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A multiplicative weights update algorithm for MINLP

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EURO Journal on Computational Optimization

Abstract

We discuss an application of the well-known multiplicative weights update (MWU) algorithm to non-convex and mixed-integer non-linear programming. We present applications to: (a) the distance geometry problem, which arises in the positioning of mobile sensors and in protein conformation; (b) a hydro unit commitment problem arising in the energy industry, and (c) a class of Markowitz’ portfolio selection problems. The interest of the MWU with respect to one of its closest competitors (classic multi-start) is that it provides a relative approximation guarantee on a certain quality measure of the solution.

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Notes

  1. An exact reformulation (formally defined in Liberti 2009 as a surjective mapping from the optima of the exact reformulation to the optima of the original problem) is not the same as an exact pointwise reformulation. Intuitively speaking, solving an exact reformulation of a problem directly yields a solution of the problem itself, which is not generally the case for an exact pointwise reformulation.

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Acknowledgments

We are very grateful to Dr. Pascale Bendotti (EDF) for useful suggestions about the HUC problem. Luca Mencarelli is sponsored by a Ph.D. Fellowship from the FP7 Marie Curie ITN “MINO” project. Youcef Sahraoui is sponsored by a CIFRE Ph.D. Fellowship with Éléctricité De France (EDF). Leo Liberti was partly sponsored by the ANR Bip:Bip project under contract ANR-10-BINF-0003, and completed this work during a visiting term at IMECC, University of Campinas (SP), Brazil, sponsored by the Chaires Françaises dans l’état de São Paulo (CFSP) program, a collaboration between the French Consulate in São Paulo, and the three main universities in São Paulo: UNICAMP, USP, and UNESP.

Author information

Authors and Affiliations

  1. CNRS LIX, École Polytechnique, 91128, Palaiseau, France

    Luca Mencarelli, Youcef Sahraoui & Leo Liberti

  2. OSIRIS, EDF R&D, 92141, Clamart, France

    Youcef Sahraoui

Authors
  1. Luca Mencarelli
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  2. Youcef Sahraoui
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  3. Leo Liberti
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Corresponding author

Correspondence to Leo Liberti.

Appendices

Appendix 1: Parameter values for the HUC instances

See Table 8.

Table 8 Inflows \(I_h (\mathrm {m}^3/\mathrm {s})\) and prices \(\Pi _h\) (currency\(\mathrm {/MWh}\)) for time periods \(h\in \{1,\ldots ,\bar{h}=164 \}\) for the 3 provided instances
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Appendix 2: Detailed results for the MVPS problem

Tables 9, 10, 11, 12 and 13 report the numeric results for each transaction cost function. Their columns are as follows:

Table 9 Comparative results of MS and MWU for the transaction cost function (a)
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Table 10 Comparative results of MS and MWU for the transaction cost function (b)
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Table 11 Comparative results of MS and MWU for the transaction cost function (c)
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Table 12 Comparative results of MS and MWU for the transaction cost function (d)
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Table 13 Comparative results of MS and MWU for the transaction cost function (e)
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  • instance name;

  • maximum risk level \(\sigma \);

  • number n of assets quoted on the financial market;

  • objective value for the MWU algorithm;

  • CPU time (in seconds) for the MWU algorithm;

  • objective value for the MWU algorithm with the local branching constraint;

  • CPU time (in seconds) for the MWU algorithm with the local branching constraint;

  • objective value for the MS algorithm with the local branching constraint;

  • CPU time (in seconds) for the MS algorithm with the local branching constraint;

  • relative objective value improvement from MS to MWU computed as

    $$\begin{aligned} \Gamma = \frac{{\textsf {val}}({\textsf {MWU}})-{\textsf {val}}({\textsf {MS}})}{|{\textsf {val}}({\textsf {MS}})|}; \end{aligned}$$
    (7.1)

  • time improvement ratio \(\Lambda \) from MS to MWU (see Eq. (4.2));

  • relative objective value improvement from MS to MWU with the local branching constraint (see Eq. (7.1));

  • time improvement ratio \(\Lambda \) from MS to MWU with the local branching constraint (see Eq. (4.2)).

The comparison metrics are summarized in the last three lines with the sum (\(\sum \)), average (avg), and the standard deviation (std) across all 20 instances.

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Mencarelli, L., Sahraoui, Y. & Liberti, L. A multiplicative weights update algorithm for MINLP. EURO J Comput Optim 5, 31–86 (2017). https://doi.org/10.1007/s13675-016-0069-8

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  • DOI: https://doi.org/10.1007/s13675-016-0069-8

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