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On the Nonemptiness and Boundedness of Solution Sets of Weakly Homogeneous Optimization Problems

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Abstract

In this paper, we introduce a new class of optimization problems whose objective functions are weakly homogeneous relative to the constraint sets. By using the normalization argument in asymptotic analysis, we prove two criteria for the nonemptiness and boundedness of the solution set of a weakly homogeneous optimization problem. Moreover, we discuss the existence and stability of the solution sets of linearly parametric optimization problems. Several examples are given to illustrate the derived results.

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References

  1. Ansari, Q.H., Lalitha, C.S., Mehta, M.: Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization. Chapman and hall/CRC (2014)

  2. Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)

    MATH  Google Scholar 

  3. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, Boston (1992)

    MATH  Google Scholar 

  4. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer-Verlag, New York (2003)

    MATH  Google Scholar 

  5. Flores-Bazán, F.: Asymptotically bounded multifunctions and the MCP beyond copositivity. J. Convex Anal. 17, 253–276 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Gowda, M.S., Sossa, D.: Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones. Math. Program. 177, 149–171 (2019)

    Article  MathSciNet  Google Scholar 

  7. Gowda, M.S., Pang, J.-S.: Some existence results for multivalued complementarity problems. Math. Oper. Res. 17, 657–669 (1992)

    Article  MathSciNet  Google Scholar 

  8. Gowda, M.S., Pang, J.-S.: On the boundedness and stability of solutions to the affine variational inequality problem. SIAM J. Control Optim. 32, 421–441 (1994)

    Article  MathSciNet  Google Scholar 

  9. Hieu, V.T.: On the solution existence and stability of polynomial optimization problems. Optim. Lett. 16, 1513–1529 (2023)

    Article  MathSciNet  Google Scholar 

  10. Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study. Springer, New York (2005)

    MATH  Google Scholar 

  11. Ma, X.X., Zheng, M.M., Huang, Z.H.: A note on the nonemptiness and compactness of solution sets of weakly homogeneous variational inequalities. SIAM J. Optim. 30, 132–148 (2020)

    Article  MathSciNet  Google Scholar 

  12. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (2009)

    MATH  Google Scholar 

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Acknowledgments

The author would like to express his deep gratitude to the two referees for their corrections and valuable comments which helped to improve the manuscript. This work has been supported by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Actions, grant agreement 813211 (POEMA).

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  1. LIP6, Sorbonne University, Paris, France

    Vu Trung Hieu

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  1. Vu Trung Hieu
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Correspondence to Vu Trung Hieu.

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Hieu, V.T. On the Nonemptiness and Boundedness of Solution Sets of Weakly Homogeneous Optimization Problems. Set-Valued Var. Anal 30, 1105–1116 (2022). https://doi.org/10.1007/s11228-022-00637-0

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  • DOI: https://doi.org/10.1007/s11228-022-00637-0

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