Projected-Search Methods for Bound-Constrained Optimization

Projected-search methods for bound-constrained minimization are based on performing a line search along a continuous piecewise-linear path obtained by projecting a search direction onto the feasible region. A potential benefit of a projected-search method is that many changes to the active set can be made at the cost of computing a single search direction. As … Read more

Stability and accuracy of Inexact Interior Point methods for convex quadratic programming

We consider primal-dual IP methods where the linear system arising at each iteration is formulated in the reduced (augmented) form and solved approximately. Focusing on the iterates close to a solution, we analyze the accuracy of the so-called inexact step, i.e., the step that solves the unreduced system, when combining the effects of both different … Read more

Interior-point algorithms for convex optimization based on primal-dual metrics

We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms we analyse are so-called short-step algorithms and they match the current best iteration complexity bounds for primal-dual symmetric interior-point algorithm of Nesterov and Todd, for symmetric cone programming problems with given self-scaled barriers. Our results apply to … Read more

A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems, with Convergence Proofs

We present an infeasible primal-dual interior point method for semidefinite optimization problems, making use of constraint reduction. We show that the algorithm is globally convergent and has polynomial complexity, the first such complexity result for primal-dual constraint reduction algorithms for any class of problems. Our algorithm is a modification of one with no constraint reduction … Read more

An interior point method with a primal-dual quadratic barrier penalty function for nonlinear semidefinite programming

In this paper, we consider an interior point method for nonlinear semidefinite programming. Yamashita, Yabe and Harada presented a primal-dual interior point method in which a nondifferentiable merit function was used. By using shifted barrier KKT conditions, we propose a differentiable primal-dual merit function within the framework of the line search strategy, and prove the … Read more

Using the primal-dual interior point algorithm within the branch-price-and-cut method

Branch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using … Read more

Bounds on Eigenvalues of Matrices Arising from Interior-Point Methods

Interior-point methods feature prominently among numerical methods for inequality-constrained optimization problems, and involve the need to solve a sequence of linear systems that typically become increasingly ill-conditioned with the iterations. To solve these systems, whose original form has a nonsymmetric 3×3 block structure, it is common practice to perform block Gaussian elimination and either solve … Read more

Addressing rank degeneracy in constraint-reduced interior-point methods for linear optimization

In earlier work (Tits et al., SIAM J. Optim., 17(1):119–146, 2006; Winternitz et al., COAP, doi=10.1007/s10589-010-9389-4, 2011), 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 constraints … Read more

Parallel solver for semidefinite programming problem having sparse Schur complement matrix

SemiDefinite Programming (SDP) problem is one of the most central problems in mathematical programming. SDP provides a practical computation framework for many research fields. Some applications, however, require solving large-scale SDPs whose size exceeds the capacity of a single processor in terms of computational time and available memory. SDPARA (SemiDefinite Programming Algorithm paRAllel version) developed … Read more

A high-performance software package for semidefinite programs: SDPA 7

The SDPA (SemiDefinite Programming Algorithm) Project launched in 1995 has been known to provide high-performance packages for solving large-scale Semidefinite Programs (SDPs). SDPA Ver. 6 solves efficiently large-scale dense SDPs, however, it required much computation time compared with other software packages, especially when the Schur complement matrix is sparse. SDPA Ver. 7 is now completely … Read more