530 research outputs found
Experimental investigations in combining primal dual interior point method and simplex based LP solvers
The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or the stalling problem of the SSX, indeed it reaches the 'near optimum' solution very quickly. The SSX algorithm, in contrast, is not affected by the boundary conditions which slow down the convergence of the PD. If the solution to the LP problem is non unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby cross over from PD to SSX can take place at any stage of the PD optimization run. The cross over to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery
Some recent advances in projection-type methods for variational inequalities
AbstractProjection-type methods are a class of simple methods for solving variational inequalities, especially for complementarity problems. In this paper we review and summarize recent developments in this class of methods, and focus mainly on some new trends in projection-type methods
Predictor-corrector interior-point algorithm for sufficient linear complementarity problems based on a new type of algebraic equivalent transformation technique
We propose a new predictor-corrector (PC) interior-point algorithm (IPA) for solving linear complementarity problem (LCP) with P_* (Īŗ)-matrices. The introduced IPA uses a new type of algebraic equivalent transformation (AET) on the centering equations of the system defining the central path. The new technique was introduced by Darvay et al. [21] for linear optimization. The search direction discussed in this paper can be derived from positive-asymptotic kernel function using the function Ļ(t)=t^2 in the new type of AET. We prove that the IPA has O(1+4Īŗ)ān logā”ć(3nĪ¼^0)/Īµć iteration complexity, where Īŗ is an upper bound of the handicap of the input matrix. To the best of our knowledge, this is the first PC IPA for P_* (Īŗ)-LCPs which is based on this search direction
Interior Point Methods for Massive Support Vector Machines
We investigate the use of interior point methods for solving quadratic
programming problems with a small number of linear constraints where
the quadratic term consists of a low-rank update to a positive semi-de nite
matrix. Several formulations of the support vector machine t into this
category. An interesting feature of these particular problems is the vol-
ume of data, which can lead to quadratic programs with between 10 and
100 million variables and a dense Q matrix. We use OOQP, an object-
oriented interior point code, to solve these problem because it allows us
to easily tailor the required linear algebra to the application. Our linear
algebra implementation uses a proximal point modi cation to the under-
lying algorithm, and exploits the Sherman-Morrison-Woodbury formula
and the Schur complement to facilitate e cient linear system solution.
Since we target massive problems, the data is stored out-of-core and we
overlap computation and I/O to reduce overhead. Results are reported
for several linear support vector machine formulations demonstrating the
reliability and scalability of the method
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