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An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

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Abstract

In this work, we develop an adaptive, multivariate partitioning algorithm for solving nonconvex, Mixed-Integer Nonlinear Programs (MINLPs) with polynomial functions to global optimality. In particular, we present an iterative algorithm that exploits piecewise, convex relaxation approaches via disjunctive formulations to solve MINLPs that is different than conventional spatial branch-and-bound approaches. The algorithm partitions the domains of variables in an adaptive and non-uniform manner at every iteration to focus on productive areas of the search space. Furthermore, domain reduction techniques based on sequential, optimization-based bound-tightening and piecewise relaxation techniques, as a part of a presolve step, are integrated into the main algorithm. Finally, we demonstrate the effectiveness of the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare our algorithm with state-of-the-art global optimization solvers. With our novel approach, we solve several large-scale instances, some of which are not solvable by state-of-the-art solvers. We also succeed in reducing the best known optimality gap for a hard, generalized pooling problem instance.

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Notes

  1. In the case of a higher order univariate monomial, i.e., \(x_i^5\), apply a reduction of the form \(x_i^2x_i^2x_i \Rightarrow \tilde{x}_i^2x_i \Rightarrow \tilde{\tilde{x}}_ix_i\).

  2. Global optimum is defined numerically by a tolerance, \(\epsilon \).

  3. To keep the algorithm notation simple, this detail is omitted from Algorithm 2.

  4. Exhaustiveness of the partitioning scheme implies AMP will eventually partition all other domains small enough such that AMP will pick an active partition with the global optimal whose length is \(\le \epsilon ^l_i+\epsilon ^u_i\).

  5. See [8] for more details on strategies for choosing the variables for partitioning.

  6. meyer15 is a generalized pooling problem instance. These problems are typically considered hard (bilinear) MINLP for global optimization [9, 36].

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Acknowledgements

The work was funded by the Center for Nonlinear Studies (CNLS) at LANL and the LANL’s directed research and development project “POD: A Polyhedral Outer-approximation, Dynamic-discretization optimization solver”. Work was carried out under the auspices of the U.S. DOE under Contract No. DE-AC52-06NA25396.

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  1. Center for Nonlinear Studies, Los Alamos National Laboratory, New Mexico, USA

    Harsha Nagarajan, Russell Bent & Kaarthik Sundar

  2. Department of Industrial Engineering, Clemson University, Clemson, USA

    Mowen Lu & Site Wang

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A: Appendix

A: Appendix

1.1 A.1: Sensitivity analysis of \(\Delta \)

One of the important details of MINLP algorithms and approaches is their parameterization. As seen in the earlier sections, AMP is no different. The quality of the solutions depend heavily on the choice of \(\Delta \). However, in spite of this problem specific dependence, it is often interesting to identify reasonable default values. Table 8 presents computational results on all instances for different choices of \(\Delta \). From these results, AMP is most effective when \(\Delta \) is between 4 and 10.

Table 8 This table shows a sensitivity analysis of AMP’s performance to the choice of \(\Delta \)
Full size table

1.2 A.2: Logarithmic and linear encodings of partition variables

In Sect. 2, the discussion on piecewise convex relaxations described formulations that encoded the partition variables with a linear number of variables and a logarithmic number of variables [52]. Table 9 compares the performance of AMP using both formulations. Despite fewer variables in the logarithmic formulation, this encoding is only effective on a few problems, generally on problems that require a significant number of partitions. These results suggest that when the logarithmic encoding has nearly the same number of partition variables as the linear encoding, the linear encoding is more effective.

Table 9 This table compares the logarithmic formulation of partition variables with the linear representation
Full size table

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Nagarajan, H., Lu, M., Wang, S. et al. An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs. J Glob Optim 74, 639–675 (2019). https://doi.org/10.1007/s10898-018-00734-1

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