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The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

Author

Listed:
  • Ruyu Wang

    (School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China)

  • Wenling Zhao

    (School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China)

  • Daojin Song

    (School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China)

  • Yaozhong Hu

    (Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada)

Abstract

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.

Suggested Citation

  • Ruyu Wang & Wenling Zhao & Daojin Song & Yaozhong Hu, 2024. "The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems," Mathematics, MDPI, vol. 12(2), pages 1-23, January.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:352-:d:1323833
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    References listed on IDEAS

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    1. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
    2. Nagurney, Anna, 2021. "Supply chain game theory network modeling under labor constraints: Applications to the Covid-19 pandemic," European Journal of Operational Research, Elsevier, vol. 293(3), pages 880-891.
    3. Wu, Zili, 2018. "Characterizations of weakly sharp solutions for a variational inequality with a pseudomonotone mapping," European Journal of Operational Research, Elsevier, vol. 265(2), pages 448-453.
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