Abstract
A parameterized nonlinear programming problem is considered in which the objective and constraint functions are twice continuously differentiable. Under the assumption that certain multiplier vectors appearing in generalized second-order necessary conditions for local optimality actually satisfy the weak sufficient condition for local optimality based on the augmented Lagrangian, it is shown that the optimal value in the problem, as a function of the parameters, is directionally differentiable. The directional derivatives are expressed by a minimax formula which generalizes the one of Gol’shtein in convex programming.
Research sponsored in part by the Air Force Office of Scientific Research, AFSC, United States Air Force, under grant no. F49620-82-K-0012.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
F.H. Clarke, “Generalized gradients and applications”, Transactions of the American Mathematical Society 205 (1975) 247–262.
J. Gauvin, “The generalized gradient of a marginal value function in mathematical programming”, Mathematics of Operations Research 4 (1979) 458–463.
J. Gauvin and F. Dubeau, “Differential properties of the marginal function in mathematical programming”, Mathematical Programming Study, to appear.
J. Gauvin and J.W. Tolle, “Differential stability in nonlinear programming”, SIAM Journal on Control and Optimization 15 (1977) 294–311.
E.G. Gol’shtein, Theory of convex programming, Translations of Mathematical Monographs, 36 (American Mathematical Society, Providence, RI, 1972).
M.R. Hestenes, Calculus of variations and optimal control theory (Wiley, New York, 1966).
M.R. Hestenes, Optimization theory: The finite-dimensional case (Wiley, New York, 1975).
W.W. Hogan, “Directional derivatives for extremal-value functions with applications to the completely convex case”, Operations Research 21 (1973) 188–209.
O.L. Mangasarian and S. Fromovitz, “The Fritz John necessary conditions in the presence of equality and inequality constraints”, Journal of Mathematical Analysis and Applications 17 (1967) 37–47.
G.P. McCormick, “Second-order conditions for constrained minima”, SIAM Journal on Applied Mathematics 15 (1967) 641–652.
S.M. Robinson, “Stability theory for systems of inequalities, part II: Differentiable nonlinear systems”, SIAM Journal on Numerical Analysis 13 (1976) 497–513.
R.T. Rockafellar, Convex analysis, (Princeton University Press, Princeton, NJ, 1970).
R.T. Rockafellar, “A dual approach to solving nonlinear programming problems by unconstrained optimization”, Mathematical Programming 5 (1973) 345–373.
R.T. Rockafellar, “Augmented Lagrange multiplier functions and duality in nonconvex programming”, SIAM Journal on Control 12 (1974) 268–285.
R.T. Rockafellar, The theory of subgradients and its applications: Convex and nonconvex functions (Haldermann Verlag, West Berlin, 1981).
R.T. Rockafellar, “Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming”, Mathematical Programming Study 17 (1982) 28–66.
R.T. Rockafellar, “Marginal values and second-order necessary conditions for optimality”, Mathematical Programming 26 (1983) 245–286.
G. Salinetti and R.J.B. Wets, “On the convergence of convex sets in finite dimensions”, SIAM Review 21 (1979) 18–33.
R.J.B. Wets, “On a compactness theorem for epiconvergent sequences of functions”, forthcoming.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 The Mathematical Programming Society, Inc.
About this chapter
Cite this chapter
Rockafellar, R.T. (1984). Directional differentiability of the optimal value function in a nonlinear programming problem. In: Fiacco, A.V. (eds) Sensitivity, Stability and Parametric Analysis. Mathematical Programming Studies, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121219
Download citation
DOI: https://doi.org/10.1007/BFb0121219
Received:
Revised:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00912-9
Online ISBN: 978-3-642-00913-6
eBook Packages: Springer Book Archive