skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: An interior point method for semidefinite programming

Abstract

Semidefinite programming is of rising importance in min-max eigenvalue problems and relaxations for NP-complete problems. We present a new interior point algorithm based on the linearization XZ {minus} {mu}I which is very successful in linear programming. We show that, although the original {Delta}X is not symmetric, by using the symmetric part of {Delta}X convergence can be guaranteed. The algorithm works for arbitrary combinations of equalities and inequalities and, contrary to linearizations such as XZ + ZX {minus} {mu}I, does not make use of Kronecker products. In numerical experiments the algorithm exhibits fast quadratic convergence.

Authors:
; ; ;
Publication Date:
OSTI Identifier:
36122
Report Number(s):
CONF-9408161-
TRN: 94:009753-0397
Resource Type:
Conference
Resource Relation:
Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
Country of Publication:
United States
Language:
English
Subject:
99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; NONLINEAR PROBLEMS; NUMERICAL SOLUTION; MATRICES; NONLINEAR PROGRAMMING; ALGORITHMS

Citation Formats

Helmberg, C., Rendl, F., Vanderbei, R.J., and Wolkowicz, H. An interior point method for semidefinite programming. United States: N. p., 1994. Web.
Helmberg, C., Rendl, F., Vanderbei, R.J., & Wolkowicz, H. An interior point method for semidefinite programming. United States.
Helmberg, C., Rendl, F., Vanderbei, R.J., and Wolkowicz, H. 1994. "An interior point method for semidefinite programming". United States. doi:.
@article{osti_36122,
title = {An interior point method for semidefinite programming},
author = {Helmberg, C. and Rendl, F. and Vanderbei, R.J. and Wolkowicz, H.},
abstractNote = {Semidefinite programming is of rising importance in min-max eigenvalue problems and relaxations for NP-complete problems. We present a new interior point algorithm based on the linearization XZ {minus} {mu}I which is very successful in linear programming. We show that, although the original {Delta}X is not symmetric, by using the symmetric part of {Delta}X convergence can be guaranteed. The algorithm works for arbitrary combinations of equalities and inequalities and, contrary to linearizations such as XZ + ZX {minus} {mu}I, does not make use of Kronecker products. In numerical experiments the algorithm exhibits fast quadratic convergence.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = 1994,
month =
}

Conference:
Other availability
Please see Document Availability for additional information on obtaining the full-text document. Library patrons may search WorldCat to identify libraries that hold this conference proceeding.

Save / Share:
  • In this talk we describe ongoing work on an interior point method for general nonlinearly constrained optimization. Our approach is a primal dual method related to those used successfully in linear and convex programming, but is based in part on the trust region method for equality con strained optimization due to Byrd and Omojokun, and developed further by Nocedal and Plantenga. This allows us to use an indefinite Hessian matrix based on exact second derivatives or approximations. Such an approach is expected to be most effective when satisfaction of nonlinear constraints is a major issue. We describe the operation ofmore » the method and present some global convergence results.« less
  • Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm that uses the primal-dual interior point method to solve the linear programming relaxations is described. A point which is a good warm start for a simplex-based cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added.more » Problems considered include clustering problems, max-cut problems, and quadratic 0, 1 problems.« less
  • The semidefinite programming problem (SDP) is: min tr CX s.t. tr A{sub i}X = b{sub i}, i = 1, {hor_ellipsis}, m, and X {>=} 0. Here C and A{sub i} are fixed symmetric matrices and X {>=} 0 is a semidefinite constraint on the unknown symmetric matrix variable X. The dual of SDP is: max b{sup T} y s.t. Z + {Sigma}{sub i=1}{sup m} y{sub i}A{sub i} = C and Z {>=} 0. Interior point methods for SDP have been developed by Nesterov and Nemirovskii, Alizadeh, Vandenberghe and Boyd, and others. Primal-dual methods use Newton steps for three equations: primalmore » feasibility, dual feasibility, and complementarity/centering. The last condition is the matrix equation XZ = {mu}I, where {mu} {yields} 0. Different ways of rewriting this equation as a symmetric matrix equation lead to different algorithmic variants, all involving Kronecker products of matrices. We describe a new primal-dual interior point algorithm for SDP, which iterates in the space of commuting matrices X and Z. Effectively X and Z are replaced by QXQ{sup T} and QZQ{sup T} respectively, where Q is orthogonal and X and Z are now diagonal. The complementarity/centering condition then reduces to an elementwise condition as in LP. The price paid for the diagonalization is the appearance of the nonlinear variable Q in the feasibility equations. Numerical results using a predictor-corrector implementation are presented. The following question is also addressed: what ranks of the primal and dual solution matrices are possible, under various nondegeneracy assumptions?« less
  • The primal barrier method of interior points developed for linear programming can be extended to solve general nonlinear optimization problems. Our fundamental concept is to recast the idea of affine scaling and view it as the result of using an ellipsoidally shaped trust region on iterative subproblems. The standard logarithmic barrier term added to the objective is retained in our nonlinear formulation to promote centering with respect to bound constraints. Our nonlinear algorithm iteratively solves quadratic subproblems subject to linearized equality constraints and an ellipsoidal trust region. Iterates must remain in the interior of the bound inequalities, but need notmore » satisfy equality constraints until convergence. The algorithm reduces to the primal barrier interior point method for linear programming problems. In the past two years M. Lalee, Nocedal and Plantenga have implemented the Byrd-Omojokun algorithm for solving large-scale equality constrained problems using trust regions, and with this software test results have been obtained for the new interior point algorithm.« less
  • It has been observed that some interior point algorithms for linear programming can be extended, in a sense step by step, to optimization problems over more general domains, such as the cone of positive semidefinite matrices and the so-called {open_quotes}ice cream{close_quotes} cone. Most of the algorithms that have been extended, however are mostly primal or dual algorithms. It turns out that analogous extensions of primal-dual methods are more challenging. We discuss such extensions and complexity issues related to such generalizations.