Numerical Optimization Methods in Economics
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.
KeywordsCentral 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
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.
- Allgower, E.L., and K. Georg. 1979. Introduction to numerical continuation methods. New York: John Wiley & Sons. Reprinted by SIAM Publications, 2003.Google Scholar
- 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
- 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
- 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
- Fletcher, R. 1987. Practical methods of optimization. Chichester: John Wiley & Sons.Google Scholar
- 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
- Frisch, R.A.K. 1955. The logarithmic potential method of convex programming. Technical Report, University Institute of Economics, University of Oslo, Norway.Google Scholar
- 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
- Holland, J.H. 1975. Adaptation in natural and artifical systems. Ann Arbor: University of Michigan Press.Google Scholar
- Judd, K.L. 1998. Numerical methods in economics. Cambridge, MA: MIT Press.Google Scholar
- Kendrick, D.A., P.R. Mercado, and H.M. Amman. 2006. Computational economics. Princeton: Princeton University Press.Google Scholar
- Khachiyan, L.G. 1979. A polynomial algorithm in linear programming. Soviet Mathematics Doklady 20: 191–194.Google Scholar
- Markowitz, H.M. 1952. Portfolio selection. Journal of Finance 7: 77–91.Google Scholar
- Miranda, M.J., and P. Fackler. 2002. Applied computational economics and finance. Cambridge, MA: MIT Press.Google Scholar
- Nocedal, J., and S.J. Wright. 2006. Numerical optimization. New York: Springer.Google Scholar
- 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
- Rosenthal, R.E. 2006. GAMS – A user’s guide. Washington, DC: GAMS Development Corporation. Online. Available at http://www.gams.com/docs/gams/GAMSUsersGuide.pdf. Accessed 7 Feb 2007.
- Simon, C.P., and L. Blume. 1994. Mathematics for economists. New York: W. W. Norton.Google Scholar
- Sturmfels, B. 2002. Solving systems of polynomial equations, vol. 97, CBMS Regional Conference Series in Mathematics, Providence: American Mathematical Society.Google Scholar