Optimization of a Liquefied Natural Gas Portfolio by SDDP techniques

Size: px

Start display at page:

Download "Optimization of a Liquefied Natural Gas Portfolio by SDDP techniques"

Aileen Mathews
5 years ago
Views:

1 Opimizaion of a Liquefied Naural Gas Porfolio by SDDP echniques J. Frédéric Bonnans INRIA-Saclay and CMAP, École Polyechnique Commun work wih Zhihao Cen (now a Arelys) Thibaul Chrisel (Toal) CEA EDF Inria workshop: sochasic opimizaion and dynamic programming, 28 June 2012 J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

2 Main work Large size sochasic, dynamic and ineger opimizaion problem coninuous relaxaion (SP): develop an algorihm (L-Shape based) for risk neural problem and risk averse problem (CVaR); sudy (firs order) sensiiviy analysis w.r.. random process parameers; ineger problem (SMIP): propose an heurisic mehod based on cuing plane mehod; highligh he difficuly (by dual programming). J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

3 Ouline 1 Conex and problem Moivaions Mahemaical formulaion 2 Coninuous relaxaion I pricing L-Shape mehod (modified) Discreizaion and error analysis Numerical es on risk neural opimizaion 3 Coninuous relaxaion II sensiiviy analysis Sensiiviy analysis Convergence resul Numerical resul 4 Ineger problem Firs heurisic mehod Cuing plane mehod Numerical resul J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

4 Ouline 1 Conex and problem Moivaions Mahemaical formulaion 2 Coninuous relaxaion I pricing L-Shape mehod (modified) Discreizaion and error analysis Numerical es on risk neural opimizaion 3 Coninuous relaxaion II sensiiviy analysis Sensiiviy analysis Convergence resul Numerical resul 4 Ineger problem Firs heurisic mehod Cuing plane mehod Numerical resul J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

5 Moivaion I: LNG shipping porfolio opimizaion Basic rules Long erm buying and selling conracs (min-max amouns per monh and per year); Roue: beween wo pors able o receive ships of a given size (in discree number); Seller-buyer price formulas based on various (spo) commodiies indexes uncerain income; Discree decisions: how many ships on each roue, each monh? J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

6 Moivaion I: LNG shipping porfolio opimizaion Por Cargo size Annual QC. Monhly QC. Price formula (in $/MMBu) Caribbean 2.9, 3.4 [48.0, 54.0] [0, 6.0] NA NG OIL if OIL 75 Scandinavia 2.9 [24.0, 30.0] [0, 3.0] 0.07OIL oherwise 0.9NA NG if NA NG 5 Norh Africa 2.9, 3.4 {100.0} [0, 12.0] 0.8NA NG oherwise Norh American 2.9, 3.4 [84.0, 88.0] [0, 8.0] NA NG Europe 2.9, 3.4 [68.0, 76.0] [0, 8.0] EU NG Asia 2.9, 3.4 {20.0} [0, 4.0] 0.08OIL 0.8 J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

7 Moivaion II: swing opion Swing opion A swing opion can be view as a one-dimensional porfolio: only one roue. { ( T 1 ) P(Q min, Q max ) = sup E j=0 e r j (S j K j )q j, T 1 } q j : (Ω, F j ) [0, 1], q j [Q min, Q max ]. j=0 (1) Proposiion (Bardou e al. 2010) If (Q min, Q max ) Z 2, bang bang propery on opimal conrol : q j {0, 1} a.s. Exising mehods: Mone-Carlo + Longsaff-Schwarz: Barrera-Eseve e al. 2006; discreizaion (quanizaion): Bardou e al. 2009; PDE s numerical mehods: Kluge hesis 2006; oher numerical mehods for BSDE. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

8 Mahemaical formulaion (ξ ) L 2 (Ω, F, P; R d ) Markov process : ξ +1 = f(w, ξ, α ) = 0,..., T 1; (2) ξ 0 = ξ 0, (W ) R d i.i.d., independen of ξ ; F = σ (ξ s, 0 s ). inf u,x [ T 1 ] E c (ξ =0 )u + g(ξ T, x T ) s.. u : (Ω, F ) U R n ; x +1 = x + A u, x 0 = 0, x T X T R m ; (3) where c (ξ) Lipschiz; g(ξ, x) convex and l.s.c. w.r.. x, Lipschiz; U, X T nonempy, convex, compac; ( T 1 =0 A U ) XT. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

9 Dynamic programming (DP.) formulaion Following he Markov propery of (x, ξ ), define Bellman (cos-o-go) funcion Q(, x, ξ ) := essinf { [ T 1 E ] c s(ξ s= s )u s + g(ξ T, x T ) F : u s : (Ω, F s ) U s, x + The dynamic programming principle reads as: T 1 s= A su s X T }. Q(, x, ξ ) = essinf c (ξ )u + Q( + 1, x +1, ξ ) s.. u U, x +1 = x + A u ; where Q( + 1, x +1, ξ ) := E [ ] Q( + 1, x +1, ξ +1 ) F, and final cos (sage T): g(ξ Q(T, x T, ξ T ) = T, x T ) if x T X T, + oherwise. (4) (5) (6) J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

10 Difficulies and main soluion mehods / ideas High dimension x : dual decomposiion (L-Shape ype mehod); Numerical condiional expecaion compuaions: space discreizaion (ree mehod, quanizaion ree); projecion (leas square regression, Tsisiklis); kernel esimaion... Ineger opimizaion problem: branch and bound mehod; cuing plane mehod; meaheurisic mehod: local search... J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

11 Ouline 1 Conex and problem Moivaions Mahemaical formulaion 2 Coninuous relaxaion I pricing L-Shape mehod (modified) Discreizaion and error analysis Numerical es on risk neural opimizaion 3 Coninuous relaxaion II sensiiviy analysis Sensiiviy analysis Convergence resul Numerical resul 4 Ineger problem Firs heurisic mehod Cuing plane mehod Numerical resul J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

12 Dual programming Proposiion Q(, x, ξ ) (Q(, x, ξ 1 )) is convex, lower semi-coninuous w.r.. x. By Moreau Fenchel heorem, we define opimaliy cu Q(, x, ξ ) = Q (, x, ξ ) x, x Q (, x, ξ ), x. Linear programming (very efficien). L-Shape mehod (Van Silke and Wes 69) for linear recourse problem based on dual programming. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

13 Discreizaion Scenario ree Non-combinaion ree for non-markov case. Ex: ree reducion mehod (Heisch and Römisch 2003). Combinaion ree for Markov case. Ex: vecorial quanizaion ree mehod (Pagès e al. 2000). J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

14 L-Shape (modified) Forward pass Parial sampling (ou-of-sample es) Q(, x, ξ ) =essinf c (ξ )u + ξ i T (ξ ) λi Q( + 1, x +1, ξ); i (7) where T (ξ ) = {(ξ i, λi )} is he Delaunay riangulaion of ξ.. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

15 L-Shape (modified) Forward pass Parial sampling (ou-of-sample es) Q(, x, ξ ) =essinf c (ξ )u + ξ i T (ξ ) λi Q( + 1, x +1, ξ); i (7) essinf c (ξ )u + ξ i T (ξ ) λi ϑ( + 1, x +1, ξ, i O +1 ), (8) (feasibiliy cu) (opimaliy cu) s.. x +1 = x + A u, { u U ad (x ) = u : x + ϑ( + 1, x +1, ξ i, O +1 ) T 1 s= } A s u s X T, u s U s, ], (λ x, λ 0 ) O j ; +1 [ j ˆp ij ξ (λx x +1 + λ 0 ) +1 where T (ξ ) = {(ξ i, λi )} is he Delaunay riangulaion of ξ. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

16 Algorihm I Forward pass (upper bound) Sep 1 : Iniialize Sep 2 : Forward pass he opimaliy cus on ree; he firs sage conrol u 0. Simulae M f random process following (2). For m = 1,..., M f do: for = 1,..., T 1 do: calculae he Delaunay riangle T (ξ m ), solve subproblem (8); compue v m opimal value of scenario ξ m. Compue he empirical saisic : v = 1 M f Mf m=1 vm and s = 1 M f Mf m=1 (vm v) 2. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

17 L-Shape Backward pass Q(, x, ξ i ) =essinf c (ξ i )u + Q( + 1, x +1, ξ i ); (9) (feasibiliy cu) (opimaliy cu) essinf c (ξ i )u + ϑ( + 1, x +1, ξ i, O +1 ), (10) s.. x +1 = x + A u, ( dual value λ x O i ) { T 1 } u U ad (x ) = u : x + A s u s X T, u s U s, s= [ ] ϑ( + 1, x +1, ξ, i O +1 ) j ˆp ij ξ (λx x +1 + λ 0 ), (λ x, λ 0 ) O j J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

18 Algorihm II Backward pass (lower bound) Sep 3 : Backward pass For = T 1,..., 1 do: for m = 1,..., M b do: = 0: solve subproblem (10) on all verices in Γ ; compue new opimaliy cus and add hem O. solve subproblem (10), backward value v i. Sep 4 : Check sop condiion If v i [v ϱs, v + ϱs] and v i v i 1 ɛ v i STOP; else go o Sep 2. where ϱ > 0 is a parameer. In sandard Mone-Carlo mehod, ϱ = 1.96 (corresponds o 95% of confidence inerval). J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

19 Discreizaion vecorial quanizaion mehod Opimal dual quanizaion : inf { F D p (ξ, Γ) p : Γ = { ξ 1,..., ξ N} R d }, { (Fp D (ξ, Γ)) p N = min λ i ξ ξ i p : λ R N i=1 N λ iξ i N } = proj i=1 conv(γ) (ξ), λ i = 1, λ 0. i=1 combinaion ree (11). Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

20 Converge resul and sabiliy of opimal value Proposiion The upper bound (forward value) and he lower bound (backward value) converge if oal quanized poins number N = T =0 N. Theorem Assume ha he soluion se is nonempy and bounded. Then, L > 0 such ha: val(ξ) val( ξ) L( F D p (ξ, Γ) p + D f (ξ, ˆξ)) (12) where D f (ξ, ˆξ) := T 1 inf u E[u ˆF ] q (13) u S(ξ) =0 where ( ξ ) is he finie sae Markov chain buil based on he quanizaion ree, (ˆξ ) is he quanizaion process, ˆF = σ(ˆξ s : 0 s ); p 1 + q 1 = 1; val( ) (resp. S( )) is he opimal value (resp. he opimal soluion se). J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

21 Convergence rae of disorion Dual quanizaion case Theorem (Pagès and Wilberz 10) Assume ha ξ L p+η (Ω, F, P; R d ) for some η > 0. Then ( ) lim N N p d inf Fp D (ξ, Γ) p p #(Γ) N = J p,d ϕ d d+p (14) where µ(dξ) = ϕ(µ) λ d (dξ) + υ (λ d Lebesgue measure on R d ), and υ λ d. The consan J p,d corresponds o he case of he uniform disribuion on [0, 1] d (or any Borel se of Lebesgue measure 1). Quanizaion ree: inf Fp D (ξ, Γ) p = O(N 1/d ) where N = T =0 N. #(Γ) N J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

22 Resul on LNG porfolio coninuous relaxaion problem Quanizaion size: N = 36000; Algorihm parameer: M f = 3000 o 6000, M b = 8 o 12; Process parameers σ 1 = σ 2 = σ 3 = 40%; ρ 12 = 0.7, ρ 13 = 0.2, ρ 23 = 0.4. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

23 Resul on LNG porfolio coninuous relaxaion problem Quanizaion size: N = 36000; Algorihm parameer: M f = 3000 o 6000, M b = 8 o 12; Process parameers σ 1 = σ 2 = σ 3 = 40%; ρ 12 = 0.7, ρ 13 = 0.2, ρ 23 = 0.4. forward ime-consumpion backward ime-consumpion J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

24 Ouline 1 Conex and problem Moivaions Mahemaical formulaion 2 Coninuous relaxaion I pricing L-Shape mehod (modified) Discreizaion and error analysis Numerical es on risk neural opimizaion 3 Coninuous relaxaion II sensiiviy analysis Sensiiviy analysis Convergence resul Numerical resul 4 Ineger problem Firs heurisic mehod Cuing plane mehod Numerical resul J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

25 Markov process (price) model A discree-ime one facor model: where ln (F s+1 )i = σ i (Fs) W i i s 1 2 (σi ) 2, i = 1,..., d; 1 i ξ = (F )i = (F 0 )i exp σ i Ws i 1 (15) 2 (σi ) 2, i = 1,..., d; s=0 F s, s he forward conrac price a ime s wih mauriy ; W i follows N(0, 1), corr(w i, W j ) = ρ ij. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

26 Sensiiviy analysis Problem inf [ T 1 ] E c (ξ =0 )u s.. u : (Ω, F ) U R n ; x +1 = x + A u, x 0 = 0, x T X T R m. (16) no final cos funcion g, he crierion is linear w.r.. (u ). v (F 0, σ) he opimal value funcion; U (F 0, σ) he opimal soluion se. Sensiiviy esimaes (firs order) δ : sensiiviy w.r.. F 0 ; ν : sensiiviy w.r.. σ. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

27 Differeniabiliy and sensiiviy esimaes By Danskin s heorem Corollary If µ (ξ(f 0, σ)) λ Lebesgue, hen he opimal funcion v (F 0, σ) is Fréche differeniable a almos every poin (F 0, σ). Thus, a almos every poin (F 0, σ), wih one opimal soluion u U : T 1 δ(df 0 ) := Dv (F 0, σ; df 0, 0) = E D F c df 0 u ; =0 (17) T 1 ν(dσ) := Dv (F 0, σ; 0, dσ) = E D σ c dσ u. =0 J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

28 Discreized problem A sequence of opimal quanizaion ˆξ m, lim m N m = : Discreized problem I quanized problem v m Q (F 0, σ) = min [ T 1 ] E (ξ =0 )u m s.. u m : (Ω, σ(ˆξ m, x m )) U, x m +1 = xm + A u m, x m 0 = 0, x m T Xm T a.s. Discreized problem II scenario problem v m S (F 0, σ) = min E [ T 1 =0 c (ˆξ m )u m s.. u m : (Ω, σ(ˆξ m, x m )) U, x m +1 = xm + A u m, x m 0 = 0, x m T Xm T a.s. ] (18) (19) Boh problems are in finie dimension. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

29 Differeniabiliy and approximae formula Lemma If µ (ξ(f 0, σ)) λ Lebesgue, hen he opimal funcions of he discreized problems v,m Q (F 0, σ) and v,m S (F 0, σ) are Fréche differeniable a almos every poin (F 0, σ). Thus, a almos every poin (F 0, σ), wih one opimal soluion u,m Q δ m Q (df 0) = Dv,m Q ν m Q (dσ) = Dv,m Q T 1 (F 0, σ; df 0, 0) = E =0 T 1 (F 0, σ; 0, dσ) = E Same resul for scenario problem δ m S, νm S. =0 D F c df 0 (u,m Q D σ c dσ (u,m Q ) ) ; U,m Q :. (20) J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

30 Convergence resul Lemma where u,m Q u,m u S u in L 2 (Ω, (F ), P; R n T ) (21) u,m Q is a subsequence of opimal sraegy of quanized problem (18); u,m S is a subsequence of opimal sraegy of scenario problem (19). Corollary A almos every poin (F 0, σ), we have where δ m Q δ m S δ m Q δ; νm Q ν; δ m S δ; ν m S ν; and νm are sensiiviy esimaes of quanized problem (18); Q and ν m S are sensiiviy esimaes of scenario problem (19). (22) J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

31 Swing opion Price formula : c (ξ ) = K ξ ; Srike : K = F 0 ; Process parameers : T = 50; σ = 30%/ T = 0.042; Forward price : F = sin(2π /T); 0 Swing parameers : Q min = 20; Q max = 30. Comparison mehods sensiiviy esimae cond. expecaion dp. mehod I Danskin quanizaion ree L-Shape II Danskin quanizaion ree discreizaion + dp. III Danskin PDE discreizaion + dp. IV Finie difference PDE discreizaion + dp. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

32 Sensiiviy resul I Opimal value ree+l-shape ree+discre. pde.+discre. ub. σ(ub.) lb. v Sensiiviy ν ree+l-shape ree+discre. pde.+discre. fd.+ pde. mean sd. dev. mean sd. dev. mean sd. dev. ν J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

33 Sensiiviy resul II Sensiiviy δ Figure: Sensiiviy values wih respec o F 0 obained by 4 mehods. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

34 Ouline 1 Conex and problem Moivaions Mahemaical formulaion 2 Coninuous relaxaion I pricing L-Shape mehod (modified) Discreizaion and error analysis Numerical es on risk neural opimizaion 3 Coninuous relaxaion II sensiiviy analysis Sensiiviy analysis Convergence resul Numerical resul 4 Ineger problem Firs heurisic mehod Cuing plane mehod Numerical resul J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

35 Example Example Q O x 0 Q(x 0 ) = min E[ξu 0 + θ] s.. u 0 [0, 3] Z, x 1 = x 0 + 2u 0, x 1 6, θ x 1 3, 2 θ x (23) where ξ = { 2, 0.5} ha each akes probabiliy 0.5; x 0 {0, 1, 2, 3}. 1 2 Figure: Bellman value funcion Q(, x, ξ ) (resp Q(, x, ξ 1 )) is generally non-convex, lower semiconinuous w.r.. x. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

36 Firs heurisic mehod (Birge and Louveaux) Compue he opimal ineger soluion using he Bellman funcion of he coninuous (relaxaion) problem: min c (ξ )u + Q con ( + 1, x +1, ξ ) s.. x +1 = x + A u, u U in,ad (x ) = { u : x + T 1 s= A s u s X T, u s U s Z n }, (opimaliy cu) Q con ( + 1, x +1, ξ ) λ x x +1 + λ 0, (λ x, λ 0 ) O +1 (ξ ). Q O x J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

37 Cuing plane mehod Theorem The mixed ineger program min{cx + hy : (x, y) S} where S = { x Z n +, y R m + : Ax + Gy b } and he linear program min{cx + hy : (x, y) conv(s)} have same opimal value. Furhermore, if (x, y ) is an opimal soluion of (MIP), hen i is an opimal soluion of (LP). (MIP) (LP) Objecive : build conv(s). lifing and projecing : Balas, Ceria and Cornuéjols, Sherali and Adams, ec.; Gomory cu: Gomory, Balas e al., ec.; disjuncive cu: Balas e al., Sen e al., ec.; Fenchel cu: Boyd, Naimo, ec.; oher mehods. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

38 Feasible se and noaions { U in,ad (ˆx ) = u : u s U s Z n, x + T 1 s= A s u s X T }; Example ˆx 0 = 1: u O Figure: feasibiliy se x 0. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

39 Feasible se and noaions { U in,ad (ˆx ) = u : u s U s Z n, convu in,ad (ˆx ); x + T 1 s= A s u s X T }; Example ˆx 0 = 1: u O Figure: feasibiliy se x 0. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

40 Feasible se and noaions { U in,ad (ˆx ) = u : u s U s Z n, convu in,ad (ˆx ); (X, U) in,ad = x + T 1 s= A s u s X T }; { (x, u ) : u s U s Z n, T 1 1 A u X T, x = A s u s }; =0 s=0 Example ˆx 0 = 1: u O Figure: feasibiliy se x 0. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

41 Feasible se and noaions { U in,ad (ˆx ) = u : u s U s Z n, convu in,ad (ˆx ); (X, U) in,ad = x + T 1 s= A s u s X T }; { (x, u ) : u s U s Z n, T 1 1 A u X T, x = A s u s }; =0 conv(x, U) in,ad ; s=0 Example ˆx 0 = 1: u O Figure: feasibiliy se x 0. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

42 Feasible se and noaions { U in,ad (ˆx ) = u : u s U s Z n, convu in,ad (ˆx ); (X, U) in,ad = x + T 1 s= A s u s X T }; { (x, u ) : u s U s Z n, T 1 1 A u X T, x = A s u s }; =0 conv(x, U) in,ad ; s=0 U in,ad (ˆx ) = proj U (conv(x, U) in,ad { } x = ˆx ); Example ˆx 0 = 1: u O Figure: feasibiliy se x 0. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

43 Feasible se and noaions { U in,ad (ˆx ) = u : u s U s Z n, convu in,ad (ˆx ); (X, U) in,ad = x + T 1 s= A s u s X T }; { (x, u ) : u s U s Z n, T 1 1 A u X T, x = A s u s }; =0 conv(x, U) in,ad ; s=0 U in,ad (ˆx ) = proj U (conv(x, U) in,ad { } x = ˆx ); Example ˆx 0 = 1: u convu in,ad (ˆx ) U in,ad (ˆx ) In general, he inclusion is sric. O Figure: feasibiliy se x 0. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

44 Limiaion of dual ype algorihm Using dual ype algorihm, he bes approximaion is he convex envelop. Q O x Objecive Approximae he convex envelop as good as possible. I is impossible o provide he exac opimal soluion / opimal value. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

45 Algorihm We solve he coninuous relaxaion problem wih he cuing plane generaed: min c (ξ )u + ϑ s.. x s+1 = x s + A s u s, s =,..., T 1, u s U s, x T X T, (feasibiliy cu) λ x s x s + λ u s u s λ 0 s, (λ x s, λ u s, λ 0 s) I s, s =,..., T 1, (opimaliy cu) ϑ λ x x +1 + λ u u + λ 0, (λ x, λ u, λ 0 ) O +1 (ξ ); (24) If u is ineger, OK. Oherwise, sudy where find he u U. u convu in,ad (x ); u U in,ad (ˆx ) \ convu in,ad (ˆx ); u U in,ad (x )( (x, u ) conv(x, U)in,ad ). J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

46 Algorihm (con.) Case III : u U in,ad (x )( (x, u Example ξ = 2, x 0 = 3: ) conv(x, U)in,ad ). u 0 3 min 2u 0 + θ s.. u 0 3, x 1 6, x 1 = 2u 0 + 3, θ x 1 3, 2 θ x O x 0 Ineger cuing plane can be generaed. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

47 Improvemen on example Q O x Red line: convex envelop of Bellman funcion of ineger problem; Blue line: Bellman funcion of coninuous relaxaion problem; Black line: approximaion by cuing plane mehod. J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

48 Numerical resul of LNG porfolio T = 6 u.b. σ(u.b.) l.b. de. counerpar value Coninuous Ineger (heurisic) Ineger (cu) Table: Comparison of opimal value of coninuous problem and ineger problem J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

49 Tha is he end! J. Frédéric Bonnans (INRIA-Saclay and CMAP) (INRIA, CMAP-X, Opimizaion Toal) of a LNG Porfolio June 28, / 39

Solving multi-stage stochastic mixed integer linear programs by the dual dynamic programming approach

Solving multi-stage stochastic mixed integer linear programs by the dual dynamic programming approach Noname manuscrip No. (will be insered by he edior) Solving muli-sage sochasic mixed ineger linear programs by he dual dynamic programming approach Zhihao Cen he dae of receip and accepance should be insered