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

Abstract

A constraint-reduced Mehrotra-Predictor-Corrector algorithm for convex quadratic programming is proposed. (At each iteration, such algorithms use only a subset of the inequality constraints in constructing the search direction, resulting in CPU savings.) The proposed algorithm makes use of a regularization scheme to cater to cases where the reduced constraint matrix is rank deficient. Global and local convergence properties are established under arbitrary working-set selection rules subject to satisfaction of a general condition. A modified active-set identification scheme that fulfills this condition is introduced. Numerical tests show great promise for the proposed algorithm, in particular for its active-set identification scheme. While the focus of the present paper is on dense systems, application of the main ideas to large sparse systems is briefly discussed.

An infeasible-start framework for convex quadratic optimization, with application to constraint-reduced interior-point and other methods

“…2.3 below). It is suggested in [28], in the context of a "constraint-reduced" primal-dual interior-point method that requires an initial primal-feasible point, that an extension to handle problems for which a primal-feasible initial point is not available can be constructed by involving the following penalized relaxed primal-dual pair, 6 for which a primal-feasible point (x, z) is readily available:…”

“…Third, it is shown how at a negligible additional cost per iteration, when the CQP is primal-infeasible, a certificate of infeasibility can be produced. 3 Fourth, promising numerical results are obtained, with the algorithm of [28] as the base iteration, in comparison with those obtained with popular schemes. Finally, application of the proposed framework to other base iterations is considered, and the case of a revised primal simplex iteration is briefly tested.…”

“…While the most popular interior-point algorithms do not require that an initial feasible point be provided, simplex algorithms do: such feasible points, when not readily available, are typically obtained by solving an auxiliary linear optimization problem ("phase 1"). Like simplex algorithms, recently proposed "constraint-reduced" interior-point algorithms, the latest of which (see, e.g., [28]) were observed to often largely outperform other approaches when the problem at hand is severely "imbalanced" (i.e., with most inequality constraints being inactive at the solution; e.g., m n − p), do require a primal-feasible initial point. While a two-phase approach could again be employed here, an important drawback of that approach is that, in the first phase, the objective function is altogether ignored, leading to likely computational waste.…”

“…5, issuance of an infeasibility certificate in cases when (P) is infeasible is investigated. Section 6 introduces a penalty-parameter updating rule that satisfies the required specifications, discusses implementation details, and reports numerical results with the base iteration from [28] on randomly generated problems and support-vector machine training problems with comparison to popular optimization solvers. Application of the framework to other base iterations is considered in Sect.…”

Nonsmooth Newton’s Method: Some Structure Exploitation

“…Throughout the paper, we investigate the influence of the reduction strategy on the performance of the aforementioned two-steps exploitation; however, a detailed optimization of the reduction policy is beyond the scope of this paper. We point out that the proposed method could be combined with constraint-reduction approaches as, e.g., those presented in [5,12]. Once the original problem (1), say V d = r for brevity, has been transformed, a reduced symmetric linear system, sayVd =r, is to be solved.…”