Matrix Relaxations in Combinatorial Optimization

Rendl, Franz

doi:10.1007/978-1-4614-1927-3_17

Cited by 5 publications

(2 citation statements)

References 54 publications

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“…It is NP-hard to compute the stability number Karp (1972) and it is even hard to approximate it Håstad (1999), therefore upper bounds on the stability number are of interest. One possible upper bound is the Lovász theta function ϑ(G), see for example Rendl (2012). The Lovász theta function is defined as the optimal value of the SDP…”

Section: The Stable Set Problem and An Sdp Relaxationmentioning

confidence: 99%

Improving ADMMs for solving doubly nonnegative programs through dual factorization

et al. 2020

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Alternating direction methods of multipliers (ADMMs) are popular approaches to handle large scale semidefinite programs that gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable. It is well known that first order methods are not suitable to compute high precision optimal solutions, however an optimal solution of moderate precision often suffices to get high quality lower bounds on the primal optimal objective function value. We present methods to obtain such bounds by either perturbing the dual objective function value or by constructing a dual feasible solution from a dual approximate optimal solution. Both procedures can be used as a post-processing phase in our ADMMs. Numerical results for DNNs that are relaxations of the stable set problem are presented. They show the impact of using the factorization of the dual variable in order to improve the progress towards the optimal solution within an iteration of the ADMM. This decreases the number of iterations as well as the CPU time to solve the DNN to a given precision. The experiments also demonstrate that within a computationally cheap post-processing, we can compute bounds that are close to the optimal value even if the DNN was solved to moderate precision only. This makes ADMMs applicable also within a branch-and-bound algorithm.

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Section: The Stable Set Problem and An Sdp Relaxationmentioning

confidence: 99%

Improving ADMMs for solving doubly nonnegative programs through dual factorization

et al. 2020

View full text Add to dashboard Cite

show abstract

“…It is NP-hard to compute the stability number [8] and it is even hard to approximate it [5], therefore upper bounds on the stability number are of interest. One possible upper bound is the Lovász theta function ϑ(G), see for example [12]. The Lovász theta function is defined as the optimal value of the SDP…”

Section: The Stable Set Problem and An Sdp Relaxationmentioning

confidence: 99%

Improving ADMMs for Solving Doubly Nonnegative Programs through Dual Factorization

Cerulli,

De Santis,

Gaar

et al. 2019

Preprint

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Augmented Lagrangian methods are among the most popular first-order approaches to handle large scale semidefinite programs. In particular, alternating direction methods of multipliers (ADMMs), which are a variant of augmented Lagrangian methods, gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable.It is well known that first order methods are not suitable to compute high precision optimal solutions, however an optimal solution of moderate precision often suffices to get high quality lower bounds on the primal optimal objective function value. We present methods to obtain such bounds by either perturbing the dual objective function value or by constructing a dual feasible solution from a dual approximate optimal solution. Both procedures can be used as a post-processing phase in our ADMMs.Numerical results for DNNs that are relaxations of the stable set problem are presented. They show the impact of using the factorization of the dual variable in order to improve the progress towards the optimal solution within an iteration of the ADMM. This decreases the number of iterations as well as the CPU time to solve the DNN to a given precision. The experiments also demonstrate that within a computationally cheap post-processing, we can compute bounds that are close to the optimal value even if the DNN was solved to moderate precision only. This makes ADMMs applicable also within a branch-and-bound algorithm.

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Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem

2013

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Matrix Relaxations in Combinatorial Optimization

Cited by 5 publications

References 54 publications

Improving ADMMs for solving doubly nonnegative programs through dual factorization

Improving ADMMs for solving doubly nonnegative programs through dual factorization

Improving ADMMs for Solving Doubly Nonnegative Programs through Dual Factorization

Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem

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