Abstract
The second derivative of an envelope cannot be expressed only by second derivatives of the constituent functions. By taking account of this fact, we derive new second order necessary optimality conditions for minimization of a sup-type function. The conditions involve an extra term besides the second derivative of the Lagrange function. Furthermore, we will comment on the relationship between the extra term and a kind of second order directional derivative of the sup-type function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Ben-Tal, M. Teboulle and J. Zowe, “Second order necessary optimality conditions for semi-infinite programming problems,” in: R. Hettich, ed.,Semi-infinite programming, Lecture Notes in Control and Inform. Sci., Vol. 15 (Springer, Berlin, 1979) pp. 17–30.
A. Ben-Tal and J. Zowe, “A unified theory of first and second order conditions for extremum problems in topological vector spaces,”Mathematical Programming Study 19 (1982) 39–76.
E.W. Cheney,Introduction to Approximation Theory (McGraw-Hill, New York, 1966).
I.V. Girsanov,Lectures on mathematical theory of extremum problems, Lecture Notes in Econom. and Math. Systems, Vol. 67 (Springer, Berlin, 1972).
R. Hettich,Semi-infinite programming, Lecture Notes in Control and Inform. Sci., Vol. 15 (Springer, Berlin, 1979).
R. Hettich and H.Th. Jongen, “Semi-infinite programming conditions of optimality and applications,” in: J. Stoer, ed.,Optimization Techniques II, Lecture Notes in Control and Inform. Sci., Vol. 7 (Springer, Berlin, 1978) pp. 1–11.
J.B. Hiriart-Urruty, “Approximate first-order and second-order directional derivatives of a marginal function in convex optimization,”Journal of Optimization Theory and Applications 48 (1986) 127–140.
H.Th. Jongen and G. Zwier, “On the local structure of the feasible set in semi-infinite optimization,” in: B. Brosowski, ed.,Parametric Optimization and Approximation (Birkhäuser, Basel, 1983) pp. 185–202.
H.Th. Jongen and G. Zwier, “On regular semi-infinite optimization,” in: E.J. Anderson and A.B. Philpott, eds.,Infinite Programming, Lecture Notes in Econom. and Math. Systems, Vol. 259 (Springer, 1985) pp. 53–64.
H. Kawasaki, “An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems,”Mathematical Programming 41 (1988) 73–96.
H. Kawasaki, “The upper and lower second order directional derivative of a sup-type function,”Mathematical Programming 41 (1988) 327–339.
H. Maurer, “First and second order sufficient optimality conditions in mathematical programming and optimal control,”Mathematical Programming Study 14 (1981) 163–177.
S.M. Robinson, “Stability theory for systems of inequalities, Part II: differentiable nonlinear systems,”SIAM Journal Numerical Analysis 13 (1976) 487–513
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
A. Shapiro, “Second-order derivatives of extremal-value functions and optimality conditions for semi-infinite programs,”Mathematics of Operations Research 10 (1985) 207–219.
W. Wetterling, “Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben,”Numerische Mathematik 15 (1970) 122–136.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kawasaki, H. Second order necessary optimality conditions for minimizing a sup-type function. Mathematical Programming 49, 213–229 (1990). https://doi.org/10.1007/BF01588788
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01588788