A Primal-Dual Slack Approach to Warmstarting Interior-Point Methods for Linear Programming

On Interior-Point Warmstarts for Linear and Combinatorial Optimization

“…Second, suggested by the complexity bound in Theorem 2.1 and confirmed by our numerical experiments in [9], we find that an initial product (ξ 0 ) T ψ 0 of magnitude similar to the maximum residual norm δ = (r 0 b , r 0 c , 1) ∞ does not impact the (theoretical) worst-case behavior but is usually large enough in practice to not cause numerical instabilities when resolving problem (1.2) after some problem perturbation. For the case of warmstarting problem (1.4) after adding cuts, in which all residuals r 0 b = (r 0 c ,r 0 b ) and r 0 c = (r 0 c ,r 0 c ) are zero after initialization of the new variables (x,ȳ,z) according to the discussion in subsection 3.2, we similarly use the original constraint violation and let…”

“…Lemma 2.3 imposes a sufficient condition, verified later, for the step sizesᾱ k and the reduction of initial infeasibilities and remaining residuals to be bounded from below by some positive constant. In support of the use of this condition on the solution of the linear system (9), which, close to optimality, becomes very ill-conditioned, we recall that this systematic ill-conditioning does not noticeably impair the accuracy of the computed primal-dual steps [44,45,47]. Nonetheless, such a condition may not be satisfied in practice for any reasonable constant η independent of the problem dimension.…”

“…While the initialization scheme in (3.10) can also be used to simply adjust the original variables (x, z) in a more traditional warmstarting approach without introducing slacks, our computational experiments in this and the earlier paper [9] have shown that the use of additional slack variables increased both numerical stability and computational performance of our method by relaxing the original nonnegativity constraints and thus enabling larger primal-dual steps to more quickly eliminate the initial problem infeasibilities.…”

“…In our own paper [9], we showed how the good warmstart performance of this method for problem perturbations can be maintained for the simpler primaldual slack approach…”

Theoretical and computational issues for improving the performance of linear optimization methods

“…Of course, the expanded boundaries must be driven to zero again during the solution of the problem. Warmstarting techniques in this group have been proposed by Benson and Shanno (2007) and Engau et al (2009Engau et al ( , 2010. They dier from each other by how the space of pairwise products is expanded.…”