Abstract
When using interior methods for solving semidefinite programming (SDP), one needs to solve a system of linear equations at each iteration. For problems of large size, solving the system of linear equations can be very expensive. In this paper, based on a semismooth equation reformulation using Fischer′s function, we propose a filter method with trust region for solving large‐scale SDP problems. At each iteration we perform a number of conjugate gradient iterations, but do not need to solve a system of linear equations. Under mild assumptions, the convergence of this algorithm is established. Numerical examples are given to illustrate the convergence results obtained.
Highlights
Semidefinite programming SDP is convex programming over positive semidefinite matrices
We use the following common notation for SDP problems: Xn and Rm denote the space of n×n real symmetric matrices and the space of vectors with m dimensions, respectively; X 0 X 0 denotes that X ∈ Xn is positive semidefinite positive definite, and X 0 X ≺ 0 is used to indicate that X ∈ Xn is negative semidefinite negative definite
Numerical results show that our algorithm is attractive for large-scale SDP problems
Summary
Semidefinite programming SDP is convex programming over positive semidefinite matrices. Since some algorithms for linear optimization can be extended to many general SDP problems, that aroused much interest in SDP. For large-scale SDP problems, IPMs become very slow. We will extend filter-trust-region methods for solving linear or nonlinear programming 16 to large-scale SDP problems and use Lipschitz continuity. We use the following common notation for SDP problems: Xn and Rm denote the space of n×n real symmetric matrices and the space of vectors with m dimensions, respectively; X 0 X 0 denotes that X ∈ Xn is positive semidefinite positive definite , and X 0 X ≺ 0 is used to indicate that X ∈ Xn is negative semidefinite negative definite.
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