Second order necessary optimality conditions for minimizing a sup-type function

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.

[1]  R. P. Hettich,et al.  Semi-infinite programming: Conditions of optimality and applications , 1978 .

[2]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[3]  S. M. Robinson.  Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems , 1976 .

[4]  J. Hiriart-Urruty,et al.  Approximate first-order and second-order directional derivatives of a marginal function in convex optimization , 1986 .

[5]  E. Cheney.  Introduction to approximation theory , 1966 .

[6]  Marc Teboulle,et al.  Second order necessary optimality conditions for semi-infinite programming problems , 1979 .

[7]  B. T. Poljak,et al.  Lectures on mathematical theory of extremum problems , 1972 .

[8]  H. Th. Jongen,et al.  On the Local Structure of the Feasible Set in Semi-Infinite Optimization , 1984 .

[9]  H. Jongen,et al.  On Regular Semi-Infinite Optimization , 1985 .

[10]  Hidefumi Kawasaki.  An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems , 1988, Math. Program..

[11]  Alexander Shapiro,et al.  Second-Order Derivatives of Extremal-Value Functions and Optimality Conditions for Semi-Infinite Programs , 1985, Math. Oper. Res..

[12]  Hidefumi Kawasaki.  The upper and lower second order directional derivatives of a sup-type function , 1988, Math. Program..

[13]  Hidefumi Kawaski,et al.  An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems , 1988 .

[14]  W. Wetterling,et al.  Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben , 1970 .

[15]  A. Ben-Tal,et al.  A unified theory of first and second order conditions for extremum problems in topological vector spaces , 1982 .

[16]  H. Maurer.  First and second order sufficient optimality conditions in mathematical programming and optimal control , 1981 .

[17]  HettichR.,et al.  Semi-infinite programming , 1979 .