# Progressive Hedging and Sample Average Approximation for the Two-stage Stochastic Traveling Salesman Problem

### Pablo Adasme, Janny Leung, Ismael Soto

#### Abstract

In this paper, we propose an adapted version of the progressive hedging algorithm (PHA) (Rockafellar and Wets, 1991; Lokketangen and Woodruff, 1996; Watson and Woodruff, 2011) for the two-stage stochastic traveling salesman problem (STSP) introduced in (Adasme et al., 2016). Thus, we compute feasible solutions for small, medium and large size instances of the problem. Additionally, we compare the PHA method with the sample average approximation (SAA) method for all the randomly generated instances and compute statistical lower and upper bounds. For this purpose, we use the compact polynomial formulation extended from (Miller et al., 1960) in (Adasme et al., 2016) as it is the one that allows us to solve large size instances of the problem in short CPU time with CPLEX. Our preliminary numerical results show that the results obtained with the PHA algorithm are tight when compared to the optimal solutions of small and medium size instances. Moreover, we obtain significantly better feasible solutions than CPLEX for large size instances with up to 100 nodes and 10 scenarios in significantly low CPU time. Finally, the bounds obtained with SAA method provide an average reference interval for the stochastic problem.

#### References

- Adasme, P., Andrade, R., Letournel, M., and Lisser, A. (2013). A polynomial formulation for the stochastic maximum weight forest problem. ENDM, 41:29-36.
- Adasme, P., Andrade, R., Letournel, M., and Lisser, A. (2015). Stochastic maximum weight forest problem. Networks, 65(4):289-305.
- Adasme, P., Andrade, R., Leung, J., and Lisser, A. (2016). A two-stage stochastic programming approach for the traveling salesman problem. ICORES-2016.
- Ahmed, S. and Shapiro, A. (2002). The sample average approximation method for stochastic programs with integer recourse. Georgia Institute of Technology.
- Bertazzi, L. and Maggioni, F. (2014). Solution approaches for the stochastic capacitated traveling salesmen location problem with recourse. J Optim Theory Appl, 166(1):321-342.
- Bertsimas, D., Brown, D., and Caramanis, C. (2011). Theory and applications of robust optimization. SIAM Reviews, 53:464-501.
- Escoffier, B., Gourves, L., Monnot, J., and Spanjaard, O. (2010). Two-stage stochastic matching and spanning tree problems: Polynomial instances and approximation. Eur J Oper Res, 205:19-30.
- Flaxman, A. D., Frieze, A., and Krivelevich, M. (2006). On the random 2-stage minimum spanning tree. Random Struct Algor, 28:24-36.
- Gaivoronski, A., Lisser, A., Lopez, R., and Xu, H. (2011). Knapsack problem with probability constraints. J Global Optim, 49:397-413.
- Lokketangen, A. and Woodruff, D. L. (1996). Progressive hedging and tabu search applied to mixed integer (0- 1) multi stage stochastic programming. Journal of Heuristics, 2(2):111-128.
- Maggioni, F., Perboli, G., and Tadei, R. (2014). The multipath traveling salesman problem with stochastic travel costs: a city logistics computational study. Transportation Research Procedia, 1(3):528-536.
- Miller, C. E., Tucker, A. W., and Zemlin, R. A. (1960). Integer programming formulations and travelling salesman problems. J. Assoc. Comput. Mach., 7:326-329.
- Rockafellar, R. T. and Wets, R. J. B. (1991). Scenarios and policy aggregation in optimization under uncertainty. Mathematics and Operations Research, 16:119-147.
- Shapiro, A., Dentcheva, D., and Ruszczynski, A. (2009). Lectures on stochastic programming: Modeling and theory. MOS-SIAM Series on Optimization, Philadelphia.
- Watson, J. P. and Woodruff, D. L. (2011). Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems. Computational Management Science, 8:355-370.

#### Paper Citation

#### in Harvard Style

Adasme P., Leung J. and Soto I. (2017). **Progressive Hedging and Sample Average Approximation for the Two-stage Stochastic Traveling Salesman Problem** . In *Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-218-9, pages 440-446. DOI: 10.5220/0006241304400446

#### in Bibtex Style

@conference{icores17,

author={Pablo Adasme and Janny Leung and Ismael Soto},

title={Progressive Hedging and Sample Average Approximation for the Two-stage Stochastic Traveling Salesman Problem},

booktitle={Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2017},

pages={440-446},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006241304400446},

isbn={978-989-758-218-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - Progressive Hedging and Sample Average Approximation for the Two-stage Stochastic Traveling Salesman Problem

SN - 978-989-758-218-9

AU - Adasme P.

AU - Leung J.

AU - Soto I.

PY - 2017

SP - 440

EP - 446

DO - 10.5220/0006241304400446