Second-order conditions for extremum problems with nonregular equality constraints

Combining results of Avakov about tangent directions to equality constraints given by smooth operators with results of Ben-Tal and Zowe, we formulate a second-order theory for optimality in the sense of Dubovitskii-Milyutin which gives nontrivial conditions also in the case of equality constraints given by nonregular operators. Secondorder feasible and tangent directions are defined to construct conical approximations to inequality and equality constraints which within a single construction lead to first- and second-order conditions of optimality for the problem also in the nonregular case. The definitions of secondorder feasible and tangent directions given in this paper allow for reparametrizations of the approximating curves and give approximating sets which form cones. The main results of the paper are a theorem which states second-order necessary condition of optimality and several corollaries which treat special cases. In particular, the paper generalizes the Avakov result in the smooth case.

[1]  A. Ioffe,et al.  Theory of extremal problems , 1979 .

[2]  Euler-Lagrange equation in the case of nonregular equality constraints , 1991 .

[3]  Andrei Dmitruk,et al.  LYUSTERNIK'S THEOREM AND THE THEORY OF EXTREMA , 1980 .

[4]  E. Levitin,et al.  CONDITIONS OF HIGH ORDER FOR A LOCAL MINIMUM IN PROBLEMS WITH CONSTRAINTS , 1978 .

[5]  E. Polak,et al.  On Second Order Necessary Conditions of Optimality , 1969 .

[6]  K. H. Hoffmann,et al.  Higher-order necessary conditions in abstract mathematical programming , 1978 .

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

[8]  E. R. Avakov,et al.  Extremum conditions for smooth problems with equality-type constraints , 1986 .

[9]  A. A. Tret' Yakov Necessary and sufficient conditions for optimality of p-th order☆ , 1984 .

[10]  J. Zowe,et al.  Second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems , 1979 .

[11]  A. Ioffe Necessary and Sufficient Conditions for a Local Minimum. 3: Second Order Conditions and Augmented Duality , 1979 .

[12]  Urszula Ledzewicz On Abnormal Optimal Control Problems with Mixed Equality and Inequality Constraints , 1993 .

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

[14]  Urszula Ledzewicz,et al.  Extension of the local maximum principle to abnormal optimal control problems , 1993 .

[15]  E. P. Avakov Necessary extremum conditions for smooth anormal problems with equality- and inequality-type constraints , 1989 .