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Approximated perspective relaxations: a project and lift approach

Author

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  • Antonio Frangioni
  • Fabio Furini
  • Claudio Gentile

Abstract

The perspective reformulation (PR) of a Mixed-Integer NonLinear Program with semi-continuous variables is obtained by replacing each term in the (separable) objective function with its convex envelope. Solving the corresponding continuous relaxation requires appropriate techniques. Under some rather restrictive assumptions, the Projected PR ( $$\mathrm{P}^2\mathrm{R}$$ P 2 R ) can be defined where the integer variables are eliminated by projecting the solution set onto the space of the continuous variables only. This approach produces a simple piecewise-convex problem with the same structure as the original one; however, this prevents the use of general-purpose solvers, in that some variables are then only implicitly represented in the formulation. We show how to construct an Approximated Projected PR ( $$\mathrm{AP}^2\mathrm{R}$$ AP 2 R ) whereby the projected formulation is “lifted” back to the original variable space, with each integer variable expressing one piece of the obtained piecewise-convex function. In some cases, this produces a reformulation of the original problem with exactly the same size and structure as the standard continuous relaxation, but providing substantially improved bounds. In the process we also substantially extend the approach beyond the original $$\mathrm{P}^2\mathrm{R}$$ P 2 R development by relaxing the requirement that the objective function be quadratic and the left endpoint of the domain of the variables be non-negative. While the $$\mathrm{AP}^2\mathrm{R}$$ AP 2 R bound can be weaker than that of the PR, this approach can be applied in many more cases and allows direct use of off-the-shelf MINLP software; this is shown to be competitive with previously proposed approaches in some applications. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • Antonio Frangioni & Fabio Furini & Claudio Gentile, 2016. "Approximated perspective relaxations: a project and lift approach," Computational Optimization and Applications, Springer, vol. 63(3), pages 705-735, April.
  • Handle: RePEc:spr:coopap:v:63:y:2016:i:3:p:705-735
    DOI: 10.1007/s10589-015-9787-8
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    References listed on IDEAS

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    Cited by:

    1. Xiaojin Zheng & Yutong Pan & Zhaolin Hu, 2021. "Perspective Reformulations of Semicontinuous Quadratically Constrained Quadratic Programs," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 163-179, January.
    2. Carina Moreira Costa & Dennis Kreber & Martin Schmidt, 2022. "An Alternating Method for Cardinality-Constrained Optimization: A Computational Study for the Best Subset Selection and Sparse Portfolio Problems," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 2968-2988, November.
    3. Ken Kobayashi & Yuichi Takano & Kazuhide Nakata, 2021. "Bilevel cutting-plane algorithm for cardinality-constrained mean-CVaR portfolio optimization," Journal of Global Optimization, Springer, vol. 81(2), pages 493-528, October.
    4. Antonio Frangioni & Claudio Gentile & James Hungerford, 2020. "Decompositions of Semidefinite Matrices and the Perspective Reformulation of Nonseparable Quadratic Programs," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 15-33, February.
    5. Dimitris Bertsimas & Ryan Cory-Wright, 2022. "A Scalable Algorithm for Sparse Portfolio Selection," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1489-1511, May.

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