On parallelization of a stochastic dynamic programming algorithm for solving large-scale mixed 0–1 problems under uncertainty

Abstract

A parallel computing implementation of a Serial Stochastic Dynamic Programming approach referred to as the S-SDP algorithm is introduced to solve large-scale multiperiod mixed 0–1 optimization problems under uncertainty. The paper presents Inner and Outer Parallelization versions of the S-SDP algorithm, referred to as Inner P-SDP and Outer P-SDP, respectively, so that the problem solving elapsed time and gap reduction is analyzed. The basic idea of Inner P-SDP consists of parallelizing the optimization of variations of the MIP subproblems attached to the sets of scenario clusters created by the modeler-defined stages to decompose the original problem. The Outer P-SDP performs simultaneous interconnected executions of the serial algorithm, so that a wider feasibility area is explored using iterative communication to redefine search directions. Strategies are presented to analyze the performance of parallel computation based on Message-Passing Interface threads to solve stage-related subproblems versus the serial version of SDP methodology. The results of using the parallelization are remarkable, as not only faster but also better solutions than the serial version are obtained. In particular, we report up to 10 times speedup for 12 threads on the Inner P-SDP algorithm. The new approach allows problems to be solved using less computing time than a state-of-the-art MIP solver. It can thus solve very large-scale problems that could not otherwise be achieved by plain use of the solver or by the S-SDP algorithm in acceptable elapsed time, if any.

Appendices

Appendix A. Implementation details

This appendix details the serial, inner and outer parallel implementation schemes that have been used for the computational experience reported in Sect. 5.

1.1 A.1 Serial implementation overview

The S-SDP algorithm presented in Sect. 3.5 has been implemented for solving the realistic production planning problem under uncertainty introduced in Cristobal et al. (2009). As it frequently happens in tactical multistage planning problems, in our case only continuous variables in a period have nonzero coefficients in the constraints of the next one. Therefore, the linking variables between stages only occur between the leaf nodes $\{\ell \}$ of the subproblems of a stage, say $r'$, and the root nodes of the successor subproblems in the next stage, say $r\in \mathcal{S}_\ell$, if any. (In our case $\mathcal{V}_\ell =\{y_g,~g\in \tilde{\mathcal{A}}_\ell \}$ for $\ell \in \mathcal{L}_{r'},~ r'\in \mathcal{R}^{e-1},e\in \mathcal{E}\backslash \{1\}$, such that $(y_g)_i$ denotes the stock of product $i$ in set $\mathcal{I}$ related to scenario group $g$). Therefore, those continuous variables are the only ones to be iteratively perturbed. The perturbation has been performed as follows: $\xi =ran/f(iter)$, where $f(iter)=k$ being $k$ a constant. Given the relatively small number of iterations performed and the large scale of the instances, the algorithm cannot ensure the goodness of a unique searching direction; consequently, a constant factor $k$ has been chosen to preserve a wide search. However, the value of parameter $k$ changes depending on the value of the variable to be perturbed, such that small perturbations for small values and big perturbation for big values are generated, looking for a relative homogeneous change. Additionally, a single new reference level is generated at the BtF scheme. We observed in our computational experimentation with the type of problem we are considering that multiple perturbations of the same solution will lead to similar reference levels that, then, significantly increase the elapsed time. Finally, all reference levels are kept active.

Throughout the numerical experiments that are reported, the stopping criteria parameters are set up to $\epsilon _1=0.001$ and $\epsilon _2=0.0001$, $niterk=miter+1$, $miter=15$ iterations, time limit of 8 hours and 35 Gb of memory limit.

1.2 A.2 Inner parallel implementation overview

The Inner P-SDP maintains the algorithmic structure of the S-SDP algorithm; i.e., model, linking variables perturbation, reference level management and stopping criteria parameters. Note that if the time limit is reached at a S-SDP execution, the Inner P-SDP execution can be allowed continue to iterate and stop afterwards, see below.

As only the variables in a period have nonzero coefficients in the constraints of the following one, the implemented Inner P-SDP algorithm comprehends asynchronized execution parts. Thus, stage $e=1$ will be managed by $th 1$, stage $e=2$ by main threads and subsequent stages by both main and auxiliary threads. Therefore, primary communication will be needed between stages $1$ and $2$; and secondary communication will be performed between consecutive pair of stages from $2$ to $|\mathcal{E}|$ whenever needed. Global synchronization is achieved when gathering the EFV curves at the end of FtB and BtF schemes. The procedure of the Inner P-SDP follows the structure of the S-SDP algorithm detailed in Sect. 3.5. It is as follows:

Step 0: (Initialization)

Step 1: (FtB scheme: solve subproblem (5) for stage ${e=1}$)

• The subproblem is solved by main thread $th 1$.

• Primary communication:  The main threads gather the solution from $th 1$. They then follow an asynchronized execution until the end of the FtB scheme, since no primary communication is needed in-between. Set stage $e:=2$.

Step 2: (Solve subproblem (5) rooted at node ${r}$, ${\forall r\in \mathcal{R}^e}$)

• Each thread solves its corresponding subproblems rooted at node $r\in \mathcal{R}_{th}^{FtB}$ and all the available threads solve their own subproblems simultaneously. Note that for stage $e>2$ all available threads will be used, otherwise only the main threads will.

Step 3: (Generate and append the EFV-$\ell$ defining constraint to subproblem (5) rooted at node ${r'}$, ${\forall \ell \in \mathcal{L}_{r'}, \, r'\in \mathcal{R}^{e-1}}$)

• Secondary communication:  Performed only for stage $e=2$. Auxiliary threads gather the solution from their corresponding main thread so that they can follow an asynchronized execution until the end of the FtB scheme.

Step 4: (Forward to next stage ${e+1}$)

• Synchronization:    If stage $e=|\mathcal{E}|$ all threads gather solution and EFV curves.

Step 5: (Compute solution value for original model (2) and check stopping criteria)

Step 6: (BtF scheme: Compute dual vector of subproblem (5) rooted at node ${r,~\forall r\in \mathcal{R}^e, e>1}$)

• The subproblem rooted at $r\in \mathcal{R}_{\mathrm{th}}^{\mathrm{BtF}}$ is solved by its corresponding thread. Finally, if stage $e=2$ then the results of solving each subproblem rooted at $r\in \mathcal{R}_{\mathrm{th}}^{\mathrm{BtF}}$ by a main thread will be shared with its corresponding auxiliary threads.

• Secondary communication:  Each main thread and its auxiliary threads gather solution and follow a synchronized execution among them but their execution is asynchronized with respect to other main and auxiliary thread groups.

Step 7: (Generate and append the EFV-$\ell$ defining constraints in subproblem (5) rooted at node ${r'}$, ${\forall \ell \in \mathcal{L}_{r'}, \, r'\in \mathcal{R}^{e-1}}$)

• Synchronization:    In case stage $e=1$ all threads gather solution and EFV curves.

Step 8: (Backward to previous stage ${e-1}$)

1.3 A.3 Outer parallel implementation overview

Let $inisize$, $inigap$, $nthread$, $pathsize$ and $pathgap$ be pilot case-driven parameters taken as input for the execution of the outer parallelization. The global solution pool generation phase stores up to $inisize$ alternative solutions, choosing them from subproblem (5) solving at stage $e=1$ for $iter=1$ with the smallest optimality gap among those whose gap is lower than $inigap$. Then, the $nthread$ most diverge solutions among them are chosen from the pool and assigned to threads. The selection criterion is as follows: the candidate solution in the pool with the highest euclidean distance of the vector of the values of the linking variables (at stage $e=1$) with respect to the optimal solution is the first candidate; then the solution with the highest distance with respect to the optimal plus the distance with respect to the first candidate is the second candidate, and so on.

The alternative solution path selection phase (at stage $e=1$ for $iter>1$) stores up to $pathsize$ candidates with an optimality gap lower than $pathgap$ and takes the one with the highest above euclidean distance from the optimal solution for stage $e=1$.

The stopping criteria parameters are set up to the same values used at the S-SDP and Inner P-SDP. The procedure of the implemented Outer P-SDP is as follows:

Step 0: (Initialization)

Step 1: (FtB scheme: Solve subproblem (5) for stage ${e=1}$)

• Generate the global solution pool for iteration $iter=1$ from the set of optimal and quasi-optimal solutions from solving the subproblem as described above.

• For $iter>1$ and those paths that have issued a warning message on “non-improved path solution” at the end of the FtB scheme (Step 5) of the previous iteration, an alternative solution is picked up from the pool, according to the criterion presented above.

Step 2: (Solve subproblem (5) rooted at node ${r}$, ${\forall r\in \mathcal{R}^e,~e\ge 2}$)

Step 3: (Generate and append the EFV-$\ell$ defining constraint to subproblem (5) rooted at node ${r'}$, ${\forall \ell \in \mathcal{L}_{r'}, \, r'\in \mathcal{R}^{e-1}}$)

Step 4: (Forward to next stage ${e+1}$)

Step 5: (Compute solution value for original model (2) and check stopping criteria)

• The communication and synchronization phase is executed, see Sect. 4.3 and Fig. 5. Compute global iteration solution and check stopping criteria. If global incumbent solution has been updated, all threads gather the corresponding solution; if global incumbent solution and path incumbent solution have not been updated, the warning message “non-improved path solution” is issued for the appropriate path.

Step 6: (BtF scheme: Compute dual vector of subproblem (5) rooted at node ${r,~ r\in \mathcal{R}^e, e>1}$)

• Path reference levels: Generate new reference levels using the path criteria for random values.

Step 7: (Generate and append the EFV-$\ell$ defining constraints in subproblem (5) rooted at node ${r'}$, ${\forall \ell \in \mathcal{L}_{r'}, \, r'\in \mathcal{R}^{e-1}}$)

Step 8: (Backward to previous stage ${e-1}$)

Appendix B. Inner P-SDP quality analysis of the largest instances in Testbed 2

This appendix details the Inner parallelization analysis for the largest cases, c85 and c86, when using an increasing number (up to 8) of multiprocessor computers of the cluster.

Table 10 and Fig. 6 show the results in terms of elapsed time, speedup and efficiency for instances c85 and c86 when using 12, 24, 48 and 96 threads for Inner P-SDP versus S-SDP (execution using only one thread for CPLEX). The new headings are as follows: $S_{\mathrm{th}}^{\mathrm{top}}$, top speedup defined as $S_{\mathrm{th}}^{\mathrm{top}}=\frac{t_{\mathrm{IP}}^{\mathrm{serial}}}{t_{\mathrm{th}}^{\mathrm{top}}}$ and $E_{\mathrm{th}}^{\mathrm{top}}\%$, top efficiency defined as $E_{\mathrm{th}}^{\mathrm{top}}\%=100 \cdot \frac{S_{th}^{top}}{th}$, where $t_{th}^{top}=\sum _{e \in \mathcal{E}}\frac{t_e^{serial}}{\min (th,|\mathcal{R}^e| |\mathcal {Z}|)}$ and $t_e^{serial}$ is the elapsed time for stage $e$ in S-SDP such that $t_{IP}^{serial}=\sum _{e\in \mathcal{E}} t_e^{serial}$. The concepts $S_{th}^{top}$ and $E_{th}^{top}$ consider the top speedup and top efficiency that could be achieved, respectively, given the specifications of the SDP algorithm and its Inner P-SDP parallelization under ideal conditions (as if time is not lost by communication and synchronization). For example, in instance c85 $t_{IP}^{serial}=26{,}180=59+10+8,\!987+17{,}125=t_1+t_2+t_3+t_4$ and $\{\mathcal{R}^e\}=\{1, 3, 27, 486\}$; therefore, with 48 threads, $t_{48}^{top}=\frac{59}{1}+\frac{10}{3}+\frac{8{,}987}{27}+\frac{17{,}125}{48}=752$, $S_{48}^{top}=\frac{26{,}180}{752}=34.81$ and $E_{48}^{top}=100\cdot \frac{34.81}{48}=72.53~\%$.

Fig. 6

Speedup and efficiency for instances c85 and c86

Table 10 Inner P-SDP speedup and efficiency for instances c85 and c86

Note in Table 10 and Figure 6 the remarkable scalability of the Inner P-SDP algorithm. The speedup increases almost linearly with up to 48 threads and the elapsed time with 96 threads is 44 and 34 times faster than for the serial version in instances c85 and c86, respectively. When comparing the efficiency and the top efficiency, we can observe that time lost by communication and synchronization is not significant in the implementation of Inner P-SDP for instance c85 and is quite small for instance c86.