More Like this

1. Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form

   Bhojanapalli, Srinadh; Boumal, Nicolas; Jain, Prateek; Netrapalli, Praneeth (January 2018, Proceedings of the 31st Conference On Learning Theory, PMLR)

   Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer–Monteiro factorization approach for solving SDPs. For a large class of SDPs, upon random perturbation of the cost matrix, with high probability, we show that all approximate second-order stationary points are approximate global optima for the penalty formulation of appropriately rank-constrained SDPs, as long as the number of constraints scales sub-quadratically with the desired rank. Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices. We particularize our results to two applications, namely, Max-Cut and matrix completion. 

   more » « less
2. Deterministic Guarantees for Burer‐Monteiro Factorizations of Smooth Semidefinite Programs

   Boumal, Nicolas; Voroninski, Vladislav; Bandeira, Alfonso S. (January 2019, Communications on pure and applied mathematics)

   We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix X of size n. Following the Burer‐Monteiro approach, we optimize a factor Y of size n × p instead, such that X = YYT. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if p is small, but results in a nonconvex optimization problem with a quadratic cost function and quadratic equality constraints in Y. In this paper, we show that if the set of constraints on Y regularly defines a smooth manifold, then, despite nonconvexity, first‐ and second‐order necessary optimality conditions are also sufficient, provided p is large enough. For smaller values of p, we show a similar result holds for almost all (linear) cost functions. Under those conditions, a global optimum Y maps to a global optimum X = YYT of the SDP. We deduce old and new consequences for SDP relaxations of the generalized eigenvector problem, the trust‐region subproblem, and quadratic optimization over several spheres, as well as for the Max‐Cut and Orthogonal‐Cut SDPs, which are common relaxations in stochastic block modeling and synchronization of rotations. © 2018 Wiley Periodicals, Inc. 

   more » « less
3. Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps

   https://doi.org/10.1287/moor.2020.1100  

   Gutekunst, Samuel C.; Williamson, David P. (July 2021, Mathematics of Operations Research)

   null (Ed.)

   The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP. 

   more » « less
4. Max cut and semidefinite rank

   https://doi.org/10.1016/j.orl.2024.107067  

   Mirka, Renee; Williamson, David P (March 2024, Operations Research Letters)

   This paper considers the relationship between semidefinite programs (SDPs), matrix rank, and maximum cuts of graphs. Utilizing complementary slackness conditions for SDPs, we investigate when the rank 1 feasible solution corresponding to a max cut is the unique optimal solution to the Goemans-Williamson max cut SDP by showing the existence of an optimal dual solution with rank n-1 . Our results consider connected bipartite graphs and graphs with multiple max cuts. We conclude with a conjecture for general graphs. 

   more » « less
5. Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering

   https://doi.org/10.1109/CDC40024.2019.9029620  

   Miller, Jared; Zheng, Yang; Roig-Solvas, Biel; Sznaier, Mario; Papachristodoulou, Antonis (December 2019, 2019 IEEE 58th Conference on Decision and Control (CDC))

   Semidefinite programs (SDPs) often arise in relaxations of some NP-hard problems, and if the solution of the SDP obeys certain rank constraints, the relaxation will be tight. Decomposition methods based on chordal sparsity have already been applied to speed up the solution of sparse SDPs, but methods for dealing with rank constraints are underdeveloped. This paper leverages a minimum rank completion result to decompose the rank constraint on a single large matrix into multiple rank constraints on a set of smaller matrices. The re-weighted heuristic is used as a proxy for rank, and the specific form of the heuristic preserves the sparsity pattern between iterations. Implementations of rank-minimized SDPs through interior-point and first-order algorithms are discussed. The problem of subspace clustering is used to demonstrate the computational improvement of the proposed method. 

   more » « less