In this paper, we examine a class of stochastic optimization problems characterized by nondifferentiability of the objective function. It is shown that, in many cases, the expected value of the objective function is differentiable and, thus, the resulting optimization problem can be solved by using classical analytical or numerical methods. The results are subsequently applied to the solution of a problem of economic resource allocation.
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Bazaraa, M. S., Goode, J. J., andShetty, C. M.,Optimality Criteria in Nonlinear Programming without Differentiability, Operations Research, Vol. 19, No. 1, 1971.
Bazaraa, M. S.,Nonlinear Programming: Nondifferentiable Functions, Georgia Institute of Technology, Ph.D. Thesis, 1971.
Bertsekas, D. P., andMitter, S. K.,Steepest Descent for Optimization Problems with Nondifferentiable Cost Functionals, Paper Presented at the 5th Annual Princeton Conference on Information Sciences and Systems, Princeton, New Jersey, 1971.
Bertsekas, D. P., andMitter, S. K.,Descent Numerical Methods for Optimization Problems with Nondifferentiable Cost Functionals, SIAM Journal on Control, Vol. 11, No. 4, 1973.
Dem'yanov, V. F., andRubinov, A. M.,Minimization of Functionals in Normed Spaces, SIAM Journal on Control, Vol. 6, No. 1, 1968.
Heins, W., andMitter, S. K.,Conjugate Convex Functions, Duality, and Optimal Control Problems, I. Systems Governed by Ordinary Differential Equations, Information Sciences, Vol. 2, No. 2, 1970.
Chanem, M. Z. E.,Optimal Control Problems with Nondifferentiable Cost Functionals, Stanford University, Department of Engineering-Economic Systems, Ph.D. Thesis, 1970.
Luenberger, D. G.,Control Problems with Kinks, IEEE Transactions on Automatic Control, Vol. AC-15, No. 5, 1970.
Neustadt, L. W.,A General Theory of Extremals, Journal of Computational and System Science, Vol. 3, No. 1, 1969.
Rockafellar, R. T.,Conjugate Convex Functions in Optimal Control and the Calculus of Variations, Journal of Mathematical Analysis and Applications, Vol. 32, No. 1, 1970.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Halmos, P. R.,Measure Theory, Van Nostrand Reinhold Company, New York, New York, 1950.
Dunford, N., andSchwartz, J. T.,Linear Operators, Part I, John Wiley and Sons (Interscience Publishers), New York, New York, 1957.
Royden, H. L.,Real Analysis, The Macmillan Company, New York, New York, 1968.
Rockafellar, R. T.,Measurable Dependence of Convex Sets and Functions on Parameters, Journal of Mathematical Analysis and Applications, Vol. 28, No. 1, 1969.
Aumann, R. J.,Integrals of Set Valued Functions, Journal of Mathematical Analysis and Applications, Vol. 12, No. 1, 1965.
Bringland, T. F.,Trajectory Integrals of Set Valued Functions, Pacific Journal of Mathematics, Vol. 33, No. 1, 1970.
This work was supported by the National Science Foundation under Grant No. GK 29237.
Communicated by P. Varayia
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Bertsekas, D.P. Stochastic optimization problems with nondifferentiable cost functionals. J Optim Theory Appl 12, 218–231 (1973). https://doi.org/10.1007/BF00934819
- Probability Measure
- Convex Function
- Optimal Control Problem
- Directional Derivative
- Differentiability Property