Abstract
Given a mathematical programming problem depending on a parameter vectorx, we consider the associated optimal value (marginal) functionϕ(x) and the optimal set-valued multifunctionM(x). The aim of this paper is to investigate continuity and differentiability properties ofϕ(x) andM(x) at a pointx 0 where the corresponding setM(x 0) of optimal solutions isnot asingleton. We show that under certain regularity conditions the multifunctionM(x) is upper Lipschitzian atx 0 andϕ(x) possesses second-order directional derivatives.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aubin, J.P. (1984). Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9:87–111.
Ben-Tal, A. and Zowe, J. (1982). A unified theory of first- and second-order conditions for extremum problems in topological vector spaces, Math. Programming Study 19:39–76.
Danskin, J.M. (1967). The Theory of max-min and Its Applications to Weapons Allocation Problems, Springer-Verlag, New York.
Demyanov, V.F. and Malozemov, V.N. (1974). Introduction to mini-max, Wiley, New York.
Dieudonné, J. (1960). Foundations of Modern Analysis, Academic Press, New York.
Gaitsgory, V.A. and Pervozvanskii, A.A. (1986). Perturbation theory for mathematical programming problems, J. Optim. Theory Appl. 49:389–410.
Fiacco, A.V. and McCormick, G.P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York.
Fiacco, A.V. (1983). Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York.
Gauvin, J. (1977). A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming, Math. Programming 12:136–138.
Gauvin, J. and Tolle, J.W. (1977). Differential stability in nonlinear programming, SIAM J. Control Optim. 15:294–311.
Gauvin, J. and Dubeau, F. (1982). Differential properties of the marginal function in mathematical programming, Math. Programming Study 19:101–119.
Golshtein, E.G. (1972). Theory of Convex Programming, Trans. Math. Monographs 26, American Mathematical Society, Providence, RI.
Hestenes, M.R. (1975). Optimization Theory—The Finite Dimensional Case, Wiley, New York.
Ioffe, A.D. (1979). Necessary and sufficient conditions for a local minimum. 3: Second-order conditions and augmented duality, SIAM J. Control Optim. 17:266–288.
Janin, R. and Gauvin, J. (1986). Directional sensitivity for optimal solutions in nonlinear mathematical programming, Preprint.
Jittorntrum, K. (1984). Solution point differentiability without strict complementarity in nonlinear programming, Math. Programming Study 21:127–138.
Lancaster, P. (1964). On eigenvalues of matrices dependent on a parameter, Numer. Math. 6:377–387.
Mangasarian, O.L. and Fromovitz, S. (1967). The Fritz John necessary optimality conditions in the presence of equality and inequality constraints, J. Math. Anal. Appl. 7:37–47.
Robinson, S.M. (1981). Some continuity properties of polyhedral multifunctions, Math. Programming Study 14:206–214.
Robinson, S.M. (1982). Generalized equations and their solutions, Part II: Applications to nonlinear programming, Math. Programming Study 19:200–221.
Rockafellar, R.T. (1970). Convex Analysis, Princeton University Press, Princeton, NJ.
Rockafellar, R.T. (1984). Directional differentiability of the optimal value function in a nonlinear programming problem, Math. Programming Study 21:213–226.
Rockafellar, R.T. (1985). Lipschitzian properties of multifunctions, Nonlinear Anal. Theory Methods Appl. 9:867–885.
Shapiro, A. (1985). Second-order derivatives of extremal-value functions and optimality conditions for semi-infinite programs, Math. Oper. Res. 10:207–219.
Shapiro, A. (1985). Second-order sensitivity analysis and asymptotic theory of parametrized nonlinear programs, Math. Programming 33:280–299.
Shapiro, A. Sensitivity analysis of nonlinear programs and differentiability properties of metric projections, SIAM J. Control Optim. (to appear).
Wets, R.J.B. (1973). On inf-compact mathematical programs, Fifth Conference on Optimization Techniques, Part I, Lecture Notes in Computer Sciences 3, pp. 426–436, Springer-Verlag, Berlin.
Author information
Authors and Affiliations
Additional information
Communicated by J. Stoer
Rights and permissions
About this article
Cite this article
Shapiro, A. Perturbation theory of nonlinear programs when the set of optimal solutions is not a singleton. Appl Math Optim 18, 215–229 (1988). https://doi.org/10.1007/BF01443623
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01443623