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SO-I: a surrogate model algorithm for expensive nonlinear integer programming problems including global optimization applications

Abstract

This paper presents the surrogate model based algorithm SO-I for solving purely integer optimization problems that have computationally expensive black-box objective functions and that may have computationally expensive constraints. The algorithm was developed for solving global optimization problems, meaning that the relaxed optimization problems have many local optima. However, the method is also shown to perform well on many local optimization problems, and problems with linear objective functions. The performance of SO-I, a genetic algorithm, Nonsmooth Optimization by Mesh Adaptive Direct Search (NOMAD), SO-MI (Müller et al. in Comput Oper Res 40(5):1383–1400, 2013), variable neighborhood search, and a version of SO-I that only uses a local search has been compared on 17 test problems from the literature, and on eight realizations of two application problems. One application problem relates to hydropower generation, and the other one to throughput maximization. The numerical results show that SO-I finds good solutions most efficiently. Moreover, as opposed to SO-MI, SO-I is able to find feasible points by employing a first optimization phase that aims at minimizing a constraint violation function. A feasible user-supplied point is not necessary.

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Fig. 1
Fig. 2

Notes

  1. 1.

    http://www.i2c2.aut.ac.nz/Wiki/OPTI/.

  2. 2.

    http://www.iim.csic.es/~gingproc/meigo.html.

  3. 3.

    Note that SO-I is used in order to find a feasible solution from which both algorithms could start. Thus, the number of failed trials for SO-I, SO-MI, and VNS-ii are identical.

    Table 2 Constrained problems
    Table 3 Unconstrained problems
    Table 4 Application problems
  4. 4.

    The Matlab codes provided on http://www.mcs.anl.gov/~more/dfo/ have been used for creating the plots.

Abbreviations

GA:

Genetic algorithm

NOMAD:

Nonsmooth Optimization by Mesh Adaptive Direct Search

RBF:

Radial basis function

SEM:

Standard error of means

SO-I:

Surrogate Optimization-Integer

SO-MI:

Surrogate Optimization-Mixed Integer

local-SO-I:

local-Surrogate Optimization-Integer

VNS:

Variable neighborhood search

\(\mathbb R \) :

Real numbers

\(\mathbb Z \) :

Integer numbers

\(\mathbf u \) :

Discrete decision variable vector, see Eq. (1d)

\(f(\cdot )\) :

Objective function, see Eq. (1a)

\(c_j(\cdot )\) :

\(j\)th constraint function, \(j=1,\ldots , m\), see Eq. (1b)

\(m\) :

Number of constraints

\(k\) :

Problem dimension

\(j\) :

Index for the constraints

\(i\) :

Index for the variables

\(u_i^l,\,u_i^u\) :

Lower and upper bounds for the \(i\)th variable, see Eq. (1c)

\(\varOmega _b\) :

Box-constrained variable domain

\(\varOmega \) :

Feasible variable domain

\(\mathcal S \) :

Set of already evaluated points

\(n_0\) :

Number of points in initial experimental design

\(q(\cdot )\) :

Auxiliary function for minimizing constraint violation in phase 1, see Eq. (5)

\(f_p(\cdot )\) :

Objective function value augmented with penalty term, see Eq. (6)

\(f_\text {max}\) :

Objective function value of the worst feasible point found so far, see Eq. (6)

\(p\) :

Penalty factor, see Eq. (6)

\(v(\cdot )\) :

Squared constraint violation function, see Eq. (7)

\({\varvec{\chi }}_\jmath \) :

\(\jmath \)th candidate point for next sample site, \(\jmath =1,\ldots , t\)

\(n\) :

Number of already sampled points

\(s(\cdot )\) :

Radial basis function interpolant

\(V(\cdot )\) :

Weighted score, see Eq. (8)

\(\omega _R, \omega _D\) :

Weights for response surface and distance criteria, respectively

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Acknowledgments

The project has been supported by NSF CISE: AF 1116298 and CPA-CSA-T 0811729. The first author thanks the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation for the financial support. The authors thank the anonymous reviewers for their helpful comments and suggestions.

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Correspondence to Juliane Müller.

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Müller, J., Shoemaker, C.A. & Piché, R. SO-I: a surrogate model algorithm for expensive nonlinear integer programming problems including global optimization applications. J Glob Optim 59, 865–889 (2014). https://doi.org/10.1007/s10898-013-0101-y

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Keywords

  • Integer optimization
  • Derivative-free
  • Computationally expensive
  • Surrogate model
  • Response surface
  • Linear and nonlinear
  • Nonconvex
  • Radial basis functions
  • Multimodal
  • Global optimization