Higher-order extremum conditions and tangent vectors at an abnormal point

In the paper under review the problem of minimizing a functional f_0(x) subject to the constraints f_i(x)leq0, i=1,2,dots,k_0, f_i(x)=0, i=k_0+1,dots,k, with functions f_i satisfying appropriate smoothness assumptions, is considered. The concept of the i-th critical sequence of directions is introduced and used in the formulation of high-order necessary conditions for optimality. The author applies this result to abnormal cases (when the Lyusternik theorem does not describe the tangent space), focusing on the abnormal points which are not 2-regular and where the second-order necessary conditions for optimality need to be replaced with conditions of higher order. The results appear to be an extension of the author's previous work [{it Extremum conditions} (Russian), Izdat. "Faktorial", Moscow, 1997; [msn] MR1469734 (99c:49001) [/msn]]. It also relates to work of other authors on abnormal problems and higher order conditions for optimality [E. R. Avakov, Mat. Zametki {bf 45} (1989), no.~6, 3--11, 110; [msn] MR1019030 (90k:49024) [/msn]; A. F. Izmailov, Mat. Zametki {bf 66} (1999), no.~1, 89--101; [msn] MR1729079 (2001g:49037) [/msn]].