A deterministic algorithm for stochastic multistage problems or - - PowerPoint PPT Presentation

CMS and MMEI 2019 27th March 2019

A deterministic algorithm for stochastic multistage problems

• r

The problem-child algorithm

Regan Baucke, Anthony Downward, Golbon Zakeri CERMICS, Ecole des Ponts ParisTech reganbaucke.github.io

Stochastic optimisation

▶ Original problem: min

u∈U EP[C(u, Z)]

▶ Discretised problem: min

u∈U

∑

ω∈Ω

ˆ P(ω)[C(u, Z(ω))] ▶ A method:

▶ Some lower bound: (¯ Ck, uk) = M(k) ▶ Upper bound: ¯ Ck = ∑

ω∈Ω

ˆ P(ω)[C(uk, Z(ω))] ▶ Optimality gap: ¯ Ck − ¯ Ck

Stochastic optimisation

▶ Original problem: min

u∈U EP[C(u, Z)]

▶ Discretised problem: min

u∈U

∑

ω∈Ω

ˆ P(ω)[C(u, Z(ω))] ▶ A method:

▶ Some lower bound: (¯ Ck, uk) = M(k) ▶ Upper bound: ¯ Ck = ∑

ω∈Ω

ˆ P(ω)[C(uk, Z(ω))] ▶ Optimality gap: ¯ Ck − ¯ Ck

Stochastic optimisation

▶ Original problem: min

u∈U EP[C(u, Z)]

▶ Discretised problem: min

u∈U

∑

ω∈Ω

ˆ P(ω)[C(u, Z(ω))] ▶ A method:

▶ Some lower bound: (¯ Ck, uk) = M(k) ▶ Upper bound: ¯ Ck = ∑

ω∈Ω

ˆ P(ω)[C(uk, Z(ω))] ▶ Optimality gap: ¯ Ck − ¯ Ck

Stochastic optimisation

▶ Original problem: min

u∈U EP[C(u, Z)]

▶ Discretised problem: min

u∈U

∑

ω∈Ω

ˆ P(ω)[C(u, Z(ω))] ▶ A method:

▶ Some lower bound: (¯ Ck, uk) = M(k) ▶ Upper bound: ¯ Ck = ∑

ω∈Ω

ˆ P(ω)[C(uk, Z(ω))] ▶ Optimality gap: ¯ Ck − ¯ Ck

Stochastic optimisation

▶ Original problem: min

u∈U EP[C(u, Z)]

▶ Discretised problem: min

u∈U

∑

ω∈Ω

ˆ P(ω)[C(u, Z(ω))] ▶ A method:

▶ Some lower bound: (¯ Ck, uk) = M(k) ▶ Upper bound: ¯ Ck = ∑

ω∈Ω

ˆ P(ω)[C(uk, Z(ω))] ▶ Optimality gap: ¯ Ck − ¯ Ck

Stochastic optimisation

▶ Original problem: min

u∈U EP[C(u, Z)]

▶ Discretised problem: min

u∈U

∑

ω∈Ω

ˆ P(ω)[C(u, Z(ω))] ▶ A method:

▶ Some lower bound: (¯ Ck, uk) = M(k) ▶ Upper bound: ¯ Ck = ∑

ω∈Ω

ˆ P(ω)[C(uk, Z(ω))] ▶ Optimality gap: ¯ Ck − ¯ Ck

Multistage stochastic optimisation

▶ Original problem: min

u∈U(x),x∈X EP[

∑

0≤t≤T

Ct(xt(ω), ut(ω), Zt(ω))] ▶ Discretised problem: min

u∈U(x),x∈X

∑

n∈N

ˆ Pn[Cn(xn, un)] ▶ A method:

▶ Some lower bound: (¯ V k, xk

n, uk n) = M(k)

▶ Upper bound: ¯ V k = ∑

n∈N

ˆ Pn[Cn(xk

n, uk n)]

▶ Optimality gap: ¯ V k − ¯ V k

Multistage stochastic optimisation

▶ Original problem: min

u∈U(x),x∈X EP[

∑

0≤t≤T

Ct(xt(ω), ut(ω), Zt(ω))] ▶ Discretised problem: min

u∈U(x),x∈X

∑

n∈N

ˆ Pn[Cn(xn, un)] ▶ A method:

▶ Some lower bound: (¯ V k, xk

n, uk n) = M(k)

▶ Upper bound: ¯ V k = ∑

n∈N

ˆ Pn[Cn(xk

n, uk n)]

▶ Optimality gap: ¯ V k − ¯ V k

Multistage stochastic optimisation

▶ Original problem: min

u∈U(x),x∈X EP[

∑

0≤t≤T

Ct(xt(ω), ut(ω), Zt(ω))] ▶ Discretised problem: min

u∈U(x),x∈X

∑

n∈N

ˆ Pn[Cn(xn, un)] ▶ A method:

▶ Some lower bound: (¯ V k, xk

n, uk n) = M(k)

▶ Upper bound: ¯ V k = ∑

n∈N

ˆ Pn[Cn(xk

n, uk n)]

▶ Optimality gap: ¯ V k − ¯ V k

Multistage stochastic optimisation

▶ Original problem: min

u∈U(x),x∈X EP[

∑

0≤t≤T

Ct(xt(ω), ut(ω), Zt(ω))] ▶ Discretised problem: min

u∈U(x),x∈X

∑

n∈N

ˆ Pn[Cn(xn, un)] ▶ A method:

▶ Some lower bound: (¯ V k, xk

n, uk n) = M(k)

▶ Upper bound: ¯ V k = ∑

n∈N

ˆ Pn[Cn(xk

n, uk n)]

▶ Optimality gap: ¯ V k − ¯ V k

Multistage stochastic optimisation

▶ Original problem: min

u∈U(x),x∈X EP[

∑

0≤t≤T

Ct(xt(ω), ut(ω), Zt(ω))] ▶ Discretised problem: min

u∈U(x),x∈X

∑

n∈N

ˆ Pn[Cn(xn, un)] ▶ A method:

▶ Some lower bound: (¯ V k, xk

n, uk n) = M(k)

▶ Upper bound: ¯ V k = ∑

n∈N

ˆ Pn[Cn(xk

n, uk n)]

▶ Optimality gap: ¯ V k − ¯ V k

Multistage stochastic optimisation

▶ Original problem: min

u∈U(x),x∈X EP[

∑

0≤t≤T

Ct(xt(ω), ut(ω), Zt(ω))] ▶ Discretised problem: min

u∈U(x),x∈X

∑

n∈N

ˆ Pn[Cn(xn, un)] ▶ A method:

▶ Some lower bound: (¯ V k, xk

n, uk n) = M(k)

▶ Upper bound: ¯ V k = ∑

n∈N

ˆ Pn[Cn(xk

n, uk n)]

▶ Optimality gap: ¯ V k − ¯ V k

#vert(d, T) = dT−1 − 1 d − 1

Multistage stochastic optimisation: independence

▶ Restrictions: Independence Zt ▶ A method:

▶ Some lower bound: (¯ V k, xk

n, uk n) = M(k)

▶ Upper bound: ¯ V k = ∑

n∈N

ˆ Pn[Cn(xk

n, uk n)]

▶ Optimality gap: ¯ V k − ¯ V k

Multistage stochastic optimisation: independence

▶ Restrictions: Independence Zt ▶ A method:

▶ Some lower bound: (¯ V k, xk

n, uk n) = M(k)

▶ Upper bound: ¯ V k = ∑

n∈N

ˆ Pn[Cn(xk

n, uk n)]

▶ Optimality gap: ¯ V k − ¯ V k

The upper bound function The problem-child algorithm Results Saddle functions

u C(u)

C(u) = min

µ∈R

µ s.t. µ ≥ C(ˆ u) + ⟨dˆ

u, u − ˆ

u⟩, ∀ˆ u,

The upper bound function | Construction

u C(u)

C(u) = min

µ∈R

µ s.t. µ ≥ C(ˆ u) + ⟨dˆ

u, u − ˆ

u⟩, ∀ˆ u,

The upper bound function | Construction

u C(u)

C(u) = min

µ∈R

µ s.t. µ ≥ C(ˆ u) + ⟨dˆ

u, u − ˆ

u⟩, ∀ˆ u,

The upper bound function | Construction

u C(u)

¯ C(u) = min

µ∈R

µ s.t. µ ≥ C(ˆ u) + ⟨dˆ

u, u − ˆ

u⟩, ∀ˆ u,

The upper bound function | Construction

u C(u)

C(u) = min

µ∈R

µ s.t. µ ≥ G(ˆ u) + ⟨dˆ

u, u − ˆ

u⟩, ∀ˆ u.

The upper bound function | Construction

u C(u)

C(u) = min

µ∈R

µ s.t. µ ≥ G(ˆ u) + ⟨dˆ

u, u − ˆ

u⟩, ∀ˆ u.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

The upper bound function | Construction

Example

min

u∈U

1 2 C1(u) + 1 2 C2(u)

The problem-child algorithm | Two-stage problem

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

1 2C1 + 1 2C2

C1 C2

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

Let’s fjx some notation: Vn(xn) = min

xm,un

Cn(xn, un) + ∑

m∈R(n)

p(n, m)Vm(xm) s.t. xm = fm(xn, un), ∀m ∈ R(n), xm ∈ Xm, ∀m ∈ R(n), un ∈ Un(xn). We want to compute V0(x0).

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

Theorem lim

k→∞

¯ V k

0 (x0) − ¯

V k

0 (x0) = 0.

The problem-child algorithm | The multistage problem

Hydro-reservoir optimisation 20 stages 9 children per node 3 reservoirs

Results | A test problem

200 400 600 800 1,000 400 420 440 460 480 500 k Expected Cost

PC Results | Convergence plot

200 400 600 800 1,000 400 420 440 460 480 500 k Expected Cost

PC 95% Results | Convergence plot

Relative ϵ Lower Upper Time (s) Iterations LP Solves 20.0% 406.9 506.9 48 300 28,000 10.0% 413.1 457.7 137 530 50,600 5.0% 416.3 438.1 396 930 88,200 1.0% 419.0 423.1 7462 3650 346,100

Results | Convergence plot

100 200 300 400 500 600 700 800 900 1,000 390 400 410 420 k Expected Cost

PC Random Results | Convergence plot

Gap Confjdence Level 95% 97.5% 99% 99.9% 10% 440 570 760 1,240 5% 1,830 2,400 3,170 5,160 1% 48,690 63,600 84,370 137,200

Results | Convergence investigation

Gap Confjdence Level 95% 97.5% 99% 99.9% 10% 440 570 760 1,240 5% 1,830 2,400 3,170 5,160 1% 48,690 63,600 84,370 137,200

Results | Convergence investigation

Gap Confjdence Level 95% 97.5% 99% 99.9% 10% 440 570 760 1,240 5% 1,830 2,400 3,170 5,160 1% 48,690 63,600 84,370 137,200

Results | Convergence investigation

The problem child principle can be used in other settings too.

Saddle functions | Bounds for saddles

Suppose we wish to solve min

u∈U max v∈V EP[C(u, v, Z)]

We can apply the problem child algorithm in the same way.

Saddle functions | Bounds for saddles

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u C(u)

¯ C(u) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u), ∀ˆ u, + ⟨dˆ

v, v − ˆ

v⟩ ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, ˆ v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, ˆ v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, ˆ v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ G(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

¯ C(u, v) = max

µ∈R,λ∈Rn

µ + ⟨λ, u⟩ s.t. µ + ⟨λ, ˆ u⟩ ≤ C(ˆ u, ˆ v) + ⟨dˆ

v, v − ˆ

v⟩, ∀(ˆ u, ˆ v), ||λ||∗ ≤ α.

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

u v C(u, v)

Saddle functions | Bounds for saddles

Let’s fjx some rather complicated notation: Gn(xn, yn) = min

xm,un max ym,vn

Cn(xn, yn, un, vn) + ∑

m∈R(n)

p(n, m)Gm(xm, ym) s.t. xm = f x

m(xn, un), ∀m ∈ R(n),

ym = f y

m(yn, vn), ∀m ∈ R(n),

(xm, ym) ∈ Xm × Ym, ∀m ∈ R(n), (un, vn) ∈ Un(xn) × Vn(yn).

Saddle functions | Multistage formulation

Theorem

There exists a set of minimax dynamic programming equations for which G0(x0, y0) = min

u∈U(x),x∈X CVaRβ

[ ∑

0≤t≤T

Ct(xt(ω), ut(ω), Zt(ω)) ]

Saddle functions | Multistage formulation

Portfolio managment problem 12 stages 6 children per node 3 assests

Saddle functions | Multistage formulation

20 40 60 80 100 120 140 160 180 200

• 5%

0% 5% 10% k Risk-adjusted yield Upper Bound Lower Bound

3 minutes 47 seconds

Saddle functions | Multistage formulation

CVaRβ Estimation Optimal Policy β0 = 0.2 β0 = 0.6 β0 = 1.0 β0 = 0.2 1.52% 1.30% 1.05% β0 = 0.6 2.38% 2.68% 2.66% β0 = 1.0 3.14% 3.73% 4.09%

Saddle functions | Multistage formulation

Conclusions

▶ Forming upper bounds ▶ Problem-child algorithm ▶ Deterministic bounds and deterministic convergence ▶ Saddle functions Baucke, R., Downward, A., & Zakeri, G. (2018). A deterministic algorithm for solving multistage stochastic minimax dynamic programmes. reganbaucke.github.io

Conclusions

▶ Forming upper bounds ▶ Problem-child algorithm ▶ Deterministic bounds and deterministic convergence ▶ Saddle functions Baucke, R., Downward, A., & Zakeri, G. (2018). A deterministic algorithm for solving multistage stochastic minimax dynamic programmes. reganbaucke.github.io

Conclusions

▶ Forming upper bounds ▶ Problem-child algorithm ▶ Deterministic bounds and deterministic convergence ▶ Saddle functions Baucke, R., Downward, A., & Zakeri, G. (2018). A deterministic algorithm for solving multistage stochastic minimax dynamic programmes. reganbaucke.github.io

The upper bound function Construction The problem-child algorithm Two-stage problem The multistage problem Results A test problem Convergence plot Convergence investigation Saddle functions Bounds for saddles Multistage formulation