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An unconstrained minimization method for solving low-rank SDP relaxations of the maxcut problem

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Abstract

In this paper we consider low-rank semidefinite programming (LRSDP) relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem. Using the Gramian representation of a positive semidefinite matrix, the LRSDP problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints. For the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm. Finally, we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes.

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Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica “A. Ruberti”, La Sapienza Università di Roma, via Ariosto, 25, Rome, Italy

    Luigi Grippo & Laura Palagi

  2. Dipartimento di Ingegneria dell’Impresa, Università degli Studi di Roma Tor Vergata, via del Politecnico, 1, Rome, Italy

    Veronica Piccialli

Authors
  1. Luigi Grippo
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  2. Laura Palagi
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  3. Veronica Piccialli
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Corresponding author

Correspondence to Veronica Piccialli.

Additional information

This work was partially supported by MIUR, Research program PRIN 2007-9PLLN7 Nonlinear Optimization, Variational Inequalities and Equilibrium Problems, Italy.

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Grippo, L., Palagi, L. & Piccialli, V. An unconstrained minimization method for solving low-rank SDP relaxations of the maxcut problem. Math. Program. 126, 119–146 (2011). https://doi.org/10.1007/s10107-009-0275-8

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  • DOI: https://doi.org/10.1007/s10107-009-0275-8

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