Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization

2010 ◽  
Author(s):  
Levent Tunçel
Keyword(s):  
Combinatorial Optimization ◽  
Semidefinite Programming

Probabilistic Combinatorial Optimization: Moments, Semidefinite Programming, and Asymptotic Bounds

2004 ◽  
Vol 15 (1) ◽  
pp. 185-209 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Karthik Natarajan ◽  
Chung-Piaw Teo
Keyword(s):  
Combinatorial Optimization ◽  
Semidefinite Programming ◽  
Asymptotic Bounds

Semidefinite programming in combinatorial optimization

1997 ◽  
Vol 79 (1-3) ◽  
pp. 143-161 ◽  
Author(s):  
Michel X. Goemans
Keyword(s):  
Combinatorial Optimization ◽  
Semidefinite Programming

scholarly journals Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps

2021 ◽  
Author(s):  
Samuel C. Gutekunst ◽  
David P. Williamson
Keyword(s):  
Combinatorial Optimization ◽  
Semidefinite Programming ◽  
Traveling Salesman Problem ◽  
Linear Program ◽  
Traveling Salesman ◽  
Quadratic Assignment ◽  
Integrality Gap ◽  
Permutation Matrices ◽  
Semidefinite Programming Relaxations ◽  
The Traveling Salesman Problem

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.

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