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Solving Nonlinear Programming Problems with Stochastic Objective Functions

Published online by Cambridge University Press:  19 October 2009

William T. Ziemba
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In many nonlinear programming applications the objective function has an inherent uncertainty that depends upon a set of random variables that have a known distribution. If one wishes to optimize the expectation of the objective, as suggested by the expected utility theorem, then as is shown here one can often solve such problems by modifying standard nonlinear programming algorithms. To illustrate what is involved, the details and justification for the application of the interior parametric sequential unconstrained maximization technique and the generalized programming method for the solution of such problems are given. Some related problems with stochastic constraints for which the solution method applies are mentioned and an example of a portfolio selection problem is given.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 7 , Issue 3 , June 1972 , pp. 1809 - 1827
Copyright
Copyright © School of Business Administration, University of Washington 1972

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References

[1]Beale, E.M.L. “The Use of Quadratic Programming in Stochastic Linear Programming.” Santa Monica: The RAND Corporation, August 1961, p. 2404.Google Scholar
[2]Bereanu, B. “On Stochastic Linear Programming: The Laplace Transform of the Distribution of the Optimum and Applications.” Journal of Mathematical Analysis and Applications, XV, 1966, pp. 280–290.CrossRefGoogle Scholar
[3]Bereanu, B.“Numerical Methods in Stochastic Linear Programming: the Convergence of the Cartesian Integration Method.”Paper presented at the 7th Mathematical Programming Symposium, the Hague, 1970.Google Scholar
[4]Dantzig, G.B. “Linear Programming under Uncertainty.” Management Science, I, 1955, pp. 197–206.CrossRefGoogle Scholar
[5]Dantzig, G.B.Linear Programming and Extensions. Princeton, N.J.: Princeton University Press, 1963.Google Scholar
[6]Davis, P.J. and Rabinowitz, P.. Numerical Integration. Waltham, Mass: Blaisdell Publishing Co., 1967.Google Scholar
[7]Elmaghraby, S.E. “Allocation under Uncertainty When the Demand has Continuous D.F.” Management Science, X, 1960, pp. 270–294.CrossRefGoogle Scholar
[8]Fiacco, A.V., and McCormick, G.P.. “The Sequential Unconstrained Minimization Technique for Non-linear Programming: A Primal–Dual Method.” Management Science, X, 1964, pp. 360–366.CrossRefGoogle Scholar
[9]Fiacco, A.V.Non-linear Programming: Sequential Unconstrained Minimization Techniques. New York: John Wiley and Sons, Inc., 1968.Google Scholar
[10]Fishburn, P.C.Utility Theory for Decision Making. New York: John Wiley and Sons, Inc., 1970.CrossRefGoogle Scholar
[11]Fletcher, R.Optimization. New York: Academic Press Inc., 1969.Google Scholar
[12]Freund, R.J. “The Introduction of Risk into a Programming Model.” Econometrica, XXIV, 1956, pp. 253–263.CrossRefGoogle Scholar
[13]Gale, D., and Nikaidô, H.. “The Jacobian Matrix and Global Univalence of Mappings.” Mathematische Annalen, CLIX, 1965, pp. 81–93.CrossRefGoogle Scholar
[14]Geoffrion, A.M. “Stochastic Programming with Aspiration or Fractile Criteria.” Management Science, XIII, 1967, pp. 672–679.CrossRefGoogle Scholar
[15]Haber, S. “Numerical Evaluation of Multiple Integrals.” SIAM Review, XII, 1970, pp. 481–526.CrossRefGoogle Scholar
[16]Hillier, F.S.The Evaluation of Risky Interrelated Investments. Amsterdam: North-Holland Publishing Co., 1969.Google Scholar
[17]Katoaka, S. “A Stochastic Programming Model.” Econometrica, XXXI, 1963, pp. 181–196.CrossRefGoogle Scholar
[18]Kirby, M.J.L.“The Current State of Chance-Constrained Programming.”In Proceedings of the Princeton Symposium on Mathematical Programming, edited by Kuhn, H. W., 1970.Google Scholar
[19]Lintner, J. “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets.” Review of Economics and Statistics, XLVII, 1965, pp. 13–37.CrossRefGoogle Scholar
[20]Loeve, M.Probability Theory, 3rd ed., Princeton: D. Van Nostrand and Co., 1963.Google Scholar
[21]Madansky, A. “Inequalities for Stochastic Linear Programming Problems.” Management Science, VI, 1960, pp. 197–204.CrossRefGoogle Scholar
[22]Mangasarian, O.L. “Nonlinear Programming Problems with Stochastic Objective Functions.” Management Science, X, 1964, pp. 353–359.CrossRefGoogle Scholar
[23]Mangasarian, O.L., and Rosen, J. B.. “Inequalities for Stochastic Nonlinear Programming Problems.” Operations Research, XII, 1964, pp. 143–154.CrossRefGoogle Scholar
[24]Murray, W.“Ill-conditioning in Barrier and Penalty Functions Arising in Constrained Nonlinear Programs.”Paper presented at the Princeton Symposium on Mathematical Programming,1967.Google Scholar
[25]Parikh, S.C. “Notes on Stochastic Programming.” Unpublished, Department of Industrial Engineering and Operations Research, University of California, Berkeley, 1968.Google Scholar
[26]Pye, G. “Portfolio Selection and Security Prices.” Review of Economics and Statistics, XLIX, 1967, pp. 111–115.CrossRefGoogle Scholar
[27]Richter, M.K. “Cardinal Utility, Portfolio Selection and Taxation.” Review of Economic Studies, XXVII, 1959, pp. 152–166.Google Scholar
[28]Samuelson, P.A. “General Proof that Diversification Pays.” Journal of Financial and Quantitative Analysis, II, 1967, pp. 1–13.CrossRefGoogle Scholar
[29]Samuelson, P.A. “The Fundamental Approximation Theorem of Mean-Variance Analysis in Terms of Means, Variance, and Higher Moments.” Review of Economic Studies, XXXVII, 1970, pp. 537–542.CrossRefGoogle Scholar
[30]Tobin, J. “Liquidity Preference as Behavior Towards Risk.” Review of Economics Studies, XXVI, 1958, pp. 65–86.CrossRefGoogle Scholar
[31]Topkis, D.M., and Veinott, A.F. Jr, “On the Convergence of Some Feasible Direction Algorithms for Nonlinear Programming.” J. SIAM Control, V, 1967, pp. 268–279.CrossRefGoogle Scholar
[32]Walkup, D.W., and Wets, R.J.B.. “Stochastic Programs with Recourse: Special Forms.”In Proceedings of the Princeton Symposium on Mathematical Programming, edited by Kuhn, H.W., 1970.Google Scholar
[33]Warburton, A., and Ziemba, W.T.. “Convex Inversion.” Journal of Mathematical Analysis and Applications, XXXV, 1971, pp. 58–66.CrossRefGoogle Scholar
[34]Wets, R. “Programming under Uncertainty: The Complete Problem.” Z. Wahrsaheinlichkeitstheorie und Verw. Gebite, IV, 1966, pp. 316–339.CrossRefGoogle Scholar
[35]Williams, A.C. “Approximation Formulas for Stochastic Linear Programming.” J. SIAM Appl. Math., XIV, 1966, pp. 668–677.CrossRefGoogle Scholar
[36]Wolfe, P. “A Duality Theorem for Nonlinear Programming.” Quart. Appl. Math., XIX, 1961, pp. 239–244.CrossRefGoogle Scholar
[37]Zangwill, W.I.Nonlinear Programming: A Unified Approach. Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1969.Google Scholar
[38]Ziemba, W.T. “Computational Algorithms for Convex Stochastic Programs with Simple Recourse.” Operations Research, XVIII, 1970, pp. 414–431.CrossRefGoogle Scholar
[39]Ziemba, W.T. “Essays on Stochastic Programming and the Theory of Economic Policy.” Ph.D. diss. Berkeley: University of California, 1969.Google Scholar
[40]Ziemba, W.T. “Calculating Investment Portfolios when the Returns have Stable Distributions, Part I: Theory.” University of British Columbia Faculty of Commerce, Working Paper No. 133, February 1972.Google Scholar
[41]Ziemba, W.T.; Brooks-Hill, F.J.; and Parkan, C.. “Calculating Investment Portfolios when the Returns are Normally Distributed.” University of British Columbia Faculty of Commerce, Working Paper No. 118, November 1971.Google Scholar
[42]Zoutendijk, G.Methods of Feasible Directions. Amsterdam: Elsevier Publishing Co., 1960.Google Scholar
[43]Zoutendijk, G. “Nonlinear Programming Computation Methods.” In Integer and Nonlinear Programming, edited by Abadie, J.. Amsterdam: North-Holland Publishing Co., 1970.Google Scholar