Sensitivity analysis in linear programming and semidefinite programming using interior-point methods

Abstract

Abstract. We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the perturbations that allow interior-point methods to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interior-point methods compare nicely with the bounds arising from using the optimal basis. We also present explicit bounds for SDP using the MonteiroZhang family of search directions and specialize them to the AHO, H..K..M, and NT directions.

An Interior-Point Approach to Sensitivity Analysis in Degenerate Linear Programs

“…More recently, Greenberg, Holder, Roos and Terlaky [5] related the dimension of the optimal set to the dimension of the set of objective perturbations for which the optimal partition is invariant. Greenberg [4] considered the simultaneous perturbations of the right-hand side and the cost vectors from an optimal partition perspective.In [13], the authors studied perturbations of the right-hand side and the cost parameters in linear programming, motivated by how interior-point methods from a near-optimal pair of strictly feasible solutions for a problem and its dual would compare with the optimal basis approach obtained from a nondegenerate optimal basic solution for such perturbations. The proposed interior-point perspective stems from the objectives of regaining feasibility and maintaining near-optimality in a single it-…”

“…In [13], the authors studied perturbations of the right-hand side and the cost parameters in linear programming, motivated by how interior-point methods from a near-optimal pair of strictly feasible solutions for a problem and its dual would compare with the optimal basis approach obtained from a nondegenerate optimal basic solution for such perturbations. The proposed interior-point perspective stems from the objectives of regaining feasibility and maintaining near-optimality in a single it-eration of the interior-point method.…”

“…Under the assumption of a unique, nondegenerate optimal solution, the authors showed that the Newton system proposed in [13] is the "right" one in the sense that it yields asymptotically the same bounds on perturbations as those that keep the current basis optimal (after symmetrization with respect to the origin). Similar results, but changing only one of the primal or dual near-optimal solutions, were obtained by Kim, Park and Park [8].…”

“…Our goal in this paper is to investigate the quality of the bounds from the interior-point perspective in the absence of the strong assumption of nondegeneracy. This will lead to a complete analysis of the interior-point perspective proposed in [13]. In doing so, we need something to compare our interior-point bounds with.…”

On Warm Starts for Interior Methods

“…For related discussions concerning linear programming, see, e.g. Jansen et al [11], Kim, Park and Park [12], Gondzio and Grothey [9], Yildirim and Todd [18,19], Yildirim and Wright [20], Gonzalez-Lima, Wei and Wolkowicz [10]. For extensions to linear semidefinite programming, see Yildirim [17].…”