Adapting derivative free optimization methods to engineering models with discrete variables

In this paper we extend Continuous Derivative Free (CDF) algorithms that solve optimization models with continuous variables to the solution of optimization models with both continuous and discrete variables. The algorithm fits naturally to the solution of discretized models arising from continuous models. Roughly speaking, the finer the discretization, the closer the discretized solution is to its continuous counterpart. The algorithm also finds stationary points of real problems with continuous and discrete variables. Encouraging results are reported on an access point communication problem and on models solved with a Field Programmable Gate Array (FPGA) device, which generally forces a fixed point discretization of the problem.

[1]  Aimo A. Törn,et al.  Stochastic Global Optimization: Problem Classes and Solution Techniques , 1999, J. Glob. Optim..

[2]  Marco Sciandrone,et al.  An Algorithm Model for Mixed Variable Programming , 2005, SIAM J. Optim..

[3]  M. Cesana,et al.  Radio Planning of Wireless Local Area Networks , 2007, IEEE/ACM Transactions on Networking.

[4]  Charles Audet,et al.  Pattern Search Algorithms for Mixed Variable Programming , 2000, SIAM J. Optim..

[5]  I. Coope,et al.  Frame Based Methods for Unconstrained Optimization , 2000 .

[6]  James M. Whitacre,et al.  Adaptation and self-organization in evolutionary algorithms , 2009, ArXiv.

[7]  Juan C. Burguillo,et al.  A Combined Global & Local Search (CGLS) Approach to Global Optimization , 2006, J. Glob. Optim..

[8]  Hugo de Garis,et al.  A Robust Algorithm for Solving Nonlinear Programming Problems , 2002, Int. J. Comput. Math..

[9]  Scott C. Douglas,et al.  Fixed-Point Algorithms for the Blind Separation of Arbitrary Complex-Valued Non-Gaussian Signal Mixtures , 2007, EURASIP J. Adv. Signal Process..

[10]  Sven Leyffer,et al.  Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming , 2001, Comput. Optim. Appl..

[11]  Jean-Pierre Deschamps,et al.  High speed fixed point dividers for FPGAs , 2009, 2009 International Conference on Field Programmable Logic and Applications.

[12]  Mahmut T. Kandemir,et al.  An Automated Framework for Accelerating Numerical Algorithms on Reconfigurable Platforms Using Algorithmic/Architectural Optimization , 2009, IEEE Transactions on Computers.

[13]  M. Unbehaun,et al.  Coverage planning for outdoor wireless LAN systems , 2002, 2002 International Zurich Seminar on Broadband Communications Access - Transmission - Networking (Cat. No.02TH8599).

[14]  Luís N. Vicente,et al.  Implicitly and densely discrete black-box optimization problems , 2009, Optim. Lett..

[15]  Juan C. Burguillo,et al.  Outdoor WLAN planning via non-monotone derivative-free optimization: algorithm adaptation and case study , 2008, Comput. Optim. Appl..

[16]  Francisco Rodríguez-Henríquez,et al.  4.2 Gbit/s single-chip FPGA implementation of AES algorithm , 2003 .

[17]  Keiichi Yasumoto,et al.  Proposal of flexible implementation of genetic algorithms on FPGAs , 2007 .

[18]  M. Powell.  The BOBYQA algorithm for bound constrained optimization without derivatives , 2009 .

[19]  Charles Audet,et al.  Mesh Adaptive Direct Search Algorithms for Constrained Optimization , 2006, SIAM J. Optim..

[20]  Marco Sciandrone,et al.  On the Global Convergence of Derivative-Free Methods for Unconstrained Optimization , 2002, SIAM J. Optim..

[21]  Matthias Unbehaun,et al.  On the deployment of picocellular wireless infrastructure , 2003, IEEE Wireless Communications.

[22]  Pierre Hansen,et al.  Variable neighborhood search and local branching , 2004, Comput. Oper. Res..

[23]  CHARLES AUDET,et al.  Finding Optimal Algorithmic Parameters Using Derivative-Free Optimization , 2006, SIAM J. Optim..

[24]  Guoliang Xue,et al.  The MINPACK-2 test problem collection , 1992 .

[25]  Saïd Salhi,et al.  A variable neighborhood-based heuristic for the heterogeneous fleet vehicle routing problem , 2009, Eur. J. Oper. Res..

[26]  Pierre Hansen,et al.  Variable neighborhood search: Principles and applications , 1998, Eur. J. Oper. Res..

[27]  Charles Audet,et al.  Mesh adaptive direct search algorithms for mixed variable optimization , 2007, Optim. Lett..

[28]  Serge Gratton,et al.  An active-set trust-region method for derivative-free nonlinear bound-constrained optimization , 2011, Optim. Methods Softw..

[29]  Ubaldo M. García-Palomares,et al.  New Sequential and Parallel Derivative-Free Algorithms for Unconstrained Minimization , 2002, SIAM J. Optim..

[30]  T. Sansaloni,et al.  Area-efficient FPGA-based FFT processor , 2003 .

[31]  Said F. Al-Sarawi,et al.  Low power serial-parallel dynamic shift register , 2003 .

[32]  Angus K. M. Wu,et al.  VLSI implementation of genetic four-step search for block matching algorithm , 2003, IEEE Trans. Consumer Electron..

[33]  Ioannis Papaefstathiou,et al.  Implementation of a genetic algorithm on a virtex-ii pro FPGA , 2009, FPGA '09.

[34]  Zvi Drezner,et al.  The uncapacitated facility location problem with demand-dependent setup and service costs and customer-choice allocation , 2007, Eur. J. Oper. Res..

[35]  Humberto Rocha,et al.  Incorporating minimum Frobenius norm models in direct search , 2010, Comput. Optim. Appl..