The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Numerical Optimization Methods in Economics

  • Karl Schmedders
Reference work entry


Optimization problems are ubiquitous in economics. Many of these problems are sufficiently complex that they cannot be solved analytically. Instead economists need to resort to numerical methods. This article presents the most commonly used methods for both unconstrained and constrained optimization problems in economics; it emphasizes the solid theoretical foundation of these methods, illustrating them with examples. The presentation includes a summary of the most popular software packages for numerical optimization used in economics, and closes with a description of the rapidly developing area of mathematical programs with equilibrium constraints, an area that shows great promise for numerous economic applications.


Central path Computable general equilibrium models Computational algebraic geometry Constrained optimization Global minimum Homotopy continuation methods Interior-point methods Karush-Kuhn-Tucker conditions Kuhn–Tucker conditions Lagrange multipliers Line search methods Linear independence constraint qualification (LICQ) Linear programming Local minimum Local solution Logarithmic barrier method Mathematical programs with equilibrium constraints Metaheuristics Multidimensional optimization problems Newton’s method Newton–Raphson methods Nonlinear programming Numerical optimization methods Optimality conditions Optimization algorithms Penalty methods Polynomial functions Quadratic programs Sequential quadratic programming Simplex method for solving linear programs Simulated annealing Smale’s global Newton method Steepest descent method Taylor’s th Trust region methods Unconstrained optimization 

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I am grateful for helpful discussions with Sven Leyffer and am indebted to Ken Judd, Annette Krauss, and in particular Che-Lin Su for detailed comments on earlier drafts. I also thank the editors Larry Blume and Steven Durlauf for a careful review of my initial submission.


  1. Allgower, E.L., and K. Georg. 1979. Introduction to numerical continuation methods. New York: John Wiley & Sons. Reprinted by SIAM Publications, 2003.Google Scholar
  2. Bhatti, M.A. 2000. Practical optimization methods: With mathematica applications. New York: Springer-Verlag.CrossRefGoogle Scholar
  3. Brandimarte, P. 2006. Numerical methods in finance and economics: A MATLAB-based introduction. New York: John Wiley & Sons.CrossRefGoogle Scholar
  4. Cerny, V. 1985. A thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications 45: 41–51.CrossRefGoogle Scholar
  5. Chen, Y., B.F. Hobbs, S. Leyffer, and T.S. Munson. 2006. Leader–follower equilibria for electric power and NOx allowances markets. Computational Management Science 4: 307–330.CrossRefGoogle Scholar
  6. Conn, A.R., N.I.M. Gould, and P.L. Toint. 2000. Trust-region methods. Philadelphia: SIAM.CrossRefGoogle Scholar
  7. Cox, D., J. Little, and D. O’shea. 1997. Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra. New York: Springer-Verlag.Google Scholar
  8. Czyzyk, J., M.P. Mesnier, and J. Morè. 1998. The NEOS server. IEEE Journal on Computational Science and Engineering 5: 68–75.CrossRefGoogle Scholar
  9. Dantzig, G.B. 1949. Programming of inter-dependent activities II, mathematical model. Econometrica 17, 200–211. Also in Koopmans, T.C., ed. Activity analysis of production and allocation. New York: John Wiley & Sons, 1951.Google Scholar
  10. Dantzig, G.B. 1963. Linear programming and extensions. Princeton: Princeton University Press.CrossRefGoogle Scholar
  11. Ferris, M.C., M.P. Mesnier, and J. Moré. 2000. NEOS and Condor: Solving nonlinear optimization problems over the Internet. ACM Transactions on Mathematical Software 26: 1–18.CrossRefGoogle Scholar
  12. Fiacco, A.V., and G.P. McCormick. 1968. Nonlinear programming: Sequential unconstrained minimization techniques. New York: John Wiley & Sons, Inc. Reprinted by SIAM Publications, 1990.Google Scholar
  13. Fletcher, R. 1987. Practical methods of optimization. Chichester: John Wiley & Sons.Google Scholar
  14. Fourer, R., D.M. Gay, and B.W. Kernighan. 2003. AMPL: A modeling language for mathematical programming. Pacific Grove: Brooks/Cole–Thomson Learning.Google Scholar
  15. Frisch, R.A.K. 1955. The logarithmic potential method of convex programming. Technical Report, University Institute of Economics, University of Oslo, Norway.Google Scholar
  16. Gould, N.I.M., and S. Leyffer. 2002. An introduction to algorithms for nonlinear optimization. In Frontiers in numerical analysis, ed. J.F. Blowey, A.W. Craig, and T. Shardlow. Berlin/Heidelberg: Springer-Verlag.Google Scholar
  17. Holland, J.H. 1975. Adaptation in natural and artifical systems. Ann Arbor: University of Michigan Press.Google Scholar
  18. Judd, K.L. 1998. Numerical methods in economics. Cambridge, MA: MIT Press.Google Scholar
  19. Karmarkar, N. 1984. A new polynomial-time algorithm for linear programming. Combinatorics 4: 373–395.CrossRefGoogle Scholar
  20. Kendrick, D.A., P.R. Mercado, and H.M. Amman. 2006. Computational economics. Princeton: Princeton University Press.Google Scholar
  21. Khachiyan, L.G. 1979. A polynomial algorithm in linear programming. Soviet Mathematics Doklady 20: 191–194.Google Scholar
  22. Kirkpatrick, S., C.D. Gelatt, and M.P. Vecchi. 1983. Optimization by simulated annealing. Science 220: 671–680.CrossRefGoogle Scholar
  23. Lasserre, J.B. 2001. Global optimization with polynomials and the problem of moments. SIAM Journal of Optimization 11: 796–817.CrossRefGoogle Scholar
  24. Markowitz, H.M. 1952. Portfolio selection. Journal of Finance 7: 77–91.Google Scholar
  25. Miranda, M.J., and P. Fackler. 2002. Applied computational economics and finance. Cambridge, MA: MIT Press.Google Scholar
  26. Nocedal, J., and S.J. Wright. 2006. Numerical optimization. New York: Springer.Google Scholar
  27. Parrilo, P.A., and B. Sturmfels. 2003. Minimizing polynomial functions. Algorithmic and Quantitative Real Algebraic Geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 60: 83–99.Google Scholar
  28. Rosenthal, R.E. 2006. GAMSA user’s guide. Washington, DC: GAMS Development Corporation. Online. Available at Accessed 7 Feb 2007.
  29. Simon, C.P., and L. Blume. 1994. Mathematics for economists. New York: W. W. Norton.Google Scholar
  30. Smale, S. 1976. A convergent process of price adjustment and global Newton methods. Journal of Mathematical Economics 3: 107–120.CrossRefGoogle Scholar
  31. Sturmfels, B. 2002. Solving systems of polynomial equations, vol. 97, CBMS Regional Conference Series in Mathematics, Providence: American Mathematical Society.Google Scholar

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Karl Schmedders
    • 1
  1. 1.