# On the Use of Biased-Randomized Algorithms for Solving Non-Smooth Optimization Problems

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Non-Smooth Optimization Problems

## 3. Basic Concepts on Biased-Randomized Algorithms

Algorithm 1: Biased-Randomized Algorithm (BRA; basic sequential version). |

## 4. Applications in Logistics

## 5. Applications in Transportation

## 6. Applications in Scheduling

## 7. General Insights from Previous Numerical Experiments

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Simpson, N.C.; Hancock, P.G. Practical Operations Management; Hercher: Naperville, IL, USA, 2013. [Google Scholar]
- Papadimitriou, C.H.; Steiglitz, K. Combinatorial Optimization: Algorithms and Complexity; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 1982. [Google Scholar]
- Garey, M.R.; Johnson, D.S. Computers and Intractability; A Guide to the Theory of NP-Completeness; W. H. Freeman & Co.: New York, NY, USA, 1990. [Google Scholar]
- Khamaru, K.; Wainwright, M.J. Convergence guarantees for a class of non-convex and non-smooth optimization problems. J. Mach. Learn. Res.
**2019**, 20, 1–52. [Google Scholar] - Bagirov, A.M.; Yearwood, J. A new nonsmooth optimization algorithm for minimum sum-of-squares clustering problems. Eur. J. Oper. Res.
**2006**, 170, 578–596. [Google Scholar] [CrossRef] - Bagirov, A.; Lai, D.T.H.; Palaniswami, M. A nonsmooth optimization approach to sensor network localization. In Proceedings of the 3rd International Conference on Intelligent Sensors, Sensor Networks and Information, Melbourne, Australia, 3–6 December 2007; pp. 727–732. [Google Scholar]
- Roy, P.; Ghoshal, S.; Thakur, S. Biogeography based optimization for multi-constraint optimal power flow with emission and non-smooth cost function. Expert Syst. Appl.
**2010**, 37, 8221–8228. [Google Scholar] [CrossRef] - Lu, Y.; Zhou, J.; Qin, H.; Li, Y.; Zhang, Y. An adaptive hybrid differential evolution algorithm for dynamic economic dispatch with valve-point effects. Expert Syst. Appl.
**2010**, 37, 4842–4849. [Google Scholar] [CrossRef] - Hashimoto, H.; Ibaraki, T.; Imahori, S.; Yagiura, M. The vehicle routing problem with flexible time windows and traveling times. Discret. Appl. Math.
**2006**, 154, 2271–2290. [Google Scholar] [CrossRef] [Green Version] - Ferone, D.; Gruler, A.; Festa, P.; Juan, A.A. Enhancing and extending the classical GRASP framework with biased randomisation and simulation. J. Oper. Res. Soc.
**2019**, 70, 1362–1375. [Google Scholar] [CrossRef] - Faulin, J.; Gilibert, M.; Juan, A.A.; Vilajosana, X.; Ruiz, R. SR-1: A simulation-based algorithm for the capacitated vehicle routing problem. In Proceedings of the 2008 Winter Simulation Conference, Miami, FL, USA, 7–10 December 2008; pp. 2708–2716. [Google Scholar]
- Juan, A.A.; Faulin, J.; Ruiz, R.; Barrios, B.; Gilibert, M.; Vilajosana, X. Using oriented random search to provide a set of alternative solutions to the capacitated vehicle routing problem. In Operations Research and Cyber-Infrastructure; Springer: Berlin/Heidelberg, Germany, 2009; pp. 331–345. [Google Scholar]
- Domínguez Rivero, O.L.; Juan Pérez, A.A.; De La Nuez Pestana, I.A.; Ouelhadj, D. An ILS-biased randomization algorithm for the two-dimensional loading HFVRP with sequential loading and items rotation. J. Oper. Res. Soc.
**2016**, 67, 37–53. [Google Scholar] [CrossRef] [Green Version] - Quintero-Araujo, C.L.; Gruler, A.; Juan, A.A.; Faulin, J. Using horizontal cooperation concepts in integrated routing and facility-location decisions. Int. Trans. Oper. Res.
**2019**, 26, 551–576. [Google Scholar] [CrossRef] - Martin, S.; Ouelhadj, D.; Beullens, P.; Ozcan, E.; Juan, A.A.; Burke, E.K. A multi-agent based cooperative approach to scheduling and routing. Eur. J. Oper. Res.
**2016**, 254, 169–178. [Google Scholar] [CrossRef] [Green Version] - Quintero-Araujo, C.L.; Caballero-Villalobos, J.P.; Juan, A.A.; Montoya-Torres, J.R. A biased-randomized metaheuristic for the capacitated location routing problem. Int. Trans. Oper. Res.
**2017**, 24, 1079–1098. [Google Scholar] [CrossRef] - Belloso, J.; Juan, A.A.; Martinez, E.; Faulin, J. A biased-randomized metaheuristic for the vehicle routing problem with clustered and mixed backhauls. Networks
**2017**, 69, 241–255. [Google Scholar] [CrossRef] - Belloso, J.; Juan, A.A.; Faulin, J. An iterative biased-randomized heuristic for the fleet size and mix vehicle-routing problem with backhauls. Int. Trans. Oper. Res.
**2019**, 26, 289–301. [Google Scholar] [CrossRef] [Green Version] - Estrada-Moreno, A.; Savelsbergh, M.; Juan, A.A.; Panadero, J. Biased-randomized iterated local search for a multiperiod vehicle routing problem with price discounts for delivery flexibility. Int. Trans. Oper. Res.
**2019**, 26, 1293–1314. [Google Scholar] [CrossRef] - Calvet, L.; Ferrer, A.; Gomes, M.I.; Juan, A.A.; Masip, D. Combining statistical learning with metaheuristics for the multi-depot vehicle routing problem with market segmentation. Comput. Ind. Eng.
**2016**, 94, 93–104. [Google Scholar] [CrossRef] [Green Version] - Brandão, J.S.; Noronha, T.F.; Resende, M.G.C.; Ribeiro, C.C. A biased random-key genetic algorithm for single-round divisible load scheduling. Int. Trans. Oper. Res.
**2015**, 22, 823–839. [Google Scholar] [CrossRef] - Gonzalez-Neira, E.M.; Ferone, D.; Hatami, S.; Juan, A.A. A biased-randomized simheuristic for the distributed assembly permutation flowshop problem with stochastic processing times. Simul. Model. Pract. Theory
**2017**, 79, 23–36. [Google Scholar] [CrossRef] - Brandão, J.S.; Noronha, T.F.; Resende, M.G.C.; Ribeiro, C.C. A biased random-key genetic algorithm for scheduling heterogeneous multi-round systems. Int. Trans. Oper. Res.
**2017**, 24, 1061–1077. [Google Scholar] [CrossRef] - Gonçalves, J.F.; Resende, M.G.C.; Costa, M.D. A biased random-key genetic algorithm for the minimization of open stacks problem. Int. Trans. Oper. Res.
**2016**, 23, 25–46. [Google Scholar] [CrossRef] [Green Version] - Fikar, C.; Juan, A.A.; Martinez, E.; Hirsch, P. A discrete-event driven metaheuristic for dynamic home service routing with synchronised trip sharing. Eur. J. Ind. Eng.
**2016**, 10, 323–340. [Google Scholar] [CrossRef] - Gruler, A.; Fikar, C.; Juan, A.A.; Hirsch, P.; Contreras-Bolton, C. Supporting multi-depot and stochastic waste collection management in clustered urban areas via simulation–optimization. J. Simul.
**2017**, 11, 11–19. [Google Scholar] [CrossRef] - Pinto, B.Q.; Ribeiro, C.C.; Rosseti, I.; Plastino, A. A biased random-key genetic algorithm for the maximum quasi-clique problem. Eur. J. Oper. Res.
**2018**, 271, 849–865. [Google Scholar] [CrossRef] - Boyd, S.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Bagirov, A.M.; Taheri, S.; Ugon, J. Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems. Pattern Recognit.
**2016**, 53, 12–24. [Google Scholar] [CrossRef] - Karmitsa, N.; Bagirov, A.M.; Taheri, S. Clustering in large data sets with the limited memory bundle method. Pattern Recognit.
**2018**, 83, 245–259. [Google Scholar] [CrossRef] - Bagirov, A.; Taheri, S.; Asadi, S. A difference of convex optimization algorithm for piecewise linear regression. J. Ind. Manag. Optim.
**2019**, 15, 909–932. [Google Scholar] [CrossRef] [Green Version] - Sayah, S.; Zehar, K. Modified differential evolution algorithm for optimal power flow with non-smooth cost functions. Energy Convers. Manag.
**2008**, 49, 3036–3042. [Google Scholar] [CrossRef] - Al-Sultan, K.S. A Tabu search approach to the clustering problem. Pattern Recognit.
**1995**, 28, 1443–1451. [Google Scholar] [CrossRef] - Oonsivilai, A.; Srisuruk, W.; Marungsri, B.; Kulworawanichpong, T. Tabu Search Approach to Solve Routing Issues in Communication Networks. Int. J. Electr. Comput. Energ. Electron. Commun. Eng.
**2009**, 3, 1211–1214. [Google Scholar] - Hemamalini, S.; Simon, S.P. Artificial Bee Colony Algorithm for Economic Load Dispatch Problem with Non-smooth Cost Functions. Electr. Power Components Syst.
**2010**, 38, 786–803. [Google Scholar] [CrossRef] - Niknam, T.; Mojarrad, H.D.; Meymand, H.Z.; Firouzi, B.B. A new honey bee mating optimization algorithm for non-smooth economic dispatch. Energy
**2011**, 36, 896–908. [Google Scholar] [CrossRef] - Basu, M. Modified Particle Swarm Optimization for Non-smooth Non-convex Combined Heat and Power Economic Dispatch. Electr. Power Components Syst.
**2015**, 43, 2146–2155. [Google Scholar] [CrossRef] - Schlüter, M.; Egea, J.A.; Banga, J.R. Extended ant colony optimization for non-convex mixed integer nonlinear programming. Comput. Oper. Res.
**2009**, 36, 2217–2229. [Google Scholar] [CrossRef] [Green Version] - Corazza, M.; Fasano, G.; Gusso, R. Particle Swarm Optimization with non-smooth penalty reformulation, for a complex portfolio selection problem. Appl. Math. Comput.
**2013**, 224, 611–624. [Google Scholar] [CrossRef] [Green Version] - Clarke, G.; Wright, J. Scheduling of Vehicles from a Central Depot to a Number of Delivery Points. Oper. Res.
**1964**, 12, 568–581. [Google Scholar] [CrossRef] - Bellmore, M.; Nemhauser, G.L. The traveling salesman problem: A survey. Oper. Res.
**1968**, 16, 538–558. [Google Scholar] [CrossRef] - Panwalkar, S.S.; Iskander, W. A survey of scheduling rules. Oper. Res.
**1977**, 25, 45–61. [Google Scholar] [CrossRef] - Juan, A.A.; Faulin, J.; Ferrer, A.; Lourenço, H.R.; Barrios, B. MIRHA: Multi-start biased randomization of heuristics with adaptive local search for solving non-smooth routing problems. Top
**2013**, 21, 109–132. [Google Scholar] [CrossRef] - Resende, M.G.; Ribeiro, C.C. Greedy randomized adaptive search procedures: Advances, hybridizations, and applications. In Handbook of Metaheuristics; Springer: Berlin/Heidelberg, Germany, 2010; pp. 283–319. [Google Scholar]
- Estrada-Moreno, A.; Fikar, C.; Juan, A.A.; Hirsch, P. A biased-randomized algorithm for redistribution of perishable food inventories in supermarket chains. Int. Trans. Oper. Res.
**2019**, 26, 2077–2095. [Google Scholar] [CrossRef] - Ferone, D.; Hatami, S.; González-Neira, E.M.; Juan, A.A.; Festa, P. A biased-randomized iterated local search for the distributed assembly permutation flow-shop problem. Int. Trans. Oper. Res.
**2019**. [Google Scholar] [CrossRef] - Mazza, D.; Pages-Bernaus, A.; Tarchi, D.; Juan, A.A.; Corazza, G.E. Supporting mobile cloud computing in smart cities via randomized algorithms. IEEE Syst. J.
**2016**, 12, 1598–1609. [Google Scholar] [CrossRef] - Melo, M.T.; Nickel, S.; Saldanha-Da-Gama, F. Facility location and supply chain management–A review. Eur. J. Oper. Res.
**2009**, 196, 401–412. [Google Scholar] [CrossRef] - Ahmadi-Javid, A.; Seyedi, P.; Syam, S.S. A survey of healthcare facility location. Comput. Oper. Res.
**2017**, 79, 223–263. [Google Scholar] [CrossRef] - De Armas, J.; Juan, A.A.; Marquès, J.M.; Pedroso, J.P. Solving the deterministic and stochastic uncapacitated facility location problem: From a heuristic to a simheuristic. J. Oper. Res. Soc.
**2017**, 68, 1161–1176. [Google Scholar] [CrossRef] - Correia, I.; Melo, T. Multi-period capacitated facility location under delayed demand satisfaction. Eur. J. Oper. Res.
**2016**, 255, 729–746. [Google Scholar] [CrossRef] [Green Version] - Estrada-Moreno, A.; Ferrer, A.; Juan, A.A.; Bagirov, A.; Panadero, J. A biased-randomised algorithm for the capacitated facility location problem with soft constraints. J. Oper. Res. Soc.
**2019**, 1–17. [Google Scholar] [CrossRef] - Cordeau, J.F.; Laporte, G.; Savelsbergh, M.W.; Vigo, D. Vehicle routing. Handbooks in Operations Research and Management Science; Elsevier: Amsterdam, The Netherlands, 2007; Volume 14, pp. 367–428. [Google Scholar]
- Adewumi, A.O.; Adeleke, O.J. A survey of recent advances in vehicle routing problems. Int. J. Syst. Assur. Eng. Manag.
**2018**, 9, 155–172. [Google Scholar] [CrossRef] - Corberan, A.; Prins, C. Recent results on Arc Routing Problems: An annotated bibliography. Networks
**2010**, 56, 50–69. [Google Scholar] [CrossRef] - Corberan, A.; Laporte, G. Arc Routing: Problems, Methods, and Applications; SIAM: Philadelphia, PA, USA, 2013. [Google Scholar]
- De Armas, J.; Ferrer, A.; Juan, A.A.; Lalla-Ruiz, E. Modeling and solving the non-smooth arc routing problem with realistic soft constraints. Expert Syst. Appl.
**2018**, 98, 205–220. [Google Scholar] [CrossRef] - Gonzalez, S.; Juan, A.A.; Riera, D.; Castella, Q.; Munoz, R.; Perez, A. Development and assessment of the SHARP and RandSHARP algorithms for the arc routing problem. AI Commun.
**2012**, 25, 173–189. [Google Scholar] [CrossRef] - Pinedo, M.L. Scheduling: Theory, Algorithms, and Systems, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Ferrer, A.; Guimarans, D.; Ramalhinho, H.; Juan, A.A. A BRILS metaheuristic for non-smooth flow-shop problems with failure-risk costs. Expert Syst. Appl.
**2016**, 44, 177–186. [Google Scholar] [CrossRef] [Green Version] - Ferrer, A.; Bagirov, A.; Beliakov, G. Solving DC programs using the cutting angle method. J. Glob. Optim.
**2015**, 61, 71–89. [Google Scholar] [CrossRef]

**Figure 4.**Percentage gaps between BRAs and reference values in non-smooth optimization problems (OPs).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Juan, A.A.; Corlu, C.G.; Tordecilla, R.D.; de la Torre, R.; Ferrer, A.
On the Use of Biased-Randomized Algorithms for Solving Non-Smooth Optimization Problems. *Algorithms* **2020**, *13*, 8.
https://doi.org/10.3390/a13010008

**AMA Style**

Juan AA, Corlu CG, Tordecilla RD, de la Torre R, Ferrer A.
On the Use of Biased-Randomized Algorithms for Solving Non-Smooth Optimization Problems. *Algorithms*. 2020; 13(1):8.
https://doi.org/10.3390/a13010008

**Chicago/Turabian Style**

Juan, Angel Alejandro, Canan Gunes Corlu, Rafael David Tordecilla, Rocio de la Torre, and Albert Ferrer.
2020. "On the Use of Biased-Randomized Algorithms for Solving Non-Smooth Optimization Problems" *Algorithms* 13, no. 1: 8.
https://doi.org/10.3390/a13010008