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D. Butnariu, Y. Censor, and S. Reich. Iterative Averaging of Entropic Projections for Solving Stochastic Convex Feasibility Problems. Computational Optimization and Applications 8:21-39, 1997. 10. D. Butnariu and A. N. Iusem. On a proximal point method for convex optimization in Banach spaces. Numerical Functional Analysis and Optimization 18:723-744, 1997. 36 11. D. Butnariu and A. N. Iusem. Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer, 2000.
Then g is convex r U \ A is a set of measure zero, and g" is positive semidefinite on A. Proof. 51]) states t h a t the set U \ A is of measure zero. Fix x E U and y E ]RN arbitrarily. The function t ~-+ g(x + ty) is convex in a neighborhood of 0; consequently, its second derivative (y, (g"(x + ty))(y)) is nonnegative whenever it exists. Since g"(x) does exist, it must be t h a t g"(x) is positive semidefinite, for every x E A. "r : Fix x and y in U. By assumption and a Fubini argument, we obtain two sequences (xn) and (yn) in U with x~ --+ x, y~ --+ y, and t ~ g"(xn + t(y~ - x~)) exists almost everywhere on [0, 1].