditions and derived necessary and sufficient conditions which enabled us to unequivocally characterize local optima, in terms of the solution of the Total Least Squares and the set of active constraints. These conditions justify a set of pruning rules, which proved, in preliminary experimental results, quite successful in a tree search procedure for determining the global minimizer.

This paper addresses the problem of finding an optimal correction of an inconsistent linear system, where only the nonzero coefficients of the constraint matrix are allowed to be perturbed for reconstructing a consistent system. Using the Frobenius norm as a measure of the distance to feasibility, a nonconvex minimization problem is formulated, whose objective function is a sum of fractional functions. A branch-and-bound algorithm for solving this nonconvex program is proposed, based on suitably overestimating the denominator function for computing lower bounds. Computational experience is presented to demonstrate the efficacy of this approach.