Two-Stage Stochastic Linear Programs

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Two-Stage Stochastic Linear Programs

Operations Research

Anthony Papavasiliou

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Two-Stage Stochastic Linear Programs

1

Short Reviews

Probability Spaces and Random Variables Convex Analysis

2

Deterministic Linear Programs

3

Two-Stage Stochastic Linear Programs

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Outline

1

Short Reviews

Probability Spaces and Random Variables Convex Analysis

2

Deterministic Linear Programs

3

Two-Stage Stochastic Linear Programs

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Probability Spaces

A probability space is the triplet (Ω, A, P), where

Ω: set of all random outcomes (example: six possible results of throwing die)

A: collection of random events, where events are subsets of Ω (example: odd numbers, ≤ 4)

Mapping P : A → [0, 1] with P(∅) = 0, P(Ω) = 1 and

P(A

1

∪ A

₂

) = P(A

1

) + P(A

2

) if A

1

∩ A

₂

= ∅

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Distributions

Cumulative distribution function of a random variable ξ is defined as F

ξ

(x ) = P(ξ ≤ x )

For discrete random variables, probability mass distribution f is defined as f (ξ

^k

) = P(ξ = ξ

^k

), k ∈ K with P

k ∈K

f (ξ

^k

) = 1 For continuous random variables, density function f is defined by P(a ≤ ξ ≤ b) = R

b

a

f (ξ)d ξ = R

b

a

dF (ξ) with R

∞

−∞

dF (ξ) = 1

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Expectation, Variance, Quantiles

The expectation of a random variable is µ = P

k ∈K

ξ

^k

f (ξ

^k

) (discrete) or R

∞

−∞

ξdF (ξ) (continuous)

The variance of a random variable is E[(ξ − µ)

²

] The rth moment of ξ is ¯ ξ

^{(r )}

= E[ξ

^r

]

The α-quantile of ξ is a point η such that for 0 < α < 1,

η = min{x |F (x ) ≥ α}

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Convex Set, Extreme Point, Convex Hull

A convex combination of points x

i

, i = 1, . . . , n, is a point P

n

i=1

λ

_i

x

_i

where P

n

i=1

λ

_i

= 1, λ

_i

≥ 0, i = 1, . . . , n

X is a convex set if it contains any convex combination of points x

i

∈ X

Extreme point of a convex set: a point which cannot be

expressed as the convex combination of two distinct points in the set

Convex hull of a set of points: the set of all convex combinations

of these points

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Affine Sets and Hyperplanes

h(x ) = 0 is an affine constraint if it can be expressed as h(x ) = Ax − b

Affine space: the set h(x ) = 0

Each set h

_i

(x ) = A

_i·

x − b = 0 is a hyperplane (where A

_i·

is the i-th row of A)

Ax = 0 is the parallel subspace

The dimension of an affine space is the dimension of its parallel

subspace

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Convex Functions

f is a convex function if for all 0 ≤ λ ≤ 1 and any x

₁

, x

₂

we have f (λx

₁

+ (1 − λ)x

₂

) ≤ λf (x

₁

) + (1 − λ)f (x

₂

)

The domain of f , dom f is the set where f is finite

f is concave if −f is convex

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Epigraph

The epigraph of f is the set epi (f ) = {(x , β)|β ≥ f (x )}

f is convex if and only if its epigraph is convex

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Piecewise Linearity and Separability

A continuous function f is piecewise linear if it is linear over regions defined by linear inequalities

A function is separable when it can be written as f = P

n i=1

f

_i

(x )

Both of these properties can be exploited in computation

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Outline

1

Short Reviews

Probability Spaces and Random Variables Convex Analysis

2

Deterministic Linear Programs

3

Two-Stage Stochastic Linear Programs

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Deterministic Linear Programs

min z = c

^T

x s.t. Ax = b x ≥ 0

x ∈ R

ⁿ

, c ∈ R

ⁿ

, A ∈ R

^m×n

, b ∈ R

^m

A solution is a vector x such that Ax = b A feasible solution is a solution with x ≥ 0

An optimal solution is a feasible solution x

^?

such that

c

^T

x

^?

≤ c

^T

x for all feasible solutions x

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Linear Programming Definitions

A basis is a choice of n linearly independent columns of A, with A = [B, N] (after rearrangement)

Associated with a basis, we have a basic solution x

_B

= B

⁻¹

b, x

N

= 0, z = c

_B^T

B

⁻¹

b

Basic feasible solutions correspond to extreme points of {x|Ax = b, x ≥ 0}

Condition for basis to be feasible: B

⁻¹

b ≥ 0

optimal: c

^T_N

− c

^TB

B

⁻¹

N ≥ 0 (can you prove this?)

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Dual Problem

max π

^T

b s.t. π

^T

A ≤ c

^T

Unbounded dual ⇒ infeasible primal Unbounded primal ⇒ infeasible dual Primal and dual can be infeasible

If x is primal feasible and π is dual feasible, c

^T

x ≤ π

^T

b There exists primal optimal solution x

^?

⇔ there exists dual optimal solution π

^?

Strong duality: c

^T

x

^?

= (π

^?

)

^T

b

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Linear Programming Duality Mnemonic Table

Primal Minimize Maximize Dual Constraints ≥ b

_i

≥ 0 Variables

≤ b

_i

≤ 0

= b

_i

Free

Variables ≥ 0 ≤ c

_j

Constraints

≤ 0 ≥ c

_j

Free = c

_j

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Outline

1

Short Reviews

Probability Spaces and Random Variables Convex Analysis

2

Deterministic Linear Programs

3

Two-Stage Stochastic Linear Programs

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2-Stage Stochastic Linear Programs

Stochastic linear programs: linear programs with uncertain data

Recourse programs: some decisions (recourse actions) can be taken after uncertainty is disclosed

First-stage decisions: decisions taken before uncertainty is revealed

Second-stage decisions: decisions taken after uncertainty is revealed

Sequence of events: x → ξ(ω) → y (ω, x )

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Sequence of Events

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Mathematical Formulation

min z = c

^T

x + E

ξ

[min q(ω)

^T

y (ω)]

s.t. Ax = b

T (ω)x + Wy (ω) = h(ω) x ≥ 0, y (ω) ≥ 0

First stage decisions x ∈ R

ⁿ¹

, c ∈ R

ⁿ¹

, b ∈ R

^m¹

, A ∈ R

^m¹^×n¹

For a given realization ω, second-stage data are q(ω) ∈ R

ⁿ²

, h(ω) ∈ R

^m²

, T (ω) ∈ R

^m²^×n¹

All random variables of the problem are assembled in a single

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Example: Capacity Expansion Planning

min

x ,y ≥0 n

X

i=1

(I

_i

· x

_i

+ E

^ξ

m

X

j=1

C

_i

· T

_j

· y

_ij

(ω))

s.t.

n

X

i=1

y

ij

(ω) = D

j

(ω), j = 1, . . . , m

m

X

j=1

y

_ij

(ω) ≤ x

_i

, i = 1, . . . n

I

_i

, C

_i

: fixed/variable cost of technology i

D

_j

(ω), T

_j

: height/width of load block j

y

_ij

(ω): capacity of i allocated to j

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Example: Procrastination

x ,y ≥0

max c

₁

x + E

ξ

c

₂

y (ω) s.t. x + y (ω) ≤ T x + y (ω) ≥ T /2 y (ω) ≤ N(ω)

x , y (ω): amount of work before/after T

c

₁

, c

₂

: quality increase per day for work prior to/after deadline

T : deadline

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Deterministic Equivalent Program

Let Q(x , ξ(ω)) = min

_y

{q(ω)

^T

y |Wy = h(ω) − T (ω)x , y ≥ 0} be the second stage value function

Let V (x ) = E

^ξ

Q(x , ξ(ω)) be the expected second-stage value function

The deterministic equivalent program is min z = c

^T

x + V (x ) s. t. Ax = b

x ≥ 0

If V (x ) is given, the problem is just a linear program (we will see why

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Convexity of Q(x , ξ(ω))

Q(x , ξ(ω)) is a convex function of x

Proof: If y

₁

is optimal reaction to x

₁

and y

₂

is optimal reaction to x

₂

, then λy

1

+ (1 − λy

2

) is feasible reaction to λx

1

+ (1 − λx

2

)

What can we say about V (x )?

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Making Constraints Implicit

Q(x , ξ) can be represented as

Q(x , ξ) = min

_y

{q(ω)

^T

y + δ(y |Y (x , ξ))}, where Y (x , ξ) = {y ≥ 0 : T (ω)x + Wy = h(ω)}

and δ(·|·) is an indicator function:

δ(y |Y ) =

( 0 if y ∈ Y

+∞ otherwise

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Some Notations and Terminology

K

1

= {x : Ax = b, x ≥ 0}

K

2

(ξ) = {x : ∃y , T (ω)x + Wy (ω) = h(ω), y (ω) ≥ 0}

K

2

= {x : V (x ) < ∞}

Relatively complete recourse: K

1

⊂ K

₂

Complete recourse: posW = R

^m²

, where posW = {Wy : y ≥ 0}

Fixed recourse: W does not depend on ω

Does example 1 have complete recourse? fixed recourse?

Does example 2 have complete recourse? fixed recourse?

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