Abstract
Tenfold improvements in computation speed can be brought to the alternating direction method of multipliers (ADMM) for Semidefinite Programming with virtually no decrease in robustness and provable convergence simply by projecting approximately to the Semidefinite cone. Instead of computing the projections via “exact” eigendecompositions that scale cubically with the matrix size and cannot be warm-started, we suggest using state-of-the-art factorization-free, approximate eigensolvers, thus achieving almost quadratic scaling and the crucial ability of warm-starting. Using a recent result from Goulart et al. (Linear Algebra Appl 594:177–192, 2020. https://doi.org/10.1016/j.laa.2020.02.014), we are able to circumvent the numerical instability of the eigendecomposition and thus maintain tight control on the projection accuracy. This in turn guarantees convergence, either to a solution or a certificate of infeasibility, of the ADMM algorithm. To achieve this, we extend recent results from Banjac et al. (J Optim Theory Appl 183(2):490–519, 2019. https://doi.org/10.1007/s10957-019-01575-y) to prove that reliable infeasibility detection can be performed with ADMM even in the presence of approximation errors. In all of the considered problems of SDPLIB that “exact” ADMM can solve in a few thousand iterations, our approach brings a significant, up to 20x, speedup without a noticeable increase in ADMM’s iterations.
Highlights
Semidefinite Programming is of central importance in many scientific fields
Areas as diverse as kernel-based learning [20], dimensionality reduction [9] analysis and synthesis of state feedback policies of linear dynamical systems [6], sum of squares programming [33], optimal power flow problems [21] and fluid mechanics [15] rely on Semidefinite Programming as a crucial enabling technology
We suggest solving (P), i.e., finding a solution (x, z, y) where yis a Lagrange multiplier for the equality constraint of (P), with the approximate version of alternating direction method of multipliers (ADMM) described in Algorithm 1
Summary
Semidefinite Programming is of central importance in many scientific fields. Areas as diverse as kernel-based learning [20], dimensionality reduction [9] analysis and synthesis of state feedback policies of linear dynamical systems [6], sum of squares programming [33], optimal power flow problems [21] and fluid mechanics [15] rely on Semidefinite Programming as a crucial enabling technology.The wide adoption of Semidefinite Programming was facilitated by reliable algorithms that can solve semidefinite problems with polynomial worst-case complexity [6]. Semidefinite Programming is of central importance in many scientific fields. Areas as diverse as kernel-based learning [20], dimensionality reduction [9] analysis and synthesis of state feedback policies of linear dynamical systems [6], sum of squares programming [33], optimal power flow problems [21] and fluid mechanics [15] rely on Semidefinite Programming as a crucial enabling technology. The wide adoption of Semidefinite Programming was facilitated by reliable algorithms that can solve semidefinite problems with polynomial worst-case complexity [6]. For small to medium-sized problems, it is widely accepted that primal–dual Interior Point methods are efficient and robust and are often the method of choice. The limitations of interior point methods become evident in large-scale problems, since each iteration requires factorizations of large Hessian matrices. First-order methods avoid this bottleneck and thereby scale better to large problems, with the ability to provide modest-accuracy solutions for many large-scale problems of practical interest
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