In this section, we introduce one of the most important ideas in economic modelling, in the theory of optimization and, indeed in much of modern analysis and computatyional mathematics: that of a convex set. Almost every situation we will meet will depend on this geometric idea. As an independent idea, the notion of convexity appeared at the end of the 19 century, particularly in the works of M...

‎The proximal point algorithm‎, ‎which is a well-known tool for finding‎ ‎minima of convex functions‎, ‎is generalized from the classical‎ ‎Hilbert space framework into a nonlinear setting‎, ‎namely‎, ‎geodesic‎ ‎metric spaces of nonpositive curvature‎. ‎In this paper we propose an‎  ‎iterative algorithm for finding the common element of the‎ ‎minimizers of a finite family of convex functions a...

It should be noted that the requirement of f to be convex in the definition of f -divergence is essential. In Euclidean spaces any convex function can be represented as the pointwise supremum of a family of affine functions and vice versa, every supremum of a family of affine functions produces a convex function. Take f(x) = 12 |x− 1| as an example. We see that it can be written as a pointwise ...