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Functional inequalities and theorems of the alternative involving composite functions

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  • N. Dinh
  • G. Vallet
  • M. Volle

Abstract

We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form $$ f + g + k\circ H \ge h$$ f + g + k ∘ H ≥ h . The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function $$ f + g + k\circ H$$ f + g + k ∘ H that are satisfied in various favorable circumstances. The results are then applied to the contexts of alternative type theorems involving composite and DC functions. The results cover several Farkas-type results for convex or DC systems and are general enough to face with unpublished situations. As applications of these results, nonconvex optimization problems with composite functions, convex composite problems with conic constraints are examined at the end of the paper. There, strong duality, stable strong duality results for these classes of problems are established. Farkas-type results and stable form of these results for the corresponding systems involving composite functions are derived as well. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • N. Dinh & G. Vallet & M. Volle, 2014. "Functional inequalities and theorems of the alternative involving composite functions," Journal of Global Optimization, Springer, vol. 59(4), pages 837-863, August.
    • Handle: RePEc:spr:jglopt:v:59:y:2014:i:4:p:837-863
    DOI: 10.1007/s10898-013-0100-z
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    References listed on IDEAS

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    1. N. Dinh & V. Jeyakumar & G. M. Lee, 2005. "Sequential Lagrangian Conditions for Convex Programs with Applications to Semidefinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 125(1), pages 85-112, April.
    2. M. Volle, 2002. "Duality Principles for Optimization Problems Dealing with the Difference of Vector-Valued Convex Mappings," Journal of Optimization Theory and Applications, Springer, vol. 114(1), pages 223-241, July.
    3. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    4. Radu Ioan Bot, 2010. "Conjugate Duality in Convex Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-04900-2, July.
    5. V. Jeyakumar, 2008. "Constraint Qualifications Characterizing Lagrangian Duality in Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 31-41, January.
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    Cited by:

    1. N. Dinh & M. A. Goberna & M. A. López & T. H. Mo, 2017. "Farkas-Type Results for Vector-Valued Functions with Applications," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 357-390, May.
    2. D. H. Fang & Y. Zhang, 2018. "Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 351-376, February.
    3. Nguyen Dinh & Dang Hai Long, 2022. "A Perturbation Approach to Vector Optimization Problems: Lagrange and Fenchel–Lagrange Duality," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 713-748, August.
    4. Nguyen Dinh & Miguel A. Goberna & Dang H. Long & Marco A. López-Cerdá, 2019. "New Farkas-Type Results for Vector-Valued Functions: A Non-abstract Approach," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 4-29, July.

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