An augmented Lagrangian interior-point method using directions of negative curvature

 We describe an efficient implementation of an interior-point algorithm for non-convex problems that uses directions of negative curvature. These directions should ensure convergence to second-order KKT points and improve the computational efficiency of the procedure. Some relevant aspects of the implementation are the strategy to combine a direction of negative curvature and a modified Newton direction, and the conditions to ensure feasibility of the iterates with respect to the simple bounds. The use of multivariate barrier and penalty parameters is also discussed, as well as the update rules for these parameters. We analyze the convergence of the procedure; both the linesearch and the update rule for the barrier parameter behave appropriately. As the main goal of the paper is the practical usage of negative curvature, a set of numerical results on small test problems is presented. Based on these results, the relevance of using directions of negative curvature is discussed. Received: July 2000 / Accepted: October 2002 Published online: December 19, 2002 Key words. Primal-dual methods – Nonconvex optimization – Linesearches Research supported by Spanish MEC grant BEC2000-0167 Mathematics Subject Classification (1991): 49M37, 65K05, 90C30     

First- and second-order methods for semidefinite programming

 In this paper, we survey the most recent methods that have been developed for the solution of semidefinite programs. We first concentrate on the methods that have been primarily motivated by the interior point (IP) algorithms for linear programming, putting special emphasis in the class of primal-dual path-following algorithms. We also survey methods that have been developed for solving large-scale SDP problems. These include first-order nonlinear programming (NLP) methods and more specialized path-following IP methods which use the (preconditioned) conjugate gradient or residual scheme to compute the Newton direction and the notion of matrix completion to exploit data sparsity. Received: December 16, 2002 / Accepted: May 5, 2003 Published online: May 28, 2003 Key words. semidefinite programming – interior-point methods – polynomial complexity – path-following methods – primal-dual methods – nonlinear programming – Newton method – first-order methods – bundle method – matrix completion The author's research presented in this survey article has been supported in part by NSF through grants INT-9600343, INT-9910084, CCR-9700448, CCR-9902010, CCR-0203113 and ONR through grants N00014-93-1-0234, N00014-94-1-0340 and N00014-03-1-0401. Mathematics Subject Classification (2000): 65K05, 90C06, 90C22, 90C25, 90C30, 90C51     

Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results

 In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over different classes of SDPs show that these methods can be very efficient for some problems. Received: March 18, 2001 / Accepted: May 31, 2001 Published online: October 9, 2002 RID="⋆" ID="⋆"The author was supported by The Ministry of Education, Culture, Sports, Science and Technology of Japan. Key Words. semidefinite programming – primal-dual interior-point method – matrix completion problem – clique tree – numerical results Mathematics Subject Classification (2000): 90C22, 90C51, 05C50, 05C05     

Karush-Kuhn-Tucker systems: regularity conditions,error bounds and a class of Newton-type methods

 We consider optimality systems of Karush-Kuhn-Tucker (KKT) type, which arise, for example, as primal-dual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for KKT systems, such as quasi-regularity or semistability (equivalently, the R ₀-property). Error bounds are useful, among other things, for identifying active constraints and developing efficient local algorithms. We propose a family of local Newton-type algorithms. This family contains some known active-set Newton methods, as well as some new methods. Regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth Newton methods and sequential quadratic programming methods. Received: December 10, 2001 / Accepted: July 28, 2002 Published online: February 14, 2003 Key words. KKT system – regularity – error bound – active constraints – Newton method Mathematics Subject Classification (1991): 90C30, 65K05     

On some interior-point algorithms for nonconvex quadratic optimization

 Recently, interior-point algorithms have been applied to nonlinear and nonconvex optimization. Most of these algorithms are either primal-dual path-following or affine-scaling in nature, and some of them are conjectured to converge to a local minimum. We give several examples to show that this may be untrue and we suggest some strategies for overcoming this difficulty. Received: June 26, 2000 / Accepted: April 2002 Published online: September 5, 2002 Key words. Nonconvex quadratic optimization – local minimum – interior-point algorithms – trust region – branch-and-cut This research is supported by the National Science Foundation Grant CCR-9731273 and DMS-9703490.     

On implementing a primal-dual interior-point method for conic quadratic optimization

 Based on the work of the Nesterov and Todd on self-scaled cones an implementation of a primal-dual interior-point method for solving large-scale sparse conic quadratic optimization problems is presented. The main features of the implementation are it is based on a homogeneous and self-dual model, it handles rotated quadratic cones directly, it employs a Mehrotra type predictor-corrector extension and sparse linear algebra to improve the computational efficiency. Finally, the implementation exploits fixed variables which naturally occurs in many conic quadratic optimization problems. This is a novel feature for our implementation. Computational results are also presented to document that the implementation can solve very large problems robustly and efficiently. Received: November 18, 2000 / Accepted: January 18, 2001 Published online: September 27, 2002 Key Words. conic optimization – interior-point methods – large-scale implementation     

A feasible semismooth asymptotically Newton method for mixed complementarity problems

 Semismooth Newton methods constitute a major research area for solving mixed complementarity problems (MCPs). Early research on semismooth Newton methods is mainly on infeasible methods. However, some MCPs are not well defined outside the feasible region or the equivalent unconstrained reformulations of other MCPs contain local minimizers outside the feasible region. As both these problems could make the corresponding infeasible methods fail, more recent attention is on feasible methods. In this paper we propose a new feasible semismooth method for MCPs, in which the search direction asymptotically converges to the Newton direction. The new method overcomes the possible non-convergence of the projected semismooth Newton method, which is widely used in various numerical implementations, by minimizing a one-dimensional quadratic convex problem prior to doing (curved) line searches. As with other semismooth Newton methods, the proposed method only solves one linear system of equations at each iteration. The sparsity of the Jacobian of the reformulated system can be exploited, often reducing the size of the system that must be solved. The reason for this is that the projection onto the feasible set increases the likelihood of components of iterates being active. The global and superlinear/quadratic convergence of the proposed method is proved under mild conditions. Numerical results are reported on all problems from the MCPLIB collection [8]. Received: December 1999 / Accepted: March 2002 Published online: September 5, 2002 RID="★" ID="★" This work was supported in part by the Australian Research Council. Key Words. mixed complementarity problems – semismooth equations – projected Newton method – convergence AMS subject classifications. 90C33, 90C30, 65H10     

Solving semidefinite-quadratic-linear programs using SDPT3

 This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primal-dual path-following algorithms. The software developed by the authors uses Mehrotra-type predictor-corrector variants of interior-point methods and two types of search directions: the HKM and NT directions. A discussion of implementation details is provided and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported. Received: March 19, 2001 / Accepted: January 18, 2002 Published online: October 9, 2002 Mathematics Subject Classification (2000): 90C05, 90C22     

A primal-dual interior-point method for linear programming based on a weighted barrier function

One motivation for the standard primal-dual direction used in interior-point methods is that it can be obtained by solving a least-squares problem. In this paper, we propose a primal-dual interior-point method derived through a modified least-squares problem. The direction used is equivalent to the Newton direction for a weighted barrier function method with the weights determined by the current primal-dual iterate. We demonstrate that the Newton direction for the usual, unweighted barrier function method can be derived through a weighted modified least-squares problem. The algorithm requires a polynomial number of iterations. It enjoys quadratic convergence if the optimal vertex is nondegenerate.The research of the second author was supported in part by ONR Grants N00014-90-J-1714 and N00014-94-1-0391.     

Solving Problems with Semidefinite and Related Constraints Using Interior-Point Methods for Nonlinear Programming

 In this paper, we describe how to reformulate a problem that has second-order cone and/or semidefiniteness constraints in order to solve it using a general-purpose interior-point algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided. Received: March 10, 2001 / Accepted: January 18, 2002 Published online: September 27, 2002 Key Words. semidefinite programming – second-order cone programming – interior-point methods – nonlinear programming Mathematics Subject Classification (2000): 20E28, 20G40, 20C20