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Bounding duality gap for separable problems with linear constraints

Author

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  • Madeleine Udell

    (California Institute of Technology)

  • Stephen Boyd

    (Stanford University)

Abstract

We consider the problem of minimizing a sum of non-convex functions over a compact domain, subject to linear inequality and equality constraints. Approximate solutions can be found by solving a convexified version of the problem, in which each function in the objective is replaced by its convex envelope. We propose a randomized algorithm to solve the convexified problem which finds an $$\epsilon $$ ϵ -suboptimal solution to the original problem. With probability one, $$\epsilon $$ ϵ is bounded by a term proportional to the maximal number of active constraints in the problem. The bound does not depend on the number of variables in the problem or the number of terms in the objective. In contrast to previous related work, our proof is constructive, self-contained, and gives a bound that is tight.

Suggested Citation

  • Madeleine Udell & Stephen Boyd, 2016. "Bounding duality gap for separable problems with linear constraints," Computational Optimization and Applications, Springer, vol. 64(2), pages 355-378, June.
    • Handle: RePEc:spr:coopap:v:64:y:2016:i:2:d:10.1007_s10589-015-9819-4
    DOI: 10.1007/s10589-015-9819-4
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    References listed on IDEAS

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    1. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
    2. Jérôme Bolte & Aris Daniilidis & Adrian S. Lewis, 2011. "Generic Optimality Conditions for Semialgebraic Convex Programs," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 55-70, February.
    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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    Cited by:

    1. Nicholas Moehle & Mykel J. Kochenderfer & Stephen Boyd & Andrew Ang, 2021. "Tax-Aware Portfolio Construction via Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 364-383, May.
    2. Nicholas Moehle & Mykel J. Kochenderfer & Stephen Boyd & Andrew Ang, 2020. "Tax-Aware Portfolio Construction via Convex Optimization," Papers 2008.04985, arXiv.org, revised Feb 2021.

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