Monte-Carlo method for Two-Stage

SLP

Lecture 5

Leonidas SakalauskasInstitute of Mathematics and InformaticsVilnius, Lithuania <sakal@ktl.mii.lt>

EURO Working Group on Continuous Optimization

Content

Introduction

Monte Carlo estimators

Stochastic Differentiation

-feasible gradient approach for two-stage SLP

Interior-point method for two stage SLP

Testing optimality

Convergence analysis

Counterexample

Two-stage stochastic optimization

problem

min],|[min)(

m

y RyhxTyWyqExcxF

,, nxbAx

assume vectors q, h and matrices W, T random in general.

Two-stage stochastic optimization

problem

Say, vector h can be distributed multivariate normally:

here are correspondingly the vector of means and the covariance matrix

),( SN

S,

Two-stage stochastic optimization

problem

the random vector Z distributed with respect to is simulated as

is standard normal vector,

R is triangle matrix that

(Choletzky factorization)

),( SN

RZ T

SRRT

subject to the feasible set

nRxbxAxD ,

where

Two-stage stochastic optimization problem withcomplete recourse will be

( ) ( , ) minnx D

F x c x E Q x

],|[min),( my RyhxTyWyqxQ

It can be derived under the assumption on the existence of a solution at the second stage and continuity of measure P, that the objective function is smoothly differentiable and the gradient is

),()( xEgxFx

where

is given by the set of solutions of the dual problem

*),( uTcxg

],0|)[(max)( * sTT

u

T RuqWuuxThuxTh

Monte-Carlo samples

We assume here that the Monte-Carlo sample of a certain size N is provided for any

),,...,,( 21 NyyyY

the sampling estimator of the objective function

and sampling variance can be computed

1

1( ) ( , )

Nj

j

F x f x yN

2

2

1

1( ) ( , ) ( )

1

Nj

j

D x f x y F xN

nDx

The gradient is evaluated using the same random sample:

1

1( ) ( , ),

Nj

j

g x g x yN

nRDx

Gradient

Covariance matrix

1

1( ) , ,

N Tj j

j

A x g x y g x g x y g xN n

We use the sampling covariance matrix

later on for normalising the gradient estimator.

– feasible direction approach

Let us define the set of feasible directions as follows:

1( ) 0, 0, 0n

i n j jV x g Ag g if x

Gradient projection

Denote, as projection of vector g onto the set U.

Since the objective function is differentiable, the solution

is optimal if

x D

0V

F x

Ug

Gradient projection

the projection of G onto the set

is

where is projector

0Ax

PGG T

A

AAAAIP TT 1

Assume a certain multiplier to be given. Define the function by

0

)(: xVx

Thus, , when

for any

x g D ( ),x g

0,

1

ˆ( ) min , min( )j

j

xg j

j n

xg

g

01 jnj g

,g V x x D

Now, let a certain small value be given.0

Then we introduce the function

, ,

, if

and define the ε - feasible set

: ( )x V x

jj

gnj

x gxg

j

ˆ,minmaxˆ)(

01

01 jnj g

0)( gx )0(1 jnj g

1( ) 0, 0, 0 ( )n

i n j j xV x g Ag g if x g

The starting point can be obtained as the solution of the deterministic linear problem:

The iterative stochastic procedure of gradient search could be used further:

where is the step-length multiplier and

is the projection of gradient estimator to the ε -feasible set.

0 0

,( , ) arg min[ | , , , ].m n

x yx y c x q y A x b W y T x h y R x R

1 ( )t t t tx x G x

( )t ttx

G

( )t t

tV xG G x

Monte-Carlo sample size problem

There is no a great necessity to compute estimators with a high accuracy on starting the optimisation, because then it suffices only to approximately evaluate the direction leading to the optimum.

Therefore, one can obtain not so large samples at the beginning of the optimum search and, later on, increase the size of samples so as to get the estimate of the objective function with a desired accuracy just at the time of decision making on finding the solution to the optimisation problem.

We propose a following version for regulating the sample size:

maxmin1

1 ,,)(

~())(()(

~(

),,(maxmin NNn

xGxAxG

nNnFishnN tttTt

t

t

t

tt

Statistical testing of the optimality

hypothesis

The optimality hypothesis could be accepted for somepoint with significance , if the followingcondition is sattisfied

Next, we can use the asymptotic normality again anddecide that the objective function is estimated with apermissible accuracy , if its confidence bound doesnot exceed this value:

tt NxD /)(~

1

tx

),,()(~)())(~()( 1

2

t

t

t

t

ttTt

t

t

nNnFishn

xgxAxgnNT

Computer simulation

Two-stage stochastic linear optimisation problem.

Dimensions of the task are as follows:

the first stage has 10 rows and 20 variables;

the second stage has 20 rows and 30 variables.

http://www.math.bme.hu/~deak/twostage/ l1/20x20.1/

(2006-01-20).

Two stage stochastic

programing

The estimate of the optimal value of the objective function given in the database is 182.94234 0.066

N0=Nmin=100 Nmax=10000. Maximal number of iterations , generation of

trials was broken when the estimated confidence interval of the objective function exceeds admissible value .

Initial data were as follows:

= =0.95; 0.99, 0.1; 0.2; 0.5; 1.0.

max 100t

Frequency of stopping under

admissible interval

0

20

40

60

80

100

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

1

0,5

0,2

0,1

Change of the objective function

under admissible interval

182

182,5

183

183,5

184

184,5

1 12 23 34 45 56 67 78 89 100

0,1

0,2

0,5

1

Change of confidence interval

under admissible interval

0

1

2

3

4

5

6

7

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99

0,1

0,2

0,5

1

Change of the Monte-Carlo sample

size under admissible interval

0

200000

400000

600000

800000

1000000

1200000

1400000

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

0,1

0,2

0,5

1

Hotelling statistics under

admissible interval

0

1

2

3

4

5

6

7

8

9

10

1 11 21 31 41 51 61 71 81 91

0,1

0,2

0,5

1

Histogram of ratio under

admissible interval 1

jt

tj

N

N

0

5

10

15

20

25

30

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

0,1

0,2

0,5

1

Wrap-Up and Conclisions

The stochastic adaptive method has been developed to solve stochastic linear problems by a finite sequence of Monte-Carlo sampling estimators

The method is grounded by adaptive regulation of the size of Monte-Carlo samples and the statistical termination procedure, taking into consideration the statistical modeling accuracy

The proposed adjustment of sample size, when it is taken inversely proportional to the square of the norm of the Monte-Carlo estimate of the gradient, guarantees the convergence a. s. at a linear rate