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We show that Dykstra's splitting for projecting onto the intersection ofconvex sets can be extended to minimize the sum of convex functions and aregularizing quadratic. We give conditions for which convergence to the primalminimizer holds so that more than one convex function can be minimized at atime, the convex functions are not necessarily sampled in a cyclic manner, andthe SHQP strategy for problems involving the intersection of more than oneconvex set can be applied. When the sum does not involve the regularizingquadratic, we discuss an approximate proximal point method combined withDykstra's splitting to minimize this sum.

Dykstra splitting and an approximate proximal point algorithm for minimizing the sum of convex functions