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Global Optimality Conditions for Some Classes of Optimization Problems

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  • Z. Y. Wu

    (University of Ballarat)

  • A. M. Rubinov

    (University of Ballarat)

Abstract

We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity.

Suggested Citation

  • Z. Y. Wu & A. M. Rubinov, 2010. "Global Optimality Conditions for Some Classes of Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 164-185, April.
    • Handle: RePEc:spr:joptap:v:145:y:2010:i:1:d:10.1007_s10957-009-9616-2
    DOI: 10.1007/s10957-009-9616-2
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    References listed on IDEAS

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    1. Z. Y. Wu & V. Jeyakumar & A. M. Rubinov, 2007. "Sufficient Conditions for Global Optimality of Bivalent Nonconvex Quadratic Programs with Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 133(1), pages 123-130, April.
    2. M. Ç. Pinar, 2004. "Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 433-440, August.
    3. Jean-Philippe Vial, 1983. "Strong and Weak Convexity of Sets and Functions," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 231-259, May.
    4. VIAL, Jean-Philippe, 1983. "Strong and weak convexity of sets and functions," LIDAM Reprints CORE 529, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Vaithilingam Jeyakumar & Zhiyou Wu, 2007. "Conditions For Global Optimality Of Quadratic Minimization Problems With Lmi Constraints," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 24(02), pages 149-160.
    6. V. Jeyakumar & A. M. Rubinov & Z. Y. Wu, 2007. "Generalized Fenchel’s Conjugation Formulas and Duality for Abstract Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 441-458, March.
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    Cited by:

    1. Amar Andjouh & Mohand Ouamer Bibi, 2022. "Adaptive Global Algorithm for Solving Box-Constrained Non-convex Quadratic Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 360-378, January.
    2. Guoyin Li, 2012. "Global Quadratic Minimization over Bivalent Constraints: Necessary and Sufficient Global Optimality Condition," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 710-726, March.
    3. Xue-Gang Zhou & Xiao-Peng Yang & Bing-Yuan Cao, 2015. "Global optimality conditions for cubic minimization problems with cubic constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(3), pages 243-264, December.

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