MIXED INTEGER STOCHASTIC OPTIMIZATION: ALGORITHMS AND APPLICATIONS 1
Departamento de Matem´atica Aplicada, Estad´ıstica e Investigaci´on Operativa, UPV/EHU 2 Departamento de Econom´ıa Aplicada III (Estad´ıstica y Econometr´ıa), UPV/EHU 3 Departamento de Matem´atica Aplicada, UPV/EHU 4 Departamento de Estad´ıstica e Investigaci´on Operativa, URJC
Gloria P´erez1
Araceli Gar´ın2
Mar´ıa Merino1
Larraitz Aranburu1
Aitziber Unzueta3
Unai Aldasoro3
Laureano Escudero4
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
Stochastic Sets
Cluster Benders Decomposition
• T : set of stages • Ω: set of scenarios • G: set of scenario groups • C: set of cluster scenario submodels Break Stage Scenario Clustering The Deterministic Equivalent Model (DEM) can be decomposed into the |C| = |Gt∗+1| cluster scenario submodels (1) (M IP c) z c =
T X
Cluster Lagrangean Decomposition
wtc(actxct + cctytc)
t=1 st A1xc1 + B1y1c = b1 (1) ′ ′ c c c c c + Bcyc = bc 2 ≤ t ≤ t∗ + 1 At xt−1 + At xt + Btcyt−1 t t t c c c + [B ]cyc = bc t > t∗ + 1 [A′t] xct−1 + [At]cxct + [B′t] yt−1 t t t c c c nx c +ny t t xt ∈ {0, 1} , yt ∈ R .t∈T,
Traditional Benders decomposition requires appending feasibility and/or optimality cuts to the master problem until the iterative procedure reaches the optimal solution. The cuts are identified by solving the auxiliary scenarios submodels. Cluster Benders Decomposition (CBD) based scheme allows to identify tighter feasibility and optimality cuts that yield feasible and optimal second stage decisions in reasonable computing time. Based on the two-stage scheme given in [2] and in the break stage concept, our purpose is to extend this algorithmic scheme to the multistage case.
t=1
t∗ = 2
t∗ + 1 = 3
t=4
8 4
T ∗1 W 1
9
• Subgradient Method (SM)
2 10 5 11
• Progressive Hedging Algorithm (PHA)
T ∗3 W 3
12
• Dynamic Constrained Cutting Plane scheme (DCCP)
6 13 3
T ∗4 W 4
14
7
stoprog.org
www.euro-online.org
The plain use of CPLEX does not provide the optimal solution of the problem in affordable computing time. More than three hours of computing time, are required in the following case.
15
|C| = 4 clusters zMIP ∈ [−66.1153, −66.0315] zSM [ite] TSM z[31] = −66.0478 935 sec zV A[ite] TV A z[48] = −66.0478 1282 sec zP HA[ite] TP HA z[51] = −66.0478 1468 sec zDCCP [ite] TDCCP z[18] = −66.0478 491 sec
which are linked by the nonanticipativity constraints, ′ ∗ ∗ c c ′ ∗ t t xt − xt = 0, c 6= c , t ≤ t , g ∈ Gt, t = CT (c, g) = CT (c′, g) ′ ∗ ∗ c c ′ ∗ t t yt − yt = 0, c 6= c , t ≤ t , g ∈ Gt, t = CT (c, g) = CT (c′, g)
• Volume Algorithm (VA)
T ∗2 W 2 1
A
Cluster based Lagrangean Decomposition (CLD) scheme is presented in [7] for obtaining strong bounds, and in several of the cases the optimal solution, of two-stage stochastic mixed 0-1 problems. We experiment with several Lagrange multiplier updating schemes, such as:
• |Ω| = 200 scenarios • 2010 constraints • 4020 0-1 variables • 6000 continuous variables • 120400 non-zero elements • 1.48% matrix density
www.optimizacion bajoincertidumbre.org
Nonsymmetric BFC-MS
Risk Averse Measures
Acknowledgements
The Branch-and-Fix Coordination algorithm for large-scale Multistage Stochastic mixed-integer optimization problems under nonsymmetric scenario trees (BFC-MS) is described in [4, 5, 6].
In [1] several strategies for modelling risk environment have been analysed:
1. Eopt, Grupo de Investigaci´on en Estad´ıstica y Optimizaci´on (IT-347-10) from the Basque Country Government
1. Scenario immunization strategy
2. UFI 11/46 BETS 2011-14, University of Basque Country.
3000
2. Maximization of the well known Value-at-Risk and Conditional Value-at-Risk (CVaR)
www.coin-or.org
2000 1500 1000 500
www.ilog.com/ products/cplex
3. Maximization of the mean-risk, i.e., the expected objective minus the weighted probability of having a bad scenario occurring for the given solution provided by the model
0
Elapsed time (in secs.)
2500
BFC−MS CPLEX
P10
P5
P1
P2
P11
P12
P3
P4
P13
P6
P7
P8
P9
P14
Instances (increasing order of binary variables)
Natural Gas Application Co-operation agreement UPV/EHU-URJC-NTNU, 2010-12 Adaptation and design of the implementation of the BFC-MS algorithm for a Natural Gas network infrastructure model
System of production, distribution, processing and transportation of natural gas with a highly volatile product demand, see [6, 11]. • |T | = 4 years
4. Maximization of the objective function expected value subject to first- and second-order stochastic dominance constraints (SDC) for a set of profiles given by the pairs of threshold objective values and the probability of not reaching them 5. Maximization of the mixture of the CVaR & SDC strategies: P max p∈P γ pV p s.t. cx + q ω y ω + M ν ωp ≥ V p ∀ω ∈ Ω, p ∈ P b1 ≤ Ax ≤ b2 ω ≤ T ω x + W ω y ω ≤ hω ∀ω ∈ Ω hX 1 2 (2) ω ωp p w ν ≤ α ∀p ∈ P ω∈Ω x, y ω ≥ 0 ∀ω ∈ Ω V p ≥ φp p ∈ P ν ωp ∈ {0, 1} ∀ω ∈ Ω, p ∈ P
• |Ω| = 1000 scenarios • |G| = 1111 scenario groups • 98456 constraints
Financial Application
• 34680 0-1 variables • 22221 continuous variables • 253340 non-zero elements • 0.0045% matrix density
Nonsymmetric BFC-MS CPLEX XPRESS |C| nodes T N F zDEM TDEM TDEM TDEM 10 10 0 78995.9 182 sec — — —: Out of memory (12Gb)
An approach for stochastic modelization of different immunization strategies in fixed-income security portfolios under uncertainty is shown in [1]. Transaction costs and risk of default have been considered in the previous financial model.
3. ECO2008-00777 ECON from the Spanish Ministry of Education and Science 4. PLANIN MTM2009-14087-C04-01 from the Spanish Ministry of Science and Innovation. 5. Co-operation agreement UPV/EHU-URJC-NTNU, 2010-12. 6. Laboratory of Quantitative Economics from the University of the Basque Country (UPV/EHU), Fernando Tusell. 7. Computational Cluster ARINA, provided by SGI/IZO-SGIker MICINN,GV/EJ, ERDF and ESF) Computing Service at the UPV/EHU
(UPV/EHU,
References
[1] L. Aranburu. Stochastic modeling and solution schemes for immunization strategies. PhD thesis, Universidad del Pas Vasco, UPV/EHU, 2011. [2] L. Aranburu, L.F. Escudero, A. Gar´ın and G. P´erez. A so-called Cluster Benders Decomposition approach for solving two stage stochastic linear problems. TOP, DOI: 10.1007/ s11750-011-0242-4, 2012. [3] J. Birge and F. Louveaux. Introduction to Stochastic Programming. Springer, 1997. [4] Escudero LF, Gar´ın A, Merino M, P´erez G. On BFC-MSMIP strategies for scenario cluster partitioning, and twin node family branching selection and bounding for multistage stochastic mixed integer programming. Computers & Op. Research 2010; 37:738-753. [5] L.F. Escudero, A. Gar´ın, M. Merino, G. P´erez. An exact algorithm for solving large-scale two-stage SMIP: some theoretical and experimental aspects. EJOR, 204:105-116, 2010. [6] L.F. Escudero, A. Gar´ın, M. Merino and G. P´erez. An algorithmic framework for solving large-scale multistage stochastic mixed 0-1 problems with nonsymmetric scenario trees. Computers & Op. Research, 39:1133–1144, 2012. [7] L F. Escudero, A. Gar´ın, G. P´erez, A. Unzueta. Lagrangian decomposition for large-scale two-stage stochastic mixed 0-1 problems. TOP, DOI: 10.1007/s11750-011-0237-1, 2012. [8] XPRESS www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-Suite.aspx [9] G. P´erez and A. Gar´ın. On downloading and using COIN-OR for solving linear/integer optimization problems. Biltoki DT 2010.05. Basque Country University, 2010. [10] G. P´erez and A. Gar´ın. On downloading and using CPLEX within COIN-OR for solving lineal/integer optimization problems. Biltoki DT 2011.08. Basque Country Univ., 2011. [11] F. Rømo, A. Tomasgard, L. Hellemo, M. Fodstad, B. H. Eidesen, N. Pedersen. Optimizing the Norwegian Natural Gas Production and Transport. Interfaces 39:46-56, 2009.
III JORNADAS DE INVESTIGACION DE LA FACULTAD DE CIENCIA Y TECNOLOGIA (2012) ZIENTZIA ETA TEKNOLOGIA FAKULTATEKO III. IKERKETA JARDUNALDIAK (2012)
