Entropy-Penalized Semidefinite Programming

Abstract

Low-rank methods for semi-definite programming (SDP) have gained a lot of interest recently, especially in machine learning applications. Their analysis often involves determinant-based or Schattennorm penalties, which are difficult to implement in practice due to high computational efforts. In this paper, we propose Entropy-Penalized Semi-Definite Programming (EP-SDP) 1 , which provides a unified framework for a broad class of penalty functions used in practice to promote a low-rank solution. We show that EP-SDP problems admit an efficient numerical algorithm, having (almost) linear time complexity of the gradient computation; this makes it useful for many machine learning and optimization problems. We illustrate the practical efficiency of our approach on several combinatorial optimization and machine learning problems.Contribution. Our contributions can be summarized as follows. We show

Warm-starting quantum optimization

“…(QP) Relaxation SOCP Relaxations [74] (SDP) Relaxation [53,65,75] Entropy-penalized SDP [76] Sparse Moment SDP [73] Moment SDP [72] Classical pre-processing Quantum circuit (QC) 0.5-Approximation [77,78] Random hyperplane [53,75] Iterative [79,80] Iterative [81] Measurement in a QC We now elaborate on the example of Goemans-Williamson random-hyperplane rounding of (SDP). For a given GW cut we generate an initial state using Y -rotations with a ε ∈ (0, 0.5), as discussed in Sec.…”

“…For example, the (QP) relaxation can be seen as a second-order cone programming (SOCP) relaxation, and could be strengthened iteratively [74], until its objective-function value coincides with the objective-function value of the non-convex (QUBO), albeit at the cost of an exponential growth of the relaxation. Similarly, one could strengthen the (SDP) relaxation either by using an entropy-penalizing term [76] or by using the Moment/SOS hierarchy [72] and its sparse variant [73].…”

Warm-starting quantum optimization

“…(QP) Relaxation SOCP Relaxations [72] (SDP) Relaxation [50,63,73] Entropy-penalized SDP [74] Sparse Moment SDP [71] Moment SDP [70] Classical pre-processing Quantum circuit (QC) 0.5-Approximation [75,76] Random hyperplane [50,73] Iterative [77,78] Iterative [79] Measurement in a QC WS-QAOA WS-RQAOA…”

“…For example, the (QP) relaxation can be seen as a second-order cone programming (SOCP) relaxation, and could be strengthened iteratively [72], until its objectivefunction value coincides with the objective-function value of the non-convex (QUBO), albeit at the cost of an exponential growth of the relaxation. Similarly, one could strengthen the (SDP) relaxation either by using an entropy-penalizing term [74] or by using the Moment/SOS hierarchy [70] and its sparse variant [71], which converge faster than the SOCP hierarchy [72], from a stronger basic relaxation.…”

A Decomposition Augmented Lagrangian Method for Low-rank Semidefinite Programming

“…The non-negative constraints on the semidefinite variable can be further relaxed into the non-negative constraints on the factorized variable. To solve the original problem better, h(X) can be the entropic penality function [25,43,35] to find a low-rank solution.…”