Semidefinite Programming Methods for the Symmetric Traveling Salesman Problem

Cvetković, Dragoš; Čangalović, Mirjana; Kovačevíc-Vujčić, Vera

doi:10.1007/3-540-48777-8_10

Cited by 33 publications

(62 citation statements)

References 21 publications

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“…Our primary objective in these computational experiments was to investigate the "quality" of the cuts where "quality" was interpreted as the decrease in integrality gap per cut. We considered mixed 0-1 SDPs resulting from the pure 0-1 SDP formulation of maxcut [19], the pure 0-1 SDP formulation of the traveling salesman problem [15,13] and Lovász-Schrijver lifting of mixed 0-1 LPs. Since SDPs are inherently harder to solve and there are no general purpose warm-start strategies for SDPs, it is unlikely that the 0-1 SDP formulation will at present be the method of choice for solving these problems.…”

Section: Computational Resultsmentioning

confidence: 99%

“…We solved the {0, 1}-SDP formulation for the TSP given in (3) [15] (see also [13]). In every iteration of the cut algorithm, we chose the "most" fractional variable, i.e.…”

Section: Computational Results For the Traveling Salesman Problemmentioning

confidence: 99%

“…We illustrate this using the traveling salesman problem (TSP) and the maxcut problem. Cvetković et al [15] (see also [18,13]) proposed the following mixed 0-1 formulation for the TSP.…”

Section: Gomory Cutsmentioning

confidence: 99%

“…In the early 1990s, Nemirovski and Nesterov [37] developed interior point algorithms for solving conic programs. Since then conic programming models have found applications in many disparate fields such as robust control [12], combinatorial optimization [1,17,18,15], robust optimization [9,16], and finance [29,8,21].…”

Section: Introductionmentioning

confidence: 99%

“…For instance, combinatorial optimization problems such as the TSP [15,13] and the maxcut [19] can be formulated as mixed 0-1 SDPs. Another important source of mixed 0-1 SDPs is the semidefinite lifting of mixed 0-1 LPs [31,25,26].…”

Section: Introductionmentioning

confidence: 99%

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Cuts for mixed 0-1 conic programming

2005

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In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.

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Section: Computational Resultsmentioning

confidence: 99%

“…We solved the {0, 1}-SDP formulation for the TSP given in (3) [15] (see also [13]). In every iteration of the cut algorithm, we chose the "most" fractional variable, i.e.…”

Section: Computational Results For the Traveling Salesman Problemmentioning

confidence: 99%

“…We illustrate this using the traveling salesman problem (TSP) and the maxcut problem. Cvetković et al [15] (see also [18,13]) proposed the following mixed 0-1 formulation for the TSP.…”

Section: Gomory Cutsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cuts for mixed 0-1 conic programming

2005

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Cutting Planes for Mixed 0-1 Semidefinite Programs

Iyengar

Çezik

2001

Integer Programming and Combinatorial Optimization

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SDP Relaxations for Some Combinatorial Optimization Problems

Sotirov

2011

International Series in Operations Research &Amp; Management Science

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In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth problem. NotationThe space of p × q real matrices is denoted by R p×q , the space of k × k symmetric matrices is denoted by S k , and the space of k×k symmetric positive semidefinite matrices by S + k . We will sometimes also use the notation X 0 instead of X ∈ S + k , if the order of the matrix is clear from the context. For index sets α, β ⊂ {1, . . . , n}, we denote the submatrix that contains the rows of A indexed by α and the columns indexed by β as A(α, β). If α = β, the principal submatrix A(α, α) of A is abbreviated as A(α). The ith column of a matrix C is denoted by C :,i .We use I n to denote the identity matrix of order n, and e i to denote the ith standard basis vector. Similarly, J n and u n denote the n×n all-ones matrix and all-ones n-vector respectively, and 0 n×n is the zero matrix of order n. We will omit subscripts if the order is clear from the context. The set of n × n permutation matrices is denoted by Π n . We set E ij = e i e T j . The 'vec' operator stacks the columns of a matrix, while the 'diag' operator maps an n × n matrix to the n-vector given by its diagonal. The adjoint operator of 'diag' we denote by 'Diag'. The trace operator is denoted by 'tr'.The Kronecker product A ⊗ B of matrices A ∈ R p×q and B ∈ R r×s is defined as the pr × qs matrix composed of pq blocks of size r × s, with block ij given by a ij B, i = 1, . . . , p, j = 1, . . . , q.

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Semidefinite Programming Methods for the Symmetric Traveling Salesman Problem

Cited by 33 publications

References 21 publications

Cuts for mixed 0-1 conic programming

Cuts for mixed 0-1 conic programming

Cutting Planes for Mixed 0-1 Semidefinite Programs

SDP Relaxations for Some Combinatorial Optimization Problems

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