We study semidefinite programming (SDP) relaxations for the NP-hard problem of globally optimizing a quadratic function over the Stiefel manifold. We introduce a strengthened relaxation based on two recent ideas in the literature: (i) a tailored SDP for objectives with a block-diagonal Hessian; (ii) and the use of the Kronecker matrix product to construct SDP relaxations. Using synthetic instances on … Read more
This paper introduces a new storage-optimal first-order method (FOM), CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs , is characterized by low-rank solutions, a priori knowledge of the restriction of the SDP solution to a small subspace, and standard … Read more
We investigate the optimal piecewise linear approximation of the bivariate product $ xy $ over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the approximation, while respecting a prescribed approximation error. First, we show how to construct optimal triangulations consisting of up to five simplices. Using … Read more
Neural networks tend to achieve better accuracy with training if they are larger — even if the resulting models are overparameterized. Nevertheless, carefully removing such excess parameters before, during, or after training may also produce models with similar or even improved accuracy. In many cases, that can be curiously achieved by heuristics as simple as … Read more
We present the Branch-and-Bound Performance Estimation Programming (BnB-PEP), a unified methodology for constructing optimal first-order methods for convex and nonconvex optimization. BnB-PEP poses the problem of finding the optimal optimization method as a nonconvex but practically tractable quadratically constrained quadratic optimization problem and solves it to certifiable global optimality using a customized branch-and-bound algorithm. By … Read more
This paper begins with a class of convex quadratic programs (QPs) with bounded variables solvable by the parametric principal pivoting algorithm with $\mbox{O}(n^3)$ strongly polynomial complexity, where $n$ is the number of variables of the problem. Extension of the Hessian class is also discussed. Our research is motivated by a preprint [7] wherein the efficient … Read more
This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “L0-path” where the discontinuous L0-function provides the exact sparsity count; the “L1-path” where the L1-function provides a convex surrogate of sparsity count; … Read more
Applying an interior-point method to the central-path conditions is a widely used approach for solving quadratic programs. Reformulating these conditions in the log-domain is a natural variation on this approach that to our knowledge is previously unstudied. In this paper, we analyze log-domain interior-point methods, proving their polynomial-time convergence and illustrating their excellent practical performance. … Read more
Bilinear terms naturally appear in many optimization problems. Their inherent nonconvexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of … Read more
The constant modulus constraint is widely used in analog beamforming, hybrid beamforming, intelligent reflecting surface design, and radar waveform design. The quadratically constrained quadratic programming (QCQP) problem is also widely used in signal processing. However, the QCQP with extra constant modulus constraints was not systematically studied in mathematic programming and signal processing. For example, the … Read more
