A new primal-dual interior-point method for semi-definite programming
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: primal 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?
- OSTI ID:
- 36356
- Report Number(s):
- CONF-9408161-; TRN: 94:009753-0696
- 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
Similar Records
An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems
Stable computation of search directions for near-degenerate linear programming problems