Generalized second-order directional derivatives for a class of nonsmooth functions (Q2714295)

The paper studies the generalized second-order directional derivatives formulas of \textit{R. W. Chaney} [J. Math. Anal. Appl. 128, 495-511 (1987; Zbl 0671.49012)], for a class of functions: NEWLINE\[NEWLINEf(x)=g(y_1,\dots,y_m)=g(f_1(x),\dots,f_m(x)) \tag{1}NEWLINE\]NEWLINEwhere \(f_i(x)=\max_{1\leq j\leq K_i}h_{ij}(x), K_i\) is a positive integer, \(i=1,\dots,m\), \(g:R^m\rightarrow R\), \(g\in C^2(R^m)\), \(h_{ij}(x)\) is piecewise \(C^2\) on \(R^n,i=1,\dots,m,j=1,\dots,K_i\). It is shown that this class of nonsmooth functions in (1) includes the functions of exact penalty type for nonsmooth constrained optimization problems. The paper also provides the second order optimality conditions for minimizing \(f(x)\) in (1).