2024/01/12 by Ruyu Wang, Wenling Zhao, Wang, Ruyu +5
Computer Science · Neuroscience · #49J40 #49K40 #49M37 #90C31 #91A10 #FOS: Mathematics #Nuclear Receptors and Signaling #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2401.06351
openalex publication_date 2024/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This study delves into equilibrium problems, focusing on the identification of finite solutions for feasible solution sequences. We introduce an innovative extension of the weak sharp minimum concept from convex programming to equilibrium problems, coining this as weak sharpness for solution sets. Recognizing situations where the solution set may not exhibit weak sharpness, we propose an augmented mapping approach to mitigate this limitation. The core of our research is the formulation of augmented weak sharpness for the solution set, a comprehensive concept that encapsulates both weak sharpness and strong non-degeneracy within feasible solution sequences. Crucially, we identify a necessary and sufficient condition for the finite termination of these sequences under the premise of augmented weak sharpness for the solution set in equilibrium problems. This condition significantly broadens the scope of existing literature, which often assumes the solution set to be weakly sharp or strongly non-degenerate, especially in the context of mathematical programming and variational inequality problems. Our findings not only shed light on the termination conditions in equilibrium problems but also introduce a less stringent sufficient condition for the finite termination of various optimization algorithms. This research, therefore, makes a substantial contribution to the field by enhancing our understanding of termination conditions in equilibrium problems and expanding the applicability of established theories to a wider range of optimization scenarios.