## Abstract

Several formulations and methods used in solving an NP-hard discrete optimization problem, maximum clique, are considered in a dynamical system perspective proposing continuous methods to the problem. A compact form for a saturated linear dynamical network, recently developed for obtaining approximations to maximum clique, is given so its relation to the classical gradient projection method of constrained optimization becomes more visible. Using this form, gradient-like dynamical systems as continuous methods for finding the maximum clique are discussed. To show the one to one correspondence between the stable equilibria of the saturated linear dynamical network and the minima of objective function related to the optimization problem, La Salle's invariance principle has been extended to the systems with a discontinuous right-hand side. In order to show the efficiency of the continuous methods simulation results are given comparing saturated the linear dynamical network, the continuous Hopfield network, the cellular neural networks and relaxation labelling networks. It is concluded that the quadratic programming formulation of the maximum clique problem provides a framework suitable to be incorporated with the continuous relaxation of binary optimization variables and hence allowing the use of gradient-like continuous systems which have been observed to be quite efficient for minimizing quadratic costs.

Original language | English |
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Pages (from-to) | 268-276 |

Number of pages | 9 |

Journal | Bulletin of the Technical University of Istanbul |

Volume | 51 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1999 |

### Funding

This work was supported by TUG BITAK-CNRS, Turkish and French Scienti"c and Technical Research Councils

Funders | Funder number |
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BITAK-CNRS | |

Technical Research Councils |

## Keywords

- Continuous methods
- Discontinuous differential equations
- Gradient systems
- Gradient-projection algorithm
- La Salle's invariance principle
- Maximum clique problem
- Optimization