Decomposition for adjustable robust linear optimization subject to uncertainty polytope
Tóm tắt
We present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to polytope uncertainty. Our approach is based on replacing the polytope by the set of its extreme points and generating the extreme points on the fly within row generation or column-and-row generation algorithms. The novelty of our approach lies in formulating the separation problem as a feasibility problem instead of a max–min problem as done in recent works. Applying the Farkas lemma, we can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. We compare the two algorithms on a robust telecommunications network design under demand uncertainty and budgeted uncertainty polytope. Our results show that the relative performance of the algorithms depend on whether the budget is integer or fractional.
Tài liệu tham khảo
Audet C, Hansen P, Jaumard B, Savard G (1997) Links between linear bilevel and mixed 0–1 programming problems. J Optim Theory Appl 93:273–300
Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805
Ben-Tal A, Nemirovski A (2002) Robust optimization methodology and applications. Math Program 92:453–480
Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99(2):351–376
Bertsimas D, Caramanis C (2010) Finite adaptability in multistage linear optimization. IEEE Trans Autom Control 55(12):2751–2766
Bertsimas D, Dunning I (2014) Multistage robust mixed integer optimization with adaptive partitions. Under review
Bertsimas D, Georghiou A (2015) Design of near optimal decision rules in multistage adaptive mixed-integer optimization. Oper Res 63(3):610–627
Bertsimas D, Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive optimization. Math Program 134(2):491–531
Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52:35–53
Bertsimas D, Goyal V, Sun XA (2011a) A geometric characterization of the power of finite adaptability in multistage stochastic and adaptive optimization. Math Oper Res 36(1):24–54
Bertsimas D, Iancu D, Parrilo P (2011b) A hierarchy of near-optimal policies for multistage adaptive optimization. IEEE Trans Autom Control 56(12):2809–2824
Bertsimas D, Litvinov E, Sun XA, Zhao J, Zheng T (2013) Adaptive robust optimization for the security constrained unit commitment problem. IEEE Trans Power Syst 28(1):52–63
Bienstock D, Özbay N (2008) Computing robust basestock levels. Discret Optim 5(2):389–414
Billionnet A, Costa M-C, Poirion P-L (2014) 2-stage robust MILP with continuous recourse variables. Discrete Appl Math 170:21–32
Chen X, Zhang Y (2009) Uncertain linear programs: extended affinely adjustable robust counterparts. Oper Res 57(6):1469–1482
Chen X, Sim M, Sun P, Zhang J (2008) A linear decision-based approximation approach to stochastic programming. Oper Res 56(2):344–357
Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2):201–213
El Ghaoui L, Oustry F, Lebret H (1998) Robust solutions to uncertain semidefinite programs. SIAM J Optim 9(1):33–52
Gabrel V, Lacroix M, Murat C, Remli N (2014) Robust location transportation problems under uncertain demands. Discrete Appl Math 164(1):100–111
Goh J, Sim M (2010) Distributionally robust optimization and its tractable approximations. Oper Res 58(4-Part-1):902–917
Iancu D, Sharma M, Sviridenko M (2013) Supermodularity and affine policies in dynamic robust optimization. Oper Res 61(4):941–956
IBM-ILOG (2015) IBM-ILOG Cplex
Kouvelis P, Yu G (1997) Robust discrete optimization and its application. Kluwer Academic Publishers, London
Matsui T (1996) Np-hardness of linear multiplicative programming and related problems. J Global Optim 9(2):113–119
Mattia S (2013) The robust network loading problem with dynamic routing. Comp Opt Appl 54(3):619–643
Mattia S (2014) Private communication
Orlowski S, Pióro M, Tomaszewski A, Wessäly R (2010) SNDlib 1.0-survivable network design library. Networks 55(3):276–286
Poss M (2014) A comparison of routing sets for robust network design. Optim Let 8(5):1619–1635
Poss M, Raack C (2013) Affine recourse for the robust network design problem: between static and dynamic routing. Networks 61(2):180–198
Postek K, Den Hertog D (2014) Multi-stage adjustable robust mixed-integer optimization via iterative splitting of the uncertainty set. CentER Discussion Paper Series
Ryoo HS, Sahinidis NV (2003) Global optimization of multiplicative programs. J Global Optim 26(4):387–418
Zeng B, Zhao L (2013) Solving two-stage robust optimization problems by a constraint-and-column generation method. Oper Res Let 41(5):457461