A Spectral Bundle Method for Semidefinite Programming

Abstract

A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order. We present a method that allows to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose to replace the traditional polyhedral cutting plane model constructed by means of subgradient information by a semidefinite model that is tailored for eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completeness. We present numerical examples demonstrating the efficacy of the approach on combinatorial examples.

IQC analysis and synthesis via nonsmooth optimization

“…[10,17,[19][20][21]. We will extend a method developed by Helmberg et al [12] for semidefinite programming (SDP) to address objective functions of the form f = 1,∞ • F .…”

“…We think of x as the current iterate, g the memory element, reflecting for instance information from previous iterates, or a subgradient at x aggregated over the past as in [12], and we call S(x, g) the descent step generated at x, based on the information in x and g.…”

“…In [3] we have discussed two memoryless descent step generators. Here, we discuss an extension of the method in [12], first to nonconvex 1 •F , then to the semi-infinite case. Just as [12], the method in [10,20] and its extension to the nonconvex case [19] use nonpolyhedral inner approximations of the -subdifferential of 1 , but [12] includes information from past iterates by maintaining an aggregate subgradient from previous iterates.…”

“…Reference [12] studies the convex case f =f and therefore uses only the test in step 3 and the updating mechanism in step 4, which improves the approximationf G(x) by modifying G(x). The authors of [12] use an even more sophisticated update of G in cases whereŶ is large, but this is not mandatory in typical control applications.…”

“…The authors of [12] use an even more sophisticated update of G in cases whereŶ is large, but this is not mandatory in typical control applications. The test in step 5 is not used in [12], where the authors keep r fixed. It becomes necessary because f is nonconvex.…”