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Directional Derivatives for Extremal-Value Functions with Applications to the Completely Convex Case

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  • William Hogan

    (United States Air Force Academy, Colorado)

Abstract

Several techniques in mathematical programming involve the constrained optimization of an extremal-value function. Such functions are defined as the extremal value of a related parameterized optimization problem. This paper reviews and extends the characterization of directional derivatives for three major types of extremal-value functions. The characterization for the completely convex case is then used to construct a robust and convergent feasible direction algorithm. Such an algorithm has applications to the optimization of large-scale nonlinear decomposable systems.

Suggested Citation

  • William Hogan, 1973. "Directional Derivatives for Extremal-Value Functions with Applications to the Completely Convex Case," Operations Research, INFORMS, vol. 21(1), pages 188-209, February.
    • Handle: RePEc:inm:oropre:v:21:y:1973:i:1:p:188-209
    DOI: 10.1287/opre.21.1.188
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    Cited by:

    1. O. Stein, 2004. "On Constraint Qualifications in Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 647-671, June.
    2. Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    3. Oyama, Daisuke & Takenawa, Tomoyuki, 2018. "On the (non-)differentiability of the optimal value function when the optimal solution is unique," Journal of Mathematical Economics, Elsevier, vol. 76(C), pages 21-32.
    4. Giorgio Giorgi, 2021. "Some Classical Directional Derivatives and Their Use in Optimization," DEM Working Papers Series 204, University of Pavia, Department of Economics and Management.
    5. George E. Monahan, 1996. "Finding saddle points on polyhedra: Solving certain continuous minimax problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(6), pages 821-837, September.
    6. Rinaldi, Francesca, 2009. "Endogenous incompleteness of financial markets: The role of ambiguity and ambiguity aversion," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 880-901, December.
    7. O. Stein & A. Winterfeld, 2010. "Feasible Method for Generalized Semi-Infinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 419-443, August.
    8. Giancarlo Bigi & Mauro Passacantando, 2012. "Gap functions and penalization for solving equilibrium problems with nonlinear constraints," Computational Optimization and Applications, Springer, vol. 53(2), pages 323-346, October.
    9. Giancarlo Bigi & Mauro Passacantando, 2015. "Descent and Penalization Techniques for Equilibrium Problems with Nonlinear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 804-818, March.

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