The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

This study considers equilibrium problems, focusing on identifying 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 s...

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Vydáno v:Mathematics (Basel) Ročník 12; číslo 2; s. 352
Hlavní autoři: Wang, Ruyu, Zhao, Wenling, Song, Daojin, Hu, Yaozhong
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 01.01.2024
Témata:
Algorithms
augmented weak sharpness
Coining
Convexity
Equilibrium
equilibrium problems
feasible solution sequence
finite termination
Game theory
Mathematical analysis
Mathematical programming
Optimization
regular normal cone
Sequences
Sharpness
weak sharpness
United Kingdom
ISSN:2227-7390, 2227-7390
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Shrnutí:This study considers equilibrium problems, focusing on identifying 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. This comprehensive concept 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 the existing literature, which often assumes the solution set to be weakly sharp or strongly non-degenerate, especially in 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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math12020352