Download - Monte-Carlo method for Two-Stage SLP
Monte-Carlo method for Two-Stage
SLP
Lecture 5
Leonidas SakalauskasInstitute of Mathematics and InformaticsVilnius, Lithuania <[email protected]>
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