Hybrid approximate proximal algorithms for efficient solutions in vector optimization

We develop new hybrid approximate proximal-type algorithms to find efficient (Pareto optimal) solutions to general problems of vector optimization in finite-dimensional and infinite-dimensional spaces. In contrast to the vast majority of publications in this direction, our algorithms do not depend on the nonemptiness of ordering cones of the spaces in question and concern finding efficient (while not weakly efficient) solutions to the vector optimization problems under consideration. In particular, one of our algorithms provides a constructive iterative procedure that converges to an efficient solution for a constrained problem of minimizing a mapping from a Hilbert space to a Banach space by combining an extragradient method of solving variational inequalities and an approximate proximal point method to find roots of maximal monotone operators. We also develop an extended hybrid approximate proximal algorithm converging to efficient solutions for vector optimization problems that is based on Bregman functions.