This paper is concerned with differential sensitivity analysis for optimal values of differentiable convex programming problems whose objective and constraint functions include parameters.
It is well known that the directional derivative of optimal-value function of convex programming problem can be represented as a min-max type positively homogeneous function with respect to the direction. Based on this fact, we investigate the relation between the generalized gradient and the local convexity or concavity (the Clarke regularity) of optimal-value function. The main results obtained here are (i) characterization of a subset of the generalized gradient of optimal-value function, (ii) relation between the Clarke regularity and the generalized gradients of optimal-value functions, and (iii) sufficient conditions for the optimal-value function to be locally approximated by convex or concave function (the Clarke regularity).