Abstract:

To provide a finite characterization of feasible solution sequences for optimization problems (OP) and variational inequality problems (VIP), an augmented set value map is introduced for the solution sets of these problems. Additionally, the concepts of augmented weak sharpness with respect to feasible solution sequences are established. These novel notions extend the traditional concepts of weak sharpness and strongly non-degeneracy relative to feasible solution sequences, addressing the limitation that solution sets often lack weak sharpness or strongly non-degeneracy in many cases. When the feasible solution sets of optimization problems and variational inequality problems exhibit augmented weak sharpness, the necessary and sufficient conditions for the finite termination of feasible solution sequences are provided for each problem. These conditions extend the corresponding results found in existing literature, where solution sets are weak sharp or strongly non-degenerate. Furthermore, sufficient conditions with fewer restrictions are provided for the finite termination of various optimization algorithms.

Key words: Optimization problem, Variational inequality problem, Feasible solution sequence, Augmented weak sharpness, Finite termination