A Survey on Proximal Point Type Algorithms for Solving Vector Optimization Problems

A Strongly Convergent Proximal Point Method for Vector Optimization

“…For anyx ∈ S, we have ψ F lev k−1 (x) ≤ 0, in view of the definition of S in (A1), (8), and (11). Hence, by combining (28) with y =x and the definition of (11), we obtain, for every w ∈ C and k ≥ 0,…”

“…Other variants of the proximal gradient method were analyzed in the multiobjective and vector optimization setting in [8,17,44]. See, for instance, [28] for a review about vector proximal point method and some variants. Most recently, [34] proposed and analyzed a conjugate gradient method for solving vector optimization problems using different line-search strategies.…”

An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems

“…Since no information is carried over between iterations of the algorithm, the trajectory (i.e., the sequence) generated by the system (17) is the same as the one generated by Algorithm 3. In particular, this means that the Pareto set of the MOP is contained in the set of fixed points of the system (17). Thus, the subdivision algorithm (which was originally designed to compute attractors of dynamical systems) can be used to compute (a superset of) the Pareto set.…”

“…In [14,15], the subgradient method was generalized to the multiobjective case, but both articles report that their method is not suitable for real-life application due to inefficiency. In [16] (see also [17]), the proximal point method for single-objective optimization was generalized to convex vector optimization problems, where differentiability of the objectives is not required. In [18] (see also [19,20]), a multiobjective version of the proximal bundle method was proposed, which currently appears to be the most efficient solver.…”