Complexity analysis and numerical implementation of a short-step primal-dual algorithm for linear complementarity problems

It is known that the norm of the solution to a weighted linear least-squares problem is uniformly bounded for the set of diagonally dominant symmetric positive definite weight matrices. This result is extended to weight matrices that are nonnegative linear combinations of symmetric positive semidefinite matrices. Further, results are given concerning the strong connection between the boundedness of weighted projection onto a subspace and the projection onto its complementary subspace using the inverse weight matrix. In particular, explicit bounds are given for the Euclidean norm of the projections. These results are applied to the Newton equations arising in a primal-dual interior method for convex quadratic programming and boundedness is shown for the corresponding projection operator.

In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations isO(źnL) orO(nL), depending on how the barrier parameter is updated.

A unified treatment is given for partially and totally asynchronous parallel successive overrelaxation (SOR) algorithms for the linear complementarity problem. Convergence conditions are established and compared to previous results. Convergence of the partially asynchronous method for the symmetric linear complementarity problem can be guaranteed if the relaxation factor is sufficiently small. Unlike previous results, this relaxation factor interval does not depend explicitly on problem size.

This paper proposes an improved predictor‐corrector interior‐point algorithm for the linear complementarity problem (LCP) based on the Mizuno‐Todd‐Ye algorithm. The modified corrector steps in our algorithm cannot only draw the iteration point back to a narrower neighborhood of the center path but also reduce the duality gap. It implies that the improved algorithm can converge faster than the MTY algorithm. The iteration complexity of the improved algorithm is proved to obtain which is similar to the classical Mizuno‐Todd‐Ye algorithm. Finally, the numerical experiments show that our algorithm improved the performance of the classical MTY algorithm.

In model-predictive control (MPC), an optimization problem has to be solved at each time step, which in real-time applications makes it important to solve these efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations is solved. We propose an algorithm for computing which sequence of subproblems an active-set algorithm will solve, for every parameter of interest. These sequences can be used to set worst-case bounds on how many iterations, floating-point operations, and, ultimately, the maximum solution time the active-set algorithm requires to converge. The usefulness of the proposed method is illustrated on a set of QPs originating from MPC problems, by computing the exact worst-case number of iterations primal and dual active-set algorithms require to reach optimality.

The Linear Dynamic Complementarity Problem is a special case of the Extended Linear Complementarity Problem

In this paper, we propose a class of generalized accelerated overrelaxation (GAOR) methods, in which one special case reduces to the generalized successive overrelaxation(GSOR) methods, for solving a linear complementarity problem LCP( M, q). Sufficient conditions for convergence of the GAOR and GSOR methods are presented for cases when the system matrix M is an H–matrix, M–matrix and a strictly or irreducible diagonally dominant matrix. Moreover, when M is an L–matrix, we examine the monotone convergence of the new methods. Finally a numerical example is used to validate the results proved in this paper.

A generalized class of infeasible-interior-point methods for solving horizontal linear complementarity problems is analyzed and sufficient conditions are given for the convergence of the sequence of iterates produced by methods in this class. In particular it is shown that the largest step path following algorithms generates convergent iterates even when starting from infeasible points. The computational complexity of the latter method is discussed in detail and its local convergent rate is analyzed. The primal-dual gap of the iterates produced by this method is superlinearly convergent to zero. A variant of the method has quadratic convergence.

This text studies a kind of obstacle problem. Combining with difference principle, we transform the original problem into monotone linear complementarity problem, and propose a novel method called potential-reduction interior point algorithm for monotone linear complementarity problem. We establish global and finite convergence of the new method. The reliability and efficiency of the algorithm is demonstrated by the numerical experiments of standard linear complementarity problems and the examples of obstacle problem with free boundary.

Recently Kojima, Megiddo, and Mizuno showed theoretical convergence of primal-dual interior point algorithms with the use of new step length rules for linear programs. Their rules, which only rely on the lengths of steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions, allow taking large step lengths without performing any line search. This paper extends and modifies their analysis to interior point algorithms for positive semidefinite linear complementarily problems. Global convergence and polynomial-time convergence are presented under similar step length rules.

The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) $${0\leq x\perp(Mx+q)\geq0}$$ can be viewed as a semismooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x, Mx + q) = 0, which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm is known to converge in at most n iterations. We show in this paper that this result no longer holds when M is a P-matrix of order ź 3, since then the algorithm may cycle. P-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally, convergence occurs for a P-matrix of order 1 or 2.

A polynomial arc-search interior-point algorithm for convex quadratic programming

This report concerns the generation of support vector machine classifiers for solving the pattern recognition problem in machine learning. Several methods are proposed based on interior point methods for convex quadratic programming. Software implementations are developed by adapting the object-oriented packaging OOQP to the problem structure and by using the software package PETSc to perform time-intensive computations in a distributed setting. Linear systems arising from classification problems with moderately large numbers of features are solved by using two techniques--one a parallel direct solver, the other a Krylov-subspace method incorporating novel preconditioning strategies. Numerical results are provided, and computational experience is discussed.

A full-Newton step [formula omitted] infeasible-interior-point algorithm for linear complementarity problems

In this paper a primal-dual path-following interior-point algorithm for the monotone semidefinite linear complementarity problem is presented. The algorithm is based on Nesterov-Todd search directions and on a suitable proximity for tracing approximately the central-path. We provide an unified analysis for both long and small-update primal-dual algorithms. Finally, the iteration bounds for these algorithms are obtained.