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Duality and robust duality for special nonconvex homogeneous quadratic programming under certainty and uncertainty environment

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  • Yanjun Wang
  • Ruizhi Shi
  • Jianming Shi

Abstract

In this paper, we discuss a kind of special nonconvex homogenous quadratic programming (HQP) and the methods to solve the HQP in an environment with certainty or uncertainty. In an environment with certainty, we first establish a strong duality between the HQP and its Lagrange dual problem, with the help of the fact that the Lagrange dual problem is equivalent to a convex semidefinite programming (SDP). Then we obtain a global solution to the HQP by solving the convex SDP. Furthermore, in an environment with uncertainty, we formulate the robust counterpart of the HQP to cope with uncertainty. We also establish the robust strong duality between the robust counterpart and its optimistic counterpart under a mild assumption. Since the counterpart is equivalent to a convex SDP under the same assumption, we can obtain a global solution to the robust counterpart by solving the convex SDP under the same assumption. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Yanjun Wang & Ruizhi Shi & Jianming Shi, 2015. "Duality and robust duality for special nonconvex homogeneous quadratic programming under certainty and uncertainty environment," Journal of Global Optimization, Springer, vol. 62(4), pages 643-659, August.
    • Handle: RePEc:spr:jglopt:v:62:y:2015:i:4:p:643-659
    DOI: 10.1007/s10898-015-0281-8
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    References listed on IDEAS

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    1. Alexander Shapiro, 1982. "Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 47(2), pages 187-199, June.
    2. V. Jeyakumar & G. Y. Li, 2011. "Robust Duality for Fractional Programming Problems with Constraint-Wise Data Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 292-303, November.
    3. Gábor Pataki, 1998. "On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 339-358, May.
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    1. Nguyen Dinh & Miguel Angel Goberna & Marco Antonio López & Michel Volle, 2017. "A Unifying Approach to Robust Convex Infinite Optimization Duality," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 650-685, September.

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