On Combinatorial Optimization Problems with Mobile Sites and Resources

This paper develops a framework to solve combinatorial resource allocation problems in a setting where the sites and resources are allowed to have dynamics. The formulation draws analogy from statistical physics to define the Free Energy function which is used as a measure of coverage function. A class of dynamics for the sites and resources is prescribed which guarantees coverage. This is done by casting the problem as a control problem in which we design the resource velocities to ensure that the time derivative of the Free Energy function is non positive.

[1]  Allen Gersho,et al.  Vector quantization and signal compression , 1991, The Kluwer international series in engineering and computer science.

[2]  M. A. Dahleh,et al.  Constraints on locational optimization problems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[3]  Vincent Kanade,et al.  Clustering Algorithms , 2021, Wireless RF Energy Transfer in the Massive IoT Era.

[4]  Sonia Martínez,et al.  Coverage control for mobile sensing networks , 2002, IEEE Transactions on Robotics and Automation.

[5]  J. Loewenthal.  DECISION , 1969, Definitions.

[6]  Sanjoy K. Mitter,et al.  Control with Limited Information , 2001, Eur. J. Control.

[7]  Daniel Liberzon,et al.  On quantized control and geometric optimization , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[8]  K. Rose.  Deterministic annealing for clustering, compression, classification, regression, and related optimization problems , 1998, Proc. IEEE.

[9]  C. W. Therrien,et al.  Decision, Estimation and Classification: An Introduction to Pattern Recognition and Related Topics , 1989 .

[10]  Robert M. Gray,et al.  Multiple local optima in vector quantizers , 1982, IEEE Trans. Inf. Theory.

[11]  Alfio Quarteroni,et al.  Numerical Mathematics (Texts in Applied Mathematics) , 2006 .

[12]  Kenneth Rose.  Deterministic annealing, clustering, and optimization , 1991 .

[13]  John A. Hartigan,et al.  Clustering Algorithms , 1975 .

[14]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[15]  Said Salhi,et al.  Facility Location: A Survey of Applications and Methods , 1996 .

[16]  Nicola Elia,et al.  Stabilization of linear systems with limited information , 2001, IEEE Trans. Autom. Control..

[17]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[18]  S. Hyakin,et al.  Neural Networks: A Comprehensive Foundation , 1994 .

[19]  Francesco Bullo,et al.  COVERAGE CONTROL FOR MOBILE SENSING NETWORKS: VARIATIONS ON A THEME , 2002 .

[20]  Francesco Bullo,et al.  Coordination and Geometric Optimization via Distributed Dynamical Systems , 2003, SIAM J. Control. Optim..

[21]  P. Sharma,et al.  A deterministic annealing approach to combinatorial library design for drug discovery , 2005, Proceedings of the 2005, American Control Conference, 2005..

[22]  F. Bullo,et al.  Decentralized algorithms for vehicle routing in a stochastic time-varying environment , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[23]  Rose,et al.  Statistical mechanics and phase transitions in clustering. , 1990, Physical review letters.

[24]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[25]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.