Addressing Rank Degeneracy in Constraint-Reduced Interior-Point Methods for Linear Optimization

Abstract

In earlier work (Tits et al., SIAM J. Optim., 17(1): 119-146, 2006; Winternitz et al., COAP, 51(3):1001-1036, the present authors and their collaborators proposed primal-dual interior-point (PDIP) algorithms for linear optimization that, at each iteration, use only a subset of the (dual) inequality constraints in constructing the search direction. For problems with many more variables than constraints in primal form, this can yield a major speedup in the computation of search directions. However, in order for the Newton-like PDIP steps to be well defined, it is necessary that the gradients of the constraints included in the working set span the full dual space. In practice, in particular in the case of highly sparse problems, this often results in an undesirably large working set-or in an expensive trial-and-error process for its selection. In this paper we present two approaches that remove this non-degeneracy requirement, while retaining the convergence results obtained in the earlier work.

A constraint-reduced MPC algorithm for convex quadratic programming, with a modified active set identification scheme

“…Applying regularization to address rank-deficiency of the normal matrix due to constraint reduction was first considered in [35], in the context of linear optimization. There a similar regularization as in [9,25] is applied, while the scheme lets the regularization die out as a solution to the optimization problem is approached, to preserve fast local convergence.…”

“…First, the stopping criterion is a variation on that of [18,34], involving both λ and [λ] + instead of only λ; in fact the latter will fail when the parameter λ max (see (26)- (27)) is not large enough and may fail when second order sufficient conditions are not satisfied, while we prove below (Theorem 1(iv)) that the new criterion is eventually satisfied indeed, in that the iterate (x, λ) converges to a solution (even if it is not unique), be it on a mere subsequence. Second, our update formula for the regularization parameter ̺ in Step 2 improves on that in [35] (̺ + = min{χ, χ max } in the notation of this paper, where χ max is a userdefined constant) as it fosters a "smooth" evolution of W from the initial value of H + R, with R specified by the user, at a rate no faster than that required for local q-quadratic convergence. And third, R 0 should be selected to compensate for possible ill-conditioning of H-so as to mitigate possible early ill-conditioning of M (Q) .…”

“…Several constraint-selection rules have been proposed for constraint-reduced algorithms on various classes of optimization problems, such as linear optimization [28,34,35], convex quadratic optimization [17,18], semi-definite optimization [23,24], and nonlinear optimization [5]. In [28,34,35], the cardinality q of Q is constant and decided at the outset. Because in the non-degenerate case the set of active constraints at the solution of (P) with H = 0 is at least equal to the number n of primal variables, q ≥ n is usually enforced in that context.…”

“…In the 1990s, work toward reducing the computational burden by using only a small portion ("working set") of the constraints in the search direction computation focused mainly on linear optimization [6,15,31,38]. This was also the case for [28], which may have been the first to consider such "constraint-reduction" schemes in the context of PDIPMs (vs. purely dual interior-point methods), and for its extensions [14,34,35]. Exceptions include the works of Jung et al [17,18] and of Park et al [23,24]; in the former, an extension to CQP was considered, with an affine-scaling variant used as the "base" algorithm; in the latter, a constraint-reduced PDIPM for semi-definite optimization (which includes CQP as a special case) was proposed, for which polynomial complexity was proved.…”

“…Other improvements over [18,34] include (i) a modified stopping criterion and a proof of termination of the algorithm (in [18,34], termination is only guaranteed under uniqueness of primal-dual solution and strict complementarity), (ii) a potentially larger value of the "mixing parameter" in the definition of the primal search direction (see footnote 2(iii)), and (iii) an improved update formula (compared to that used in [35]) for the regularization parameter, which fosters a smooth evolution of the regularized Hessian W from an initial (ma-trix) value H + R, where R is specified by the user, at a rate no faster than that required for local q-quadratic convergence. In Section 2 below, we introduce the proposed algorithm (Algorithm CR-MPC) and a general condition (Condition CSR) to be satisfied by the constraintselection rule, and we state global and local quadratic convergence results for Algorithm CR-MPC, subject to Condition CSR.…”