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Scalable branching on dual decomposition of stochastic mixed-integer programming problems

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Abstract

We present a scalable branching method for the dual decomposition of stochastic mixed-integer programming. Our new branching method is based on the branching method proposed by Carøe and Schultz that creates branching disjunctions on first-stage variables only. We propose improvements to the process for creating branching disjunctions, including (1) branching on the optimal solutions of the Dantzig–Wolfe reformulation of the restricted master problem and (2) using a more comprehensive (yet simple) measure for the dispersions associated with subproblem solution infeasibility. We prove that the proposed branching process leads to an algorithm that terminates finitely, and we provide conditions under which globally optimal solutions can be identified after termination. We have implemented our new branching method, as well as the Carøe–Schultz method and a branch-and-price method, in the open-source software package DSP. Using SIPLIB test instances, we present extensive numerical results to demonstrate that the proposed branching method significantly reduces the number of node subproblems and solution times.

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL, 60439, USA

    Kibaek Kim & Brian Dandurand

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  1. Kibaek Kim
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Correspondence to Kibaek Kim.

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Code availability statement

The software code and experiment scripts are available to reproduce the results published in the manuscript at https://doi.org/10.5281/zenodo.1470838.

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The algorithms developed in the paper are also available in the most recent version of the software package DSP available in https://github.com/Argonne-National-Laboratory/DSP.

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Appendix

Appendix

We report the numerical results from the different branching methods for solving smkp instances. This set of instances is challenging for the dual decomposition because the instances have a larger number of constraints and variables in the first stage than those in the second stage. This makes each iteration of the dual decomposition take a significant amount of time.

To circumvent the issue, we set a 300-s time limit for each scenario subproblem solution and use a (possibly suboptimal) feasible solution to generate the inequalities (5b) for the Lagrangian dual problem. We emphasize that the cuts generated at a suboptimal feasible solution are valid. Moreover, we use 4-h time limit for each instance (Table 5).

Table 5 Computational results for smkp instances by using different branching methods
Full size table

We report the numerical results without any heuristic in Table 6. The BNP and CS+DW methods found the global optimal solutions at the root node for all the instances, because the primal solutions \((\hat{x}_j,\hat{y}_j)\) obtained from the root node subproblem were integer feasible. Note that the differences in total solution time were due mainly to the wall-clock time limit of 300 s for subproblem solutions. For the given time limit, CPLEX may return slightly different solutions even for the same problem. On the other hand, the CS method was not able to find any feasible solution (and thus an upper bound) for all the instances. However, we found that the lower bounds obtained by CS method are the same as the optimal objective values for all the instances. The reason is that the average point \(\bar{x}\) computed by the CS method did not represent valid primal solutions for the instances, which required solving more node subproblems for finding a primal feasible solution. This observation is consistent with results with the CS method for sslp instances, as shown in Table 2.

Table 6 Computational results for smkp instances by using CS branching methods with the fixing-first heuristic
Full size table

Table 6 reports the numerical results from using the CS method with the fixing-first heuristic. Note that we do not report the results for the other methods because the global optimal solutions were already obtained at the root node without any heuristic. With the heuristic, the CS method found feasible solutions for 12 instances, of which 8 instances were optimal. However, the CS method still failed to find a feasible solution for the other 8 instances within the 4-h time limit.

Since smkp instances have binary variables only, the simple rounding heuristic does nothing but fixing the average point for checking the feasibility. As a result, the numerical results are same as those without any heuristic and thus not reported.

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Kim, K., Dandurand, B. Scalable branching on dual decomposition of stochastic mixed-integer programming problems. Math. Prog. Comp. 14, 1–41 (2022). https://doi.org/10.1007/s12532-021-00212-y

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