Advertisement

Mathematical Optimization

  • Elisa Pappalardo
  • Panos M. Pardalos
  • Giovanni Stracquadanio
Chapter
Part of the SpringerBriefs in Optimization book series (BRIEFSOPTI)

Abstract

Mathematical Optimization is an interdisciplinary branch of applied mathematics, related to the fields of Operations Research, Computational Complexity, and Algorithm Theory. It can be informally defined as the science of finding the “best” solution from a set of available alternatives, where the notion of “best” is intrinsically related to the specific problem addressed. Nowadays, optimization problems arise in all sorts of areas and are countless in everyday life, such as in engineering, microelectronics, telecommunications, biomedicine, genetics, proteomics, economics, finance, and physics. However, in spite of the proliferation of optimization algorithms, there is no universal method suitable for all optimization problems, and the choice of the “most appropriate method” to solve the specific problem is demanded to the user. With this in mind, in this chapter we address some general questions about optimization problems and their solutions, in order to provide a background knowledge of the issues analyzed later.

Keywords

Linear Programming Problem Nonlinear Optimization Problem Local Search Heuristic Nonconvex Problem Continuous Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Androulakis, I., Maranas, C., Floudas, C.: αbb: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4), 337–363 (1995)Google Scholar
  2. 2.
    Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W., Vance, P.H.: Branch-and-price: column generation for solving huge integer programs. Oper. Res. 46(3), 316–329 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  4. 4.
    Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1998)MATHGoogle Scholar
  5. 5.
    Dorigo, M., Stützle, T.: The ant colony optimization metaheuristic: Algorithms, applications, and advances. In: Handbook of Metaheuristics, pp. 250–285. Springer, New York (2003)Google Scholar
  6. 6.
    Du, D.Z., Pardalos, P.M.: Handbook of Combinatorial Optimization, vol. 3. Springer, New York (1998)Google Scholar
  7. 7.
    Feo, T.A., Resende, M.G.: Greedy randomized adaptive search procedures. J. Glob. Optim. 6(2), 109–133 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. WH Freeman, San Francisco (1990)Google Scholar
  9. 9.
    Gendreau, M.: Handbook of Metaheuristics, vol. 146. Springer, New York (2010)CrossRefMATHGoogle Scholar
  10. 10.
    Glover, F., Laguna, M.: Tabu Search. Wiley, London (1993)Google Scholar
  11. 11.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading (1989)MATHGoogle Scholar
  12. 12.
    Hansen, N., Müller, S.D., Koumoutsakos, P.: Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evol. Comput. 11(1), 1–18 (2003)CrossRefGoogle Scholar
  13. 13.
    Horst, R., Pardalos, P.M., Van Thoai, N.: Introduction to Global Optimization. Kluwer, Dordrecht (2000)Google Scholar
  14. 14.
    Ingber, L., Petraglia, A., Petraglia, M.R., Machado, M.A.S., et al.: Adaptive simulated annealing. In: Stochastic Global Optimization and Its Applications with Fuzzy Adaptive Simulated Annealing, pp. 33–62. Springer, New York (2012)Google Scholar
  15. 15.
    Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948. IEEE, New York (1995)Google Scholar
  16. 16.
    Kirkpatrick, S., Gelatt, D.G. Jr., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, vol. 5 (1951)Google Scholar
  18. 18.
    Michalewicz, Z.: Evolution strategies and other methods. In: Genetic Algorithms + Data Structures = Evolution Programs, pp. 159–177. Springer, New York (1996)Google Scholar
  19. 19.
    Mitchell, J.E.: Branch-and-cut algorithms for combinatorial optimization problems. In: Handbook of Applied Optimization, pp. 65–77. Oxford, GB: Oxford University Press (2002)Google Scholar
  20. 20.
    Mitchell, J.E., Pardalos, P.M., Resende, M.G.: Interior point methods for combinatorial optimization. In: Handbook of Combinatorial Optimization, vol. 1, pp. 189–297. Kluwer Academic Publishers (1998)Google Scholar
  21. 21.
    Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Nesterov, Y., Nemirovskii, A.S., Ye, Y.: Interior-point Polynomial Algorithms in Convex Programming, vol. 13. Studies in Applied Mathematics, Philadelphia (1994)CrossRefMATHGoogle Scholar
  23. 23.
    Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 13(1), 271–369 (2004)MathSciNetGoogle Scholar
  24. 24.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover, Mineola (1998)MATHGoogle Scholar
  25. 25.
    Pardalos, P.M., Resende, M.G.: Interior point methods for global optimization. In: Interior Point Methods of Mathematical Programming. Citeseer (1996)Google Scholar
  26. 26.
    Pardalos, P.M., Resende, M.G.: Handbook of Applied Optimization, vol. 1. Oxford University Press, Oxford (2002)CrossRefMATHGoogle Scholar
  27. 27.
    Pardalos, P.M., Rosen, J.B.: Constrained Global Optimization: Algorithms and Applications. Springer, New York (1987)CrossRefMATHGoogle Scholar
  28. 28.
    Pelikan, M.: Bayesian optimization algorithm. In: Hierarchical Bayesian Optimization Algorithm, pp. 31–48. Springer, New York (2005)Google Scholar
  29. 29.
    Puchinger, J., Raidl, G.: Combining metaheuristics and exact algorithms in combinatorial optimization: a survey and classification. In: Artificial Intelligence and Knowledge Engineering Applications: A Bioinspired Approach, pp. 113–124 (2005)Google Scholar
  30. 30.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, London (1998)MATHGoogle Scholar
  31. 31.
    Storn, R., Price, K.: Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Elisa Pappalardo, Panos M. Pardalos, Giovanni Stracquadanio 2013

Authors and Affiliations

  • Elisa Pappalardo
    • 1
  • Panos M. Pardalos
    • 2
    • 3
  • Giovanni Stracquadanio
    • 1
  1. 1.Department of Biomedical EngineeringJohns Hopkins UniversityBaltimoreUSA
  2. 2.Center for Applied Optimization Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.Laboratory of Algorithms and Technologies for Networks Analysis (LATNA) Higher School of EconomicsNational Research UniversityMoscowRussia

Personalised recommendations