Abstract
Parallel globed optimization is one very promising area of research since due to inherent difficulty of the problems it studies, only instances of limited dimension can be solved in reasonable computer time on conventional machines. However, the use of parallel and distributed processing can substantially increase the possibilities for the success of the global optimization approach in practice. In this chapter we are concerned with the development of parallel algorithms for solving certain classes of non-convex optimization problems. We present an introductory survey of exact parallel algorithms that have been used to solve structured (partially separable) problems and problems with simple constraints, and algorithms based on parallel local search and its deterministic or stochastic refinements for solving general non-convex problems. Indefinite quadratic programming, posynomial optimization, and the general global concave minimization problem can be solved using these approaches. In addition, the minimum concave cost network flow problem and location problems with economies of scale are used in illustrating these techniques for the solution of large-scale, structured problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Archetti and F. Schoen (1982)“Asynchronous Parallel Search in Global Optimization Problems”, Lecture Notes in Control and Information Systems 38, Springer-Verlag, pp. 501–507
M. Avriel (1976) Nonlinear Programming: Analysis and Methods, Prentice- Hall, N.J.
W.P. Baritompa, Zhang Baoping, R.H. Mladineo, G.R. Wood and Z.B. Zabinsky (1995) “Towards Pure Adaptive Search”, Journal of Global Optimization 7, pp. 93–110
Roberto Battiti and Giampietro Tecchiolli (1993) “The Reactive Tabu Search”, ORSA Journal of Computing (to appear)
Roberto Battiti and Giampietro Tecchiolli (1994) “The Continuous Reactive Tabu Search: Blending Combinatorial Optimization and Stochastic Search for Global Optimization”, Preprint UTM 432 Dipartimento Di Matematica, Universta Di Trento, Via Sommarive 14,38050 Povo (Trento), Italy.
Mohtar S. Bazaraa and C. M. Shetty (1979) Nonlinear Programming - Theory and Algorithms, John Wiley & Sons.
Sonja Berner (1996) “Parallel Methods for Verified Global Optimization: Practice and Theory”, Journal of Global Optimization 9, pp. 1–22.
M. Bertocchi (1990) “A Parallel Algorithm for Global Optimization”, Optimization 21, pp. 379–386.
Bruno Betro (1991) “Bayesian Methods in Global Optimization”, Journal of Global Optimization 1, pp. 1–14.
Dimitri P. Bertsekas and John N. Tsitsiklis (1989) Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, London.
T.B. Boffey and P. Saeidi (1996) “A Parallel Branch-and-Bound Method for a Network Design Problem”, Belgian Journal of Operations Research 32, pp. 69–83.
P. Brachetti, M. De Felice Ciccoli, G. Di Pillo and S. Lucidi (1994) “A New Version of the Price’s Algorithm for Global Optimization”, Journal of Global Optimization, to appear.
Richard H. Byrd, Elisabeth Eskow, Robert B. Schnabel and Sharon L. Smith (1991) “Parallel Global Optimization: Numerical Methods, Dynamic Scheduling Methods, and Application to Molecular Configuration”, Research Report CU-CS-553-91, University of Colorado at Boulder, Department of Computer Science, Campus Box 430, Boulder, Colorado 80309-0430, USA.
Richard H. Byrd, Thomas Derby, Elisabeth Eskow, Klaas P.B. Oldenkamp and Robert B. Schnabel (1993) “A New Stochastic/ Perturbation Method for Large-Scale Global Optimization and Its Application to Water Cluster Problems”, Research Report CU-CS-652-93, University of Colorado at Boulder, Department of Computer Science, Campus Box 430, Boulder, Colorado 80309-0430, USA.
Richard H. Byrd, C.L. Dert, A.H.G. Rinnooy Kan and R.B. Schnabel (1990) “Concurrent Stochastic Methods for Global Optimization”, Mathematical Programming 46, pp. 1–29.
R.D. Chamberlain, M.N. Edelman, M.A. Franklin, E.E. Witte (1988) “Simulated Annealing on a Multiprocessor”, Proceedings of 1988 IEEE International Conference on Computer Design, pp. 540–544.
Djurdje Cvijovic and Jacek Klinowski (1995) “Taboo Search: An Approach to the Multiple Minima Problem”, Science 267, pp. 664–666
A. Dekker and E. Aarts (1991) “Global Optimization and Simulated Annealing”, Mathematical Programming 50, pp. 367–393.
L.C.W. Dixon and M. Jha (1993) “Parallel Algorithms for Global Optimization”, Journal of Optimization Theory and Application 79, pp. 385–395.
H.M.M. Ten Eikelder, M.G.A. Verhoeven, T.W.M. Vossen and E.H.L. Aarts (1995) “A Probabilistic Analysis of Local Search”. In: I. H. Osman and S. W. Otto (Eds) Metaheuristics: The State of the Art, Kluwer, Boston.
Jerry Eriksson (1991) “Parallel Global Optimization Using Interval Analysis”, Research Report UMINF-91.17, University of Umea, Institute of In-formation Processing, Department of Computing Science, S-901 87 Umea, Sweden.
Jerry Eriksson and Per Lindstrom (1995) “A Parallel Interval Method Implementation for Global Optimization Using Dynamic Load Balancing”, Reliable Computing 1, pp. 77–91
Yurij G. Evtushenko (1985) Numerical Optimization Techniques, Optimization Software, New York.
Thomas A. Feo and Mauricio G. C. Resende (1995) “Greedy Randomized Adaptive Search Procedures”, Journal of Global Optimization 6, pp. 109–133.
C. A. Floudas and P. M. Pardalos [Editors] (1992) Recent Advances in Global Optimization, Princeton University Press, Princeton.
A. Floudas and P. M. Pardalos [Editors] (1996) State of the Art in Global Optimization, Kluwer Academic Publishers.
G. Gallo and C. Sodini (1979) “Adjacent Extreme Flows and Application to Min Concave Cost Flow Problem”, Networks 9, pp. 95–121.
G. Gallo and C. Sodini (1979) “Concave Cost Minimization on Networks”, European Journal of Operational Research 3, pp. 239–249.
G. Gallo, C. Sandi and C. Sodini (1980) “An Algorithm for the Min Concave Cost Flow Problem”, European Journal of Operational Research 4, pp. 248–255.
R. S. Garfinkel and G. L. Nemhauser (1972) Integer Programming, John Wiley & Sons, N.Y.
R. Ge (1990) “A Filled Function Method for Finding a Global Minimizer of a Function of Several Variables”, Mathematical Programming 46, pp. 191–204.
V.P. Gergel, Ya. D. Sergeyev and R. G. Strongin (1993) “A Parallel Global Optimization Method and its Implementation on a Transputer System”, Optimization 26, pp. 261–275.
S. Ghannadan, A. Migdalas, H. Tuy and P. Varbrand (1996) “Tabu Meta-Heuristic Based on Local Search for the Concave Production- Transportation Problem”, Studies in Location Analysis 8, Special Issue Edited by C. R. Reeves, pp. 33–47.
F. Glover (1989) “Tabu Search - Part I”, ORSA Journal on Computing 1, pp. 190–206.
F. Glover (1990) “Tabu Search - Part II”, ORSA Journal on Computing 2, pp. 4–31.
F. Glover, M. Laguna, E. Taillard, and D. De Werra [Editors] (1993) Tabu Search, Annals of Operations Research 41, J.C. Baltzer AG, Basel, Switzerland.
F. Glover, E. Taillard, and D. De Werra (1993) “A User’s Guide to Tabu Search”, in: [36], pp. 2–28.
Jun Gu (1995) “Parallel Algorithms for Satisfiability (SAT) Problem” in [69].
G.M. Guisewite and P. M. Pardalos (1990) “Minimum Concave-Cost Network Flow Problems: Applications, Complexity, and Algorithms”, Annals of Operations Research 25, pp. 75–100.
G.M. Guisewite (1995) “Network Problem”, in [49], pp. 609–648.
Eldon Hansen (1992) Global Optimization Using Interval Analysis, Marcel Dekker, New York.
L. He and E. Polak (1993) “Multistart Method with Estimation Scheme for Global Satisfycing Problems”, Journal of Global Optimization 3, pp. 139–156.
T. Henriksen and T. Madsen (1992) “Use of a Depth-First Strategy in Par-allel Global Optimization”, Research Report 92 - 10, Technical University of Denmark, Lyngby, Denmark.
K. Holmqvist and A. Migdalas (1996) “A C++ Class library for Interval Arithmetic in Global Optimization”, in [26].
K. Holmqvist, A. Migdalas and P.M. Pardalos (1996)“Parallelized Heuristics for Combinatorial Search”, Chapter 8 in this book.
K. Holmqvist, A. Migdalas and P. M. Pardalos (1996) “Greedy Randomized Adaptive Search for the Location Problem with Economies of Scale”, in: I. Bomze, T. Csendes, R. Horst and P.M. Pardalos [Editors] Developments in Global Optimization.
Reiner Horst and Hoang Tuy (1990) Global Optimization: Deterministic Approaches, Springer Verlag, Berlin.
Reiner Horst, Panos M. Pardalos and Nguyen V. Thoai (1995) Introduction to Global Optimization, Kluwer Academic Publishers, Dordrecht.
Reiner Horst and Panos M. Pardalos [Editors] (1995) Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht.
R. Baker Kearfott (1996) Rigorous Global Search for Continuous Problems, Kluwer Academic Publishers, Dordrecht.
R. Baker Kearfott and Vladik Kreinovich [Editors] (1996) Applications of Interval Computations, Kluwer Academic Publishers, Dordrecht.
T. Larsson, A. Migdalas and M. Ronnqvist (1994) “A Lagrangean Heuristic for the Capacitated Concave Minimum Cost Network Flow Problem”, European Journal of Operational Research 78, pp. 116–129.
Anthony Leclerc (1993) “Parallel Interval Global Optimization and its Implementation in C++”, Interval Computations 3, pp. 148–163.
A. V. Levy, A. Montalvo, S. Gomez and A. Calderon (1981) “Topics in Global Optimization in Numerical Analysis”, J.P. Hennart, Lecture Notes in Mathematics 909, pp. 18–33, Springer-Verlag, Berlin.
Zhian Li, P. M. Pardalos and S. H. Levine (1992) “Space-Covering Approach and Modified Frank-Wolfe Algorithm for Optimal Nuclear Reactor Reload Design”, in [25].
Solving a Bilevel Linear Program: A Parallel Algorithm, in Optimization: Techniques and Applications, Guangzhong Liu [Editors], ICOTA’95, World Scientific, Singapore, pp. 90–96.
A. Migdalas and Maud Gothe-Lundgren (1994) Combinatorial Optimization: Problems and Algorithms, Linkoping, Sweden (in Swedish)
A. Migdalas and P. M. Pardalos [Editor] (1996) Hierarchical and Bilevel Programming, Special Issue of the Journal of Global Optimization 8, No. 3.
M. Minoux (1976) “Multiflots De Cout Minimal Avec Fonctions De Cout Concaves”, Annals of Telecommunication 31, pp. 77–92.
Jonas Mockus (1994) “Application of Bayesian Approach to Numerical Methods of Global and Stochastic Optimization” Journal of Global Optimization 4, pp. 347–365.
R. E. Moore, E. Hansen and A. Leclerc (1992) “Rigorous Methods for Global Optimization”, in: [25].
P.M. Pardalos and J.B. Rosen (1987) Global Optimization: Algorithms and Applications, Springer-Verlag, Lecture Notes in Computer Science 268.
P.M. Pardalos and G. Schnitger (1988) “Checking local optimality in constrained quadratic programming is NP-hard”, Operations Research Letters 7, pp. 33–35.
P. M. Pardalos (1989) “Parallel Search Algorithms in Global Optimization”, Applied Mathematics and Computation 29, pp. 219–229.
P.M. Pardalos and J.B. Rosen [Editors] (1990) Computational Methods in Global Optimization, Annals of Operations Research 25.
Panos M. Pardalos and G.M. Guisewite (1993) “Parallel Computing in Nonconvex Programming”, Annals of Operations Research 43, pp. 87–107.
P.M. Pardalos, G. Xue and D. Shalloway (1994) “Optimization Methods for Computing Global Minima of Nonconvex Potential Energy Functions”, Journal of Global Optimization 4, pp. 117–133.
P.M. Pardalos, A.T. Phillips and J.B. Rosen (1992) Topics in Parallel Computing in Mathematical Programming, Science Press, New York.
Panos M. Pardalos, Mauricio G. C. Resende, and K.G. Ramakrishnan [Editors] (1995) Parallel Processing of Discrete Optimization Problems, DIMACS Series in Discrete Mathematics and Theoretical Computer Sci-ence 22, American Mathematical Society.
P.M. Pardalos, Guoliang Xue and P.D. Panagiotopoulos (1995) “Parallel Algorithms for Global Optimization” in: A. Ferreira and P. M. Pardalos [Editors], Solving Irregular Problems in Parallel: State of the Art, Springer- Verlag, Berlin.
A.T. Phillips and J.B. Rosen (1989) “Guaranteed e-approximate solution for indefinite quadratic global minimization”, Naval Research Logistics Quarterly 37, pp. 499–514.
A.T. Phillips and J.B. Rosen (1990) “A parallel algorithm for partially separable non-convex global minimization”, in [65].
A.T. Phillips, J.B. Rosen and M. Van Vliet (1992) “A Parallel Stochastic Method for Solving Linearly Constrained Concave Global Minimization Problems”, Journal of Global Optimization 2, pp. 243–258.
Janos D. Pinter (1996) Global Optimization in Action, Kluwer Academic Publishers, Dordrecht.
M. Pogu and J.E. Souza de Cursi (1994) “Global Optimization by Random Perturbation of the Gradient Method with Fixed Parameter”, Journal of Global Optimization 5, pp. 159–180.
William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery (1992) Numerical Recipies in Fortran: The Art of Scientific Computing, Second Edition, Cambridge University Press, Cambridge.
W.L. Price (1978) “A Controlled Random Search Procedure for Global Optimization” in Towards Global Optimization 2, L.C.W and G.P. Szego [Editors], North-Holland.
A.H.G. Rinnooy Kan and G.T. Timmer (1987) “Stochastic Global Optimization Methods. Part I: Clustering Methods; Part II: Multi Level Methods”, Mathematical Programming 39, pp. 27–78.
H. Ratscheck and J. Rokne (1988) New Computer Methods for Global Optimization, Ellis Horwood Limited, Chichester.
Klaus Ritter and Stefan Schaffler (1994) “A Stochastic Method for Constraint Global Optimization”, SIAM Journal on Optimization 4, pp. 894–904.
H. Edwin Romeijn and Robert L. Smith (1994) “Simulated Annealing for Constrained Global Optimization”, Journal of Global Optimization 5, pp. 101–126.
Fabio Schoen (1991) “Stochastic Techniques for Global Optimization: A Survey of Recent Advances”, Journal of Global Optimization 1, pp. 207–228.
Fabio Schoen (1994) “On an New Stochastic Global Optimization Algorithm Based on Censored Observations”, Journal of Global Optimization 4, pp. 17–35.
Hans-Paul Schefel (1981) Numerical Optimization of Computer Models, Translated from the 1977 German edition, John Wiley & Sons, Chichester.
Sharon L. Smith and Robert B. Schnabel (1992) “Dynamic Scheduling Strategies for an Adaptive, Asynchronous Parallel Global Optimization Algorithms”, Research Report CU-CS-652-93, University of Colorado at Boulder, Department of Computer Science, Campus Box 430, Boulder, Colorado 80309-0430, USA.
Roman G. Strongin and Yaroslav D. Sergeyev (1992) “Global Multidimensional Optimization on Parallel Computer”, Parallel Computing 18, pp. 1259–1273.
C. Sutti (1984) “Local and Global Optimization by Parallel Algorithms for MIMD Systems”, Annals of Operations Research 1, pp. 151–164.
Zaiyong Tang (1995) “Recurrent Neural Networks for Global Optimization” in in Optimization: Techniques and Applications, Guangzhong Liu et al [Editors], ICOTA’95, World Scientific, Singapore, pp. 415–421.
Aimo Torn and Antanas Zilinskas (1987) Global Optimization, Lecture Notes in Computer Science 350, Springer-Verlag, Berlin.
Aimo Torn and Sami Viitanen (1992) “Topographical Global Optimization” in [25].
H. Tuy (1964) “Concave programming under linear constraints”, Soviet Mathematics Doklady 5, 1437–1440.
Guoliang Xue (1994) “Molecular Conformation on the CM-5 by Parallel Two-Level Simulated Annealing”, Journal of Global Optimization 4, pp. 187–208.
B. Jr. Yaged (1971) “Minimum Cost Routing for Static Network Models”, Networks 1, pp. 139–172.
Zelda B. Zabinsky, Robert L. Smith, J. Fred McDonald, H. Edwin Romeijn and David E. Kaufman (1993) “Improving Hit-and-Run for Global Optimization”, Journal of Global Optimization 3, pp. 171–192.
Chun Zhang and Hsu-Pin (Ben) Wang (1993) “Mixed-Discrete Nonlinear Optimization with Simulated Annealing”, Engineering Optimization 21, pp. 277–291.
Anatoly A. Zhigljavsky (1991) Theory of Global Random Search, Mathematics and Its Applications 65, Kluwer Academic Publishers, Dordrecht.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Kluwer Academic Publishers
About this chapter
Cite this chapter
Holmqvist, K., Migdalas, A., Pardalos, P.M. (1997). Parallel Continuous Non-Convex Optimization. In: Migdalas, A., Pardalos, P.M., Storøy, S. (eds) Parallel Computing in Optimization. Applied Optimization, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3400-2_12
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3400-2_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3402-6
Online ISBN: 978-1-4613-3400-2
eBook Packages: Springer Book Archive