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An approach to characterizing $$\epsilon $$ ϵ -solution sets of convex programs

Author

Listed:
  • N. V. Tuyen

    (Hanoi Pedagogical University 2, Xuan Hoa)

  • C.-F. Wen

    (Kaohsiung Medical University
    Kaohsiung Medical University Hospital)

  • T. Q. Son

    (Saigon University, HCMC)

Abstract

In this paper, we propose an approach to characterizing $${\epsilon} $$ ϵ -solution sets of convex programs with a given $${\epsilon} >0$$ ϵ > 0 . The results are divided into two parts. The first one is devoted to establishing the expressions of $${\epsilon} $$ ϵ -solution sets of a class of convex infinite programs. The representation is given based on the study of relationships among the following three sets: the set of Lagrange multipliers corresponding to a given $${\epsilon} $$ ϵ -solution, the set of $${\epsilon} $$ ϵ -solutions of the dual problem corresponding, and the set of $${\epsilon} $$ ϵ -Kuhn–Tucker vectors associated with the problem in consideration. The second one is devoted to some special cases: the $${\epsilon} $$ ϵ -solution sets of convex programs that have set constraints and the almost $${\epsilon} $$ ϵ -solution sets of convex programs that have finite convex constraints. Several examples are given.

Suggested Citation

  • N. V. Tuyen & C.-F. Wen & T. Q. Son, 2022. "An approach to characterizing $$\epsilon $$ ϵ -solution sets of convex programs," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 249-269, July.
    • Handle: RePEc:spr:topjnl:v:30:y:2022:i:2:d:10.1007_s11750-021-00616-y
    DOI: 10.1007/s11750-021-00616-y
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    References listed on IDEAS

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    1. V. Jeyakumar & G. M. Lee & N. Dinh, 2004. "Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 83-103, October.
    2. T. Son & D. Kim, 2013. "ε-Mixed type duality for nonconvex multiobjective programs with an infinite number of constraints," Journal of Global Optimization, Springer, vol. 57(2), pages 447-465, October.
    3. T. Son & N. Dinh, 2008. "Characterizations of optimal solution sets of convex infinite programs," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 147-163, July.
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