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On Characterizing the Solution Sets of Pseudoinvex Extremum Problems

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  • X. M. Yang

    (Chongqing Normal University
    Chongqing Key Laboratory of Operations Research and System Engineering)

Abstract

In this paper, we study the minimization of a pseudoinvex function over an invex subset and provide several new and simple characterizations of the solution set of pseudoinvex extremum problems. By means of the basic properties of pseudoinvex functions, the solution set of a pseudoinvex program is characterized, for instance, by the equality $\nabla f(x)^{T}\eta(\bar{x},x)=0$ , for each feasible point x, where $\bar{x}$ is in the solution set. Our study improves naturally and extends some previously known results in Mangasarian (Oper. Res. Lett. 7: 21–26, 1988) and Jeyakumar and Yang (J. Opt. Theory Appl. 87: 747–755, 1995).

Suggested Citation

  • X. M. Yang, 2009. "On Characterizing the Solution Sets of Pseudoinvex Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 537-542, March.
    • Handle: RePEc:spr:joptap:v:140:y:2009:i:3:d:10.1007_s10957-008-9470-7
    DOI: 10.1007/s10957-008-9470-7
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    References listed on IDEAS

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    1. Jeyakumar, V. & Lee, G.M. & Dinh, N., 2006. "Characterizations of solution sets of convex vector minimization problems," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1380-1395, November.
    2. 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.
    3. Z. L. Wu & S. Y. Wu, 2006. "Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 341-360, August.
    4. X.M. Yang & X.Q. Yang & K.L. Teo, 2003. "Generalized Invexity and Generalized Invariant Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 607-625, June.
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    Cited by:

    1. Satoshi Suzuki, 2019. "Optimality Conditions and Constraint Qualifications for Quasiconvex Programming," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 963-976, December.
    2. M. Oveisiha & J. Zafarani, 2014. "On Characterization of Solution Sets of Set-Valued Pseudoinvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 387-398, November.
    3. Xiangkai Sun & Kok Lay Teo & Liping Tang, 2019. "Dual Approaches to Characterize Robust Optimal Solution Sets for a Class of Uncertain Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 984-1000, September.
    4. Qamrul Hasan Ansari & Mahboubeh Rezaei, 2012. "Invariant Pseudolinearity with Applications," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 587-601, June.
    5. Kin Keung Lai & Shashi Kant Mishra & Sanjeev Kumar Singh & Mohd Hassan, 2022. "Stationary Conditions and Characterizations of Solution Sets for Interval-Valued Tightened Nonlinear Problems," Mathematics, MDPI, vol. 10(15), pages 1-16, August.
    6. Vsevolod I. Ivanov, 2019. "Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 144-162, April.
    7. Satoshi Suzuki & Daishi Kuroiwa, 2015. "Characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 62(3), pages 431-441, July.

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