The Multicut L-Shaped Method

1 / 25 The Multicut L-Shaped Method Operations Research Anthony Papavasiliou

Contents 2 / 25 1 The Multicut L-Shaped Method [ 5.1 of BL] 2 Example 3 Example: Capacity Expansion Planning

Table of Contents 3 / 25 1 The Multicut L-Shaped Method [ 5.1 of BL] 2 Example 3 Example: Capacity Expansion Planning

4 / 25 Extensive Form 2-Stage Stochastic Linear Program K (EF ) : min c T x + p k qk T y k s.t. Ax = b k=1 T k x + Wy k = h k, k = 1,..., K x 0, y k 0, k = 1,..., K K realizations of random vector ξ, with probabilities p k, k = 1,..., K Randomness: q k, h k, T k y k : second-stage decision given realization k

5 / 25 L-Shaped Master Problem We know that K V (x) = {min p k qk T y k Wy k = h k T k x, y k 0} k=1 is a piecewise linear function of x Define master problem as (M) : min z = c T x + θ (1) s.t. Ax = b D l x d l, l = 1,..., r (2) E l x + θ e l, l = 1,..., s (3) x 0, θ R Feasibility cuts: equation 2 Optimality cuts: equation 3

6 / 25 Multicut L-Shaped Master Problem We also know that Q(x, ξ k ) = {min qk T y k Wy k = h k T k x, y k 0} is a piecewise linear function of x The multicut L-shaped method master problem is K (M) : min z = c T x + θ k (4) s.t. Ax = b k=1 D l x d l, l = 1,..., r (5) E l(k) x + θ k e l(k), l(k) = 1,..., s k, k = 1,..., K (6) x 0

L-Shaped Optimality Cuts 7 / 25 Consider a trial first-stage decision x v Let πk v be simplex multipliers of second-stage problem: min w = qk T y s.t. Wy = h k T k x v y 0 e s+1 E s+1 x supports V (x) at x v, where E s+1 = e s+1 = K p k (πk v )T T k (7) k=1 K p k (πk v )T h k (8) k=1

8 / 25 Multicut L-Shaped Optimality Cuts Consider a trial first-stage decision x v Let πk v be simplex multipliers of second-stage problem: min w = qk T y s.t. Wy = h k T k x v y 0 e sk +1 E sk +1x supports Q(x, ξ k ) at x v, where E sk +1 = p k (πk v )T T k (9) e sk +1 = p k (πk v )T h k (10)

L-Shaped Graphical Illustration of Optimality Cuts 9 / 25 V (x) e s+1 E s+1 x x v x

10 / 25 Multicut L-Shaped Graphical Illustration of Optimality Cuts p k Q(x, ξ k ) e sk +1 E sk +1x x v x

L-Shaped Versus Multicut 11 / 25 V(x) θ v e s+1 E s+1 x x v

L-Shaped Versus Multicut 12 / 25 Q(x, ξ a ) θ v a / p a e sa +1 E s a +1 x θ v b /p b Q(x, ξ b ) e sb +1 E s b +1 x x v

Alternative Formulation of Master 13 / 25 Alternatively, cuts can support Q(x, ξ k ) instead of p k Q(x, ξ k ) K (M) : min z = c T x + p k θ k s.t. Ax = b k=1 D l x d l, l = 1,..., r E l(k) x + θ k e l(k), l(k) = 1,..., s k, k = 1,..., K x 0 E sk +1 = (π v k )T T k, e sk +1 = (π v k )T h k

14 / 25 The L-Shaped Algorithm Step 0. Set r = s = v = 0 Step 1. Solve master problem (M). Let (x v, θ v ) be an optimal solution. If s = 0 (no optimality cuts), remove θ from (M) and set θ 0 = Step 2. If x / K 2, add feasibility cut (equation 2) and return to Step 1. Otherwise, go to Step 3. K 2 = {x y : Wy = h k T k x, y 0, k = 1,..., K } Step 3. Compute E s+1, e s+1. Let w v = e s+1 E s+1 x v. If θ v w v, stop with x v an optimal solution. Otherwise, set s = s + 1, add optimality cut to equation 3 and return to Step 1.

15 / 25 The Multicut L-Shaped Algorithm Step 0. Set r = v = 0, s k = 0, k = 1,..., K Step 1. Solve master problem (M). Let (x v, θ v 1,..., θv K ) be an optimal solution. If s k = 0 (no optimality cuts for scenario k), remove θ k from (M) and set θ v k = Step 2. If x / K 2, add feasibility cut and return to Step 1. Otherwise, go to Step 3. Step 3. If 1 θ v k < p k(π v k )T (h k T k x v ), compute E sk +1, e sk +1 and set s k = s k + 1. If θ v k p k(π v k )T (h k T k x v ) for all k = 1,..., K, stop with x v an optimal solution. Otherwise, return to Step 1. 1 Typically, less than K optimality cuts are added per iteration

Table of Contents 16 / 25 1 The Multicut L-Shaped Method [ 5.1 of BL] 2 Example 3 Example: Capacity Expansion Planning

17 / 25 Example z = min E ξ (y 1 + y 2 ) s.t. 0 x 10 y 1 y 2 = ξ x y 1, y 2 0 1 p 1 = 1/3 ξ = 2 p 2 = 1/3 4 p 3 = 1/3 K 2 = R

Multicut L-Shaped Method in Example 2 18 / 25 Iteration 1, Step 1: x 1 = 0 Iteration 1, Step 3: x 1 not optimal, add cuts: θ 1 1 x 3, θ 2 2 x 3, θ 3 4 x 3 Iteration 2, Step 1: x 2 = 10, θ 2 1 = 3, θ2 2 = 8/3, θ2 3 = 2 Iteration 2, Step 3: x 2 not optimal, add cuts: θ 1 x 1 3, θ 2 x 2 3, θ 3 x 4 3 Iteration 3, Step 1: x 3 = 2, θ1 3 = 1/3, θ3 2 = 0, θ3 3 = 2/3 is optimal

19 / 25 Tradeoffs Multicut L-shaped method has: More detailed representation of value function (+) Larger master problem (-) Typically (not always), fewer iterations are required in multicut L-shaped method, but each iteration requires more time

Table of Contents 20 / 25 1 The Multicut L-Shaped Method [ 5.1 of BL] 2 Example 3 Example: Capacity Expansion Planning

Master Problem 21 / 25 (M) : min x 0 θ ω n I i x i + i=1 θ ω 0 N p ω θ ω ω=1 m λ v ωj D j + j=1 n ρ v ωi x i, v V ωk i=1

Sequence of Investment Decisions Iteration Coal (MW) Gas (MW) Nuclear (MW) Oil (MW) 1 0 0 0 0 2 0 0 0 10701.3 3 0 14309.6 0 0 4 10407 0 0 0 5 0 0 7154.8 5034.6 6 0 2329 7154.8 1410 7 0 1280.2 8518.8 1647.6 8 2102.1 3310.9 5756 0 9 8767.3 236.7 0 2291.4 10 6396 0 3919 1168.8 11 8230.5 2165 773.5 0 12 5085 1311 3919 854 22 / 25

Sequence of Value Function Approximations 23 / 25 Iteration L-shaped Multicut θ N k ω=1 pωθ ωk 1 0 0 2 0 0 3 0 14674 4 0 61181 5 0 61181 6 0 28444 7 59736 83545 8 40998 186865 9 50222 108401 10 96290 171767 11 61593 125272 12 186788 13 107349 14 124788 15 130041 16 125272

24 / 25 Observations Multi-cut converges with fewer iterations Multi-cut incurs non-zero second stage cost in iteration 3 (L-shaped method requires 7 iterations) Iterations 5 and 6 have identical N ω=1 p ωθ ωk, does not imply convergence θ k for L-shaped need not be increasing (see iteration 12, attempt to remove nuclear) Final iterations of L-shaped (12-15) oscillate around near-optimal mix