We consider the problem of estimating the discrete clustering structures under the subGaussian mixture model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integervalued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signaltonoise ratio. In addition, we show that the SDP relaxation is robust under the semirandom setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semirandom model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the “globaltolocal” mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes, the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recovermore »
Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps
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 more »
 Award ID(s):
 1908517
 Publication Date:
 NSFPAR ID:
 10298221
 Journal Name:
 Mathematics of Operations Research
 ISSN:
 0364765X
 Sponsoring Org:
 National Science Foundation
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