Revisiting Spectral Bundle Methods: Primal-dual (Sub)linear Convergence Rates

Abstract

The spectral bundle method proposed by Helmberg and Rendl [25] is well established for solving large scale semidefinite programs (SDP) thanks to its low per iteration computational complexity and strong practical performance. In this paper, we revisit this classic method showing it achieves sublinear convergence rates in terms of both primal and dual SDPs under merely strong duality. Prior to this work, only limited dual guarantees were known. Moreover, we develop a novel variant, called the block spectral bundle method (Block-Spec), which not only enjoys the same convergence rate and low per iteration complexity, but also speeds up to linear convergence when the SDP admits strict complementarity. Numerically, we demonstrate the effectiveness of both methods, confirming our theoretical findings that the block spectral bundle method can substantially speed up convergence.

A preconditioned iterative interior point approach to the conic bundle subproblem

“…Note that a higher edge density increases the cost of matrix-vector multiplications but also causes a larger offset in objective value which entails somewhat reduced precision requirements for the KKT systems to reach the same relative precision. As the experiments will show, the model size-it is selected by ConicBundle on basis of the active rank, starts with roughly twice this size after descent steps for reasons investigated in [10] and increases further over null steps-seems to be less dependent on the edge densities but grows markedly with the number of nodes.…”

“…If the subspace is kept small, this allows to find approximately optimal solutions in reasonable time. In order to reach solutions of higher precision it seems unavoidable to go beyond the full active eigenspace [10,23]. In the current implementation within the callable library ConicBundle [19], which also supports second order cone and nonnegative variables, the quadratic subproblem is solved by an interior point approach.…”

A Preconditioned Iterative Interior Point Approach to the Conic Bundle Subproblem

“…If the subspace is kept small, this allows to find approximately optimal solutions in reasonable time. In order to reach solutions of higher precision it seems unavoidable to go beyond the full active eigenspace [17,9]. In the current implementation within the callable library ConicBundle [15] the quadratic subproblem is solved by an interior point approach.…”