Stochastic nuclear outages semidefinite relaxations
Tóm tắt
This paper deals with stochastic scheduling of nuclear power plant outages. Focusing on the main constraints of the problem, we propose a stochastic formulation with a discrete distribution for random variables, that leads to a mixed 0/1 quadratically constrained quadratic program. Then we investigate semidefinite relaxations for solving this hard problem. Numerical results on several instances of the problem show the efficiency of this approach, i.e., the gap between the optimal solution and the continuous relaxation is on average equal to 53.35 % whereas the semidefinite relaxation yields an average gap of 2.76 %. A feasible solution is then obtained with a randomized rounding procedure.
Tài liệu tham khảo
Adasme P, Lisser A, Soto I (2012) A quadratic semidefinite relaxation approach for OFDMA resource allocation. Networks 59(1): 3–12
Benson SJ, Ye Y, Zhang X (2000) Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J Optim 10: 443–461
Borchers B (1999) CSDP, a C library for semidefinite programming. Optim Methods Softw 11: 613–623
Boyd S, Vandenberghe L (1996) Semidefinite programming. SIAM Rev 38: 49–95
Calafiore G, Campi MC (2004) Decision making in an uncertain environment: the scenario-based optimization approach. In: Andrysek J, Karny M Kracik J (eds) Multiple participant decision making. Advanced Knowledge International, pp 99–111
Fortet R (1960) Applications de l’algebre de boole en recherche opérationnelle. Revue Francaise de Recherche Operationnelle 4: 17–26
Fourcade F, Eve T, Socroun T (1997) Improving Lagrangian relaxation: an application to the scheduling of pressurized water reactor outages. IEEE Trans Power Syst 12(2): 919–925
Fourcade F, Johnson E, Bara M, Cortey-Dumont P (1997) Optimizing nuclear power plant refueling with mixed-integer programming. Eur J Oper Res 97: 269–280
Gaivoronski A, Lisser A, Lopez R (2011) Knapsack problem with probability constraints. J Glob Optim 49(3): 397–413
Goemans MX (1998) Semidefinite programming and combinatorial optimization. Documenta Mathematica Extra Volume ICM 1998 3: 657–666
Goemans MX, Williamson DP (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J ACM 1115–1145
Gorge A, Lisser A, Zorgati R (2012) Semidefinite Relaxations for the scheduling nuclear outages problem. In: Accepted in the 1st international conference on operations research and enterprise systems (ICORES) proceedings
Helmberg C, Rendl F (2000) A spectral bundle method for semidefinite programming. SIAM J Optim 10: 673–696
Henrion R (2004) Introduction to chance constraint programming. Tutorial paper for the stochastic programming community HomePage. http://www.wias-berlin.de/people/henrion/publikat.html
Helmberg C, Rendl F (1998) Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math Program 82: 291–315
Khemmoudj MOI, Porcheron M, Bennaceur H (2006) When constraint programming and local search solve the scheduling problem of electricité de France nuclear power plant outages. In: 12th international conference on principles and practice of constraint programming (CP), Springer LNCS 4204, Nantes, France, pp 271–283
Kosuch S, Lisser A (2011) On two-stage stochastic knapsack problems with or without probability constraint. Discrete Appl Math 159(16): 1827–1841
Laurent M, Rendl F (2005) Semidefinite programming and integer programming. In: Aardal K, Nemhauser G, Weismantel R (eds) Handbook on discrete optimization. Elsevier, Amsterdam, pp 393–514
Lisser A, Lopez R (2010) Application of semidefinite relaxation and VNS for multiuser detection in synchronous CDMA. Networks 5: 187–193
Lovász L (1979) On the Shannon capacity of a graph. IEEE Trans Inf Theory 25: 1–7
Lovász L, Schrijver A (1991) Cones of matrices and set-functions and 0-1 optimization. SIAM J Optim 1: 166–190
Nemirovski A, Shapiro A (2004) Scenario approximations of chance constraints. In: Calafiore, Campi (eds) Probabilistic and randomized methods for design under uncertainty, vol 45. Springer, London, http://www.optimization-online.org
Nesterov Y, Nemirovski A (1994) Interior point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics, Philadelphia
Porcheron M, Gorge A, Juan O, Simovic T, Dereu G (2009) Challenge ROADEF/EURO 2010: a large-scale energy management problem with varied constraints. EDF R&D report. http://challenge.roadef.org
Prekopa A (1995) Stochastic programming. Kluwer, Dordrecht, p 34
Todd MJ (2001) Semidefinite optimization. Acta Numer 10: 515–560
Wolkowicz H, Saigal R, Vandenberghe L (2000) Handbook of semidefinite programming: theory, algorithms and applications. Kluwer Academic Publishers, Dordrecht, The Netherlands