Modeling Uncertainty in Linear Programs: Stochastic and Robust Programming DGA PhD Student - PhD Thesis EDF-INRIA 10 November 2011
and motivations In real life, Linear Programs are uncertain for several reasons: Estimation errors: parameters calculated from statistic estimations. Measure errors: for measured parameters. A posteriori known parameters. (prices, demands,... ) Execution errors: limited precision to execute an optimal solution. = Goal: Predict or limit the influence of these noises. Integrate these uncertainty in the decisional process
Decisional context Several decisional contexts are possible: Static context: all the uncertainty is raised in the same time. 2 stages: before and after knowing the realization Dynamic context: decisions have to be reactualized when a random event is realized. We can decide a first solution value, and we modify it after the realization to make it feasible or to improve the objective cost. First decision must be taken: In order to be fitted to most of the cases, minimizing an average cost: Stochastic approach. To be covered against the worst realization: Robust approach.
Example: scheduling nuclear plan outages d i,k,w : decisions on the dates of the outages, decisional values. x i,k,w : decisions on the production, to fulfill the demand after decisions of outages are taken.
Deterministic linear problem min i,k C i,k reload d i,k,w + w C w flexp ( DEM w i,k PMAX i x i,k,w ) i, k, w, d i,k,w d i,k,w 1 (1) i, k, w, x i,k,w d i,k,w DAi,k d i,k+1,w (2) i, k, Lmin i,k w x i,k,w Lmax i,k (3) w, 0 DEM w i,k PMAX i x i,k,w Pflex w (4) c, w [ID c, IF c ], i A c (d i,k,w Lc,i d i,k,w Lc,i TU c,i ) Q c (5) d i,k,w : decisions on the dates of the outages, decisional values. x i,k,w : decisions on the production.
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General Formulation A general formulation for these problems are: s.t : min c T [ x + E ξ miny q(ξ) T y(ξ) ] Ax = b T (ξ)x + W (ξ)y(ξ) = h(ξ) x 0, y(ξ) 0, Datas A, b, c are deterministic, while q(ξ), T (ξ), W (ξ) depends on random events ξ. First stage decisions, here and now variables x, independant of ξ. Secund stage decisions, wait and see recourse variables y(ξ), dependant of ξ. (6)
Special case: discretization of scenarios For discrete stochasticity and we can enumerate the set of scenarios S, this leads to a big Linear Program: min x,y s c T x + s S π s q T s y s s.t : Ax = b T s x + W s y s = h s, x 0, y s 0, s S s S (7)
Stochastic formulation min i,k C i,k reload d i,k,w + π s C w,s flexp P w,s w,s i, k, w, d i,k,w d i,k,w 1 (8) i, k, w, s, x i,k,w,s d i,k,w DAi,k,s d i,k+1,w (9) i, k, s, Lmin i,k,s w x i,k,w,s Lmax i,k,s (10) c, w [ID c, IF c ], w, s 0 P w,s Pflex w,s (11) w, s, PMAX i,s x i,k,w,s + P w,s = DEM w,s (12) i,k i A c (d i,k,w Lc,i,s d i,k,w Lc,i,s TU c,i,s ) Q c (13)
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Notations
Basic ideas Q(x) is convex. We approximate Q(x) with cutting planes of its convex hull: it gives lower bounds. We generate cutting planes in a Benders decomposition scheme. Master problem with the cutting plane approximation, slave problem to generate feasibility and optimality cuts.
General Benders decomposition scheme
Integer L-shaped method
Integer L-shaped decomposition scheme
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Dynamic case: non anticipativity constraints Let two scenarios s 1 and s 2 equal on the first t 0 time steps. Decisions x i,k,t,s1 and x i,k,t,s2 could be different for t < t 0 in the last static approach. How could we have decided this at time t, when it was impossible to make the difference between these two scenarios? It is anticipative. Non anticipativity constraints: x i,k,t,s1 = x i,k,t,s2 for t t 0. = It is equivalent to consider a scenario tree to group these decisions. Decisions are indexed on nodes of this tree.
Scenario trees
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General Formulation These problems have the following dynamic structure: min x0,...,x N c 0.x 0 + E x1 [c 1.x 1 + E x2 [ + E xn [c n.x n ]]] s.t : Ax 0 b T i.x i 1 + W i.x i h i, i N x i 0 (14) N : set of nodes of the scenario tree (the root is 0) x 0 : decision before every realization. x i decision before knowing node i but assuming i 1.
General Formulation for discrete scenarios These problems have the following structure: s.t : min x0,...,x N c 0.x 0 + i N π ic i.x i Ax 0 b T i.x i 1 + W i.x i h i, i N x i 0 N : set of nodes of the scenario tree (the root is 0) x 0 : decision before every realization. x i decision before knowing node i but assuming i 1. (15)
Typical examples Finance portfolio problems for purchase/sales strategies. Electricity short term production problems: mixing between nuclear/thermal/hydraulic/renewable energies/spot market (cf Mahlke PhD thesis). Can be applied for our problem to model dynamicity, but too many recourses and variables. Difficulty because decision on a whole planning, not only on a ponctual decision at time steps. = The result is not a single solution, but a strategy, set of solution depending on the random realization.
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What is robustness? Classical definitions for robust solutions: Solution with margins? Adding slacks. ( light robustness ) Solution resistant to approximations or ignorance zones to avoid bad impacts. Solution which do not change in a significant way when the vector of decision is slightly perturbed. Solution acceptable in a large number of scenarios and which is never too bad. = The decisional context has to precise the definition.
Uncertainty sets Definition: predefinite set of random events which we want to be covered from. Classical uncertainty sets: Cartesian product of intervalls Polyhedrons Ellipsoids
Uncertainty sets for our example We assume that outage prolongations are independant random events. The uncertainty of the outage (i, k) prolongation is in [[0, DA i,k ]]. Ω worstcase = [[0, DA i,k ]] i,k Ω Γ budget = (da i,k) [[0, DA i,k ]] da i,k Γ i,k i,k We can decompose prolongations into nominal prolongation in [0, DA 0 i,k], extreme prolongations in [[1 + DA 0 i,k, DA i,k ]]: Ω N ext = (da i,k ) (ε i,k ) {0, 1}, i, k, da i,k DA 0 i,k + (DA i,k DA 0 i,k )ε i,k et ε i,k N i,k
Robust framework Considering a linear program with uncertainty ω Ω: s.t : We consider following class or problems s.t : min d,x c D (ω)d + c X (ω).x Ad B T (ω)d + W (ω)x h(ω) d, x 0 min d,x max ω Ω f (d, x, ω) Ad B T (ω)d + W (ω)x h(ω) d, x 0 where f (d, x, ω) is an objective to define, various robust criteria. (16) (17)
Game theory interpretation Min-max problem We are the first player, we play on the decisions x, d. Randomness is the opponent, it plays on the random realization ω to penalize the decision x, d. Our goal is to anticipate the worst case for us, the best choice of the opponent, after our choice on x, d.
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Easiest case: uncertainty does not influence feasibility s.t : min x C(ω)x Ax = b x 0 (18)
Worst case objective Minimization problem of the worst case which can happen, f (X, ω) = C(ω).X : s.t : Very conservative approach. min x max ω Ω C(ω)x Ax = b x 0 (19)
Maximal regret Minimization problem of the gap between the robust solution and the determistic solution assuming an alea. f (X, ω) = C(ω).X C(ω).X (ω). s.t : min x max ω Ω min x (ω) c(ω)x c(ω)x (ω) Ax = b Ax(ω) = b, ω X, X (ω) 0 Advantage: favorite objective for risk averse managers. Drawback: This leads to difficult problems. (Cutting planes algorithm of Shimizu and Aiyoshi) (20)
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Soyster s approach: column uncertainty Every column A j K j, with K j convex in R m s.t : The robust problem is equivalent to: min x cx j A j.x j b x 0 (21) with A i,j = max Aj K j A i,j s.t : min x cx j A j.x j b x 0 (22)
Soyster s approach with interval uncertainty If K j is an interval, it is equivalent to: s.t : min x cx j A i,j.x j b i, i X 0 (23) = Worst case conservative approach. = There is often a too big gap between these conservative solutions and nominal solution. = This lead to parametric approaches: we do not want anymore to be covered against ALL the risks in the uncertainty set, but having a parametrization to ponderate this gap.
Parametric approaches Idea: extreme scenario for all random event are unusual. Ben Tal and Nimerovski approach: maximal deviations on a constraint i is in an ellipsoid defined with par : Ξ i (Ω i ) = {ξ i,j [1, 1] n ξij 2 Ω i}. Bertsimas and Sim approach: Γ i for a constraint i is the total deviation to nominal values in constraint i. Uncertainty set is Φ i (Γ i ) is thus: j Φ i (Γ i ) = {ξ i,j [1, 1] n j ξ ij Γ i }. It leads to linear programs. Γ i = n is Soyster s approach.
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A special case of column uncertainty s.t : min x c.x n j=1 A j.x j B.x n+1 0 x n+1 = 1 x 0 (24) Last general approaches are used for this particuler cases.
Is is possible to use the special structure? Not a lot of papers on this field. The dual problem of an uncertain RHS Linear Program is a Linear Program with costs uncertainty. Remli PhD thesis: results for inequality constraints. Equality constraints: needs to a penalty model. Minoux: results on PERT scheduling with uncertain RHS.
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General framework Analogously to the first stage and second stage decision of Stochastic Programming. s.c : min d max ω Ω min x(ω) c D (ω).d + c X (ω).x(ω) Ad b T (ω).d + W (ω).x(ω) h(ω) d, x(ω) {0, 1} (25) In our game theory illustration, we can play second stage decision after the opponent played the stochastic event. Secund stage variable are recourse variables.
Related works Minoux, Thiele: several studies on 2 stage robust programming with RHS uncertainty Remli PhD thesis: application for 2 stage decision localisation-transportation problem. Atamturk and Zhang: Two-Stage Robust Network Flow and Design Under Demand Uncertainty. Bertsimas and Litvinov: Security Constrained Unit Commitment My future PhD work on scheduling nuclear maintenance outages?
First example min max min C i,k d i,k,w α Ω x reload d i,k,w + i,k,w,α i,k w C w flexp DEM w PMAX i x i,k,w,α i,k i, k, w, d i,k,w d i,k,w 1 (26) i, k, w, α, x i,k,w,α d i,k,w DAi,k,α d i,k+1,w (27) i, k,, α Lmin i,k w x i,k,w,α Lmax i,k (28) w, α, 0 DEM w PMAX i x i,k,w,α Pflex w (29) i,k c, α, w [ID c, IF c ], (d i,k,w Lc,i d i,k,w Lc,i TU ) Q c,i,α c (30) i Ac
Linearization min d i,k,w,c i,k C i,k reload d i,k,w + C α Ω, i, k, w, d i,k,w 1 d i,k,w (31) C w flexp DEM w PMAX i x i,k,w,α C (32) w i,k i, k, w, α, x i,k,w,α d i,k,w DAi,k,α d i,k+1,w (33) i, k,, α Lmin i,k w x i,k,w,α Lmax i,k (34) w, α, 0 DEM w PMAX i x i,k,w,α Pflex w (35) i,k c, α, w [ID c, IF c ], (d i,k,w Lc,i d i,k,w Lc,i TU ) Q c,i,α c (36) i Ac
Adding recourse to the first stage decisions min max min C i,k d i,k,w α Ω x,c, reload d i,k,w + i,k w C w flexp ( DEM w ) PMAX i x i,k,w,α + Corg i,k c i,k,αi,k i,k i,k i, k, w, d i,k,w d i,k,w 1 (37) i, k, w, i,k,w,αi,k i,k,w 1,αi,k (38) i, k, w, α Ω O, i,k,w,α = d i,k,w (39) i, k, w, w, i,k,w,αi,k d i,k,w d i,k,w (40) i, k, w, w, d i,k,w i,k,w,αi,k d i,k,w (41) i, k, w, c i,k,αi,k d i,k,w i,k,w,αi,k (42) i, k, w, c i,k,αi,k i,k,w,αi,k d i,k,w (43) i, k, w, α Ω, x i,k,w,α d i,k,w DAi,k,α d i,k+1,w (44) i, k, α Ω Lmin i,k w x i,k,w,α Lmax i,k (45) w, α, 0 DEM w PMAX i x i,k,w,α Pflex w (46) i,k c, α, w [ID c, IF c ], (d i,k,w Lc,i d i,k,w Lc,i TU ) Q c,i,α c (47) i Ac
: Requires probabilistic distribution of scenarios: unknown in general applications. Decision on the average cost: less conservative solutions. Big size of Linear Programs. Generic resolution with Bender s decomposition : Requires no probabilistic distribution of randomness: worst case approach. Importance of the choice of uncertainty sets. Different approach following the uncertain coefficient: objective / matrix /RHS. Min-max problems.
Bibliography N. Remli, Robustesse en programmation linéaire, Phd. thesis, Université Paris-Dauphine, 2011. M. Minoux, Duality, Robustness, and 2-stage robust LP decision models. Application to Robust PERT Scheduling. 2007 P. Kall, S. W. Wallace. Stochastic Programming, John Wiley & Sons, Chichester, 1994 S. Ben Salem, Gestion robuste de la production électrique à horizon court-terme, Phd. thesis, Ecole Centrale Paris, 2011