Geodesic
The second order optimality conditions for nonlinear mathematical programming with $C^{1,1}$ data
Applications of Mathematics, Tome 42 (1997) no. 4, pp. 311-320.

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To find nonlinear minimization problems are considered and standard $C^2$-regularity assumptions on the criterion function and constrained functions are reduced to $C^{1,1}$-regularity. With the aid of the generalized second order directional derivative for $C^{1,1}$ real-valued functions, a new second order necessary optimality condition and a new second order sufficient optimality condition for these problems are derived.
DOI : 10.1023/A:1023068513188
Classification : 49J45, 90C30
Keywords: nonlinear programming; constrained problems; $C^{1;1}$ functions; second order conditions 
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Liu, Liping; Křížek, Michal. The second order optimality conditions for nonlinear mathematical programming with $C^{1,1}$ data. Applications of Mathematics, Tome 42 (1997) no. 4, pp. 311-320. doi : 10.1023/A:1023068513188. http://geodesic.mathdoc.fr/articles/10.1023/A:1023068513188/

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