Totally convex functions for fixed points computation and infinite dimensional optimization (Q1575115)

For a differentiable convex function f, the Bregman distance is \[ D_f(x,y) := f(y) - f(x) + f^\circ (x,x-y), \] where \(f^\circ\) is directional derivative. The function \(f\) is called totally convex if \(\inf\{ D_f(x,y) : \|x-t\|=t\} > 0\) when \(t > 0\). These concepts are applied to find common fixed points for families of operators, using a projection constructed from \(D_f\), in Banach spaces that are not Hilbert. The discussion includes stochastic convex feasibility problems, a proximal point method, and an augmented Lagrangian method for constrained optimization. Convergence proofs are given.