Distance to Solvability/Unsolvability in Linear Optimizatio (original) (raw)

2006, Siam Journal on Optimization

In this paper we measure how much a linear optimization problem, in R n , has to be perturbed in order to loose either its solvability (i.e., the existence of optimal solutions) or its unsolvability property. In other words, if we consider as ill-posed problems those in the boundary of the set of solvable ones, this paper deals with the associated distance to ill-posedness. Our parameter space is the set of all the linear semiinfinite programming problems with a fixed, but arbitrary, index set. In this framework, which includes as a particular case the ordinary linear programming, we obtain a formula for the distance from a solvable problem to unsolvability in terms of the nominal problem's coefficients. Moreover, this formula also provides the exact expression, or a lower bound, of the distance from an unsolvable problem to solvability. 1 and practical applications, namely, stability of the feasible set ([2], [5], [17]), measures of conditioning ([9], [15]), complexity analysis of certain algorithms for computing solutions ([8], [10]), size of the feasible set ([2], [7]), and the optimal set, sensitivity of the optimal value, stability of the dual problem, metric regularity of mappings ([5], [6], [13]), etc.