Multi-stage stochastic optimization and approximations ...homepage.univie.ac.at/anna.timonina/website/docs/ISMP2012.pdf · ug)Multi-stage stochastic optimization and approximations

Multi-stage stochastic optimization andapproximations with applications

Anna Timonina, M.Sc.

University of Vienna,Abraham Wald PhD Program in Statistics and Operations Research.

Supervisor Prof. Georg Pflug

Anna Timonina, M.Sc. (University of Vienna, Abraham Wald PhD Program in Statistics and Operations Research.Supervisor Prof. Georg Pflug)Multi-stage stochastic optimization and approximations with applications1 / 21

Initial Problem Description

maxA[H1(x0, ξ1), ...,HT (xT−1, ξT );F ] : x / F , x ∈ X, ξ / F,

A(·, ·) is a multi-period acceptability functional (E,AV@R...);Ht(x t , ξt+1) are (intermediate) profit functions;ξ = (ξ1, ..., ξT ) is a stochastic process on (Ω,F = (F1, ...,FT ),P),where F is a filtration.x = (x0, ..., xT−1) are the decision functions;x / F is a non-anticipativity condition (means that xt is measurable w.r.t.Ft for all t).That is a variational problem that could be solved only in very specialcases.


Problem Approximation

maxA[H1(x0, ξ1), ...,HT (xT−1, ξT );F ] : x / F , x ∈ X, ξ / F,

How to solve this problem though in general it is unsolvable?We approximate this problem by a simpler one:

maxA[H1(x0, ξ1), ...,HT (xT−1, ξT ); F ] : x / F , x ∈ X, ξ / F,

that is a tree structured problem of huge but finite dimension, whereΩ is a finite probability space;F is a finite filtration;x is a high dimensional vector.


Filtration as a tree

ξ = (ξ1, ..., ξT ) a stochastic process defined on (Ω,F ,P);

ξ = (ξ1, ..., ξT ) a finite valued scenario process defined on (Ω, F , P).

It is not natural to construct approximations on the same probabilityspace, because concrete decision problems are given without reference on aprobability space, they are given in distributional setups.As the stochastic process ξ is given by its probability distribution only, weneed to approximate this distribution by discrete one, i.e. to generatepoints from this distribution.Hence, our aim is

1 To generate points from the given distrubution (using differentquantization algorithms);

2 To solve the multi-stage stochastic optimization program using thegenerated points.

For these purposes we represent stochastic process ξ = (ξ1, ..., ξT ) as afinitely valued tree.


Sample tree

Figure: High bushiness v.s. Low bushiness

The larger the bushiness of the tree, the closer the approximatedistribution to the true one and, hence, the closer the solution of theapproximate problem to the true solution.


Types of distances and Quantization types


The Kantorovich and the nested distances


Distance between stochastic process and a tree

Suppose that the tree is given: the tree structure, the probabilities of thenodes pn,i and the values sitting on the nodes yn,i (n is the predecessornumber and 1 ≤ i ≤ s(n), where s(n) is a total number of successors ofn). There is a stochastic process ξ = (ξ1, ..., ξT ) with a known distributionfunction F .The upper boundfor the distance can be defined as

d22 (ξ, tree) ≤

T∑t=2

∑n∈Nt

E[||ξt−yn,t ||22|ξt−1 ∈ Bt−1]P(ξt ∈ Bt),

where Bt = ξt : l t ≤ ξt ≤ utis the Voronoi diagram.Let pn,i =

∑ij=1 pn,j . For the

one-dimensional case we can find

bn,i = F−1t (b1,i1−1, b1,i1 , ..., bnt−1,it−1−1, bnt−1,it−1 ; pn,i ).


Convergence of the distance

This figure shows, that the distance between normal distributed stochasticprocess and the tree with Pages numbers and with random generatednumbers sitting on the nodes decreases when the bushiness of the treeincreases:


Optimal quantization

Optimal quantization [1, Gilles Pages, Huyen Pham] means:

1 to find optimal supportingpoints zi , i = 1, ...,N(z1 ≤ z2 ≤ ... ≤ zN):

minz=(z1,...,zN)

∫mins

d(x , zs)rdP(x)

2 given thesupporting points zi , to findthe probabilities pi , such that

dKA(P, P)→ min


QOND quantization

Consider the nested distance between stochastic process and a tree

d22 (ξ, tree) ≤

T∑t=2

∑n∈Nt

E[||ξt − yn,t ||22|ξt−1 ∈ Bt−1]P(ξt ∈ Bt),

where yn,t are the unknown values on the nodes of the tree.Obviously, if yn,t = ξt the distance equals to zero and, hence, minimized.


QOND quantization v.s. optimal quatization

If we fix the structure of the tree and find values and probabilities of thenodes with a use of optimal quantization and QOND quantizationalgorithms for the known distribution we can see that the distance betweenthese trees converges to some value greater than zero

If we increase the bushiness of the tree we receive the convergence of thedistance to 0 for both algorithms of quantization, that means the greaterthe bushiness the closer the trees to each other.



Example: inventory control problem

The grocery shop has to place regular orders one period ahead.

ξ1, ..., ξT is the demand for goods of the shop at times t = 1, ...,T ;

xt−1, t = 1, ...,T is an order of goods one period ahead;

1− lt is a storage loss;

rapid orders are possible for a price of ut > 1 per piece;

The selling price is st (st > 1) and the final inventory KT has a valuelTKT .

E[T∑t=1

(stξt − xt−1 − utMt) + lTKT ]→ maxx

subject to xt / Ft , t = 1, ...,T ,

lt−1Kt−1 + xt−1 − ξt = Kt −Mt .

Kt ≥ 0,Mt ≥ 0.


Comparison of the optimal and random log-normalsampling with the theoretical solution

If we consider the demand ξ ∼ lnN (µ,C ), we are able to calculate theexact analytical solution (violet color in the figure), and compare it withresults obtained by random generation (boxplot) and optimal quantization(green) and to see the convergence for the increasing bushiness:


Example: risk-management of CAT-events (IIASA’sproject)


Problem description

S0

zk,0, C0

D0

S1

zk,1, C1, D0i0

zk,0ξk,1, D1

... St

zk,t , Ct ,∑

j Dj ij

zk,t−1ξk,t , Dt

... ST

CT ,∑

j Dj

zT−1ξk,T

x0 x1 xt−1 xt xT−1

Risk-neutral:

(1− α)T∑t=1

ρ−tE(ct) + αE(ST )→ maxdt ,ct ,zt ,xt

Risk-averse:

(1− α)T∑t=1

ρ−tE(u(ct)) + αE(u(ST ))→ maxdt ,ct ,zt ,xt

, with u(x) =xγ − 1

γ


CAT-Modelling

a. b.

Figure: Mongolia and Mexico optimal strategy for 3 years


Future research

1 To calculate the rate of convergence theoretically and numerically;

2 To study how to find the optimal tree structure for the stochasticprocess given by the distribution;

3 To calculate the optimal tree structure for each of the examples;

4 To investigate some new applications.


Bibliography

Gilles Pages, Huyen Pham, “Optimal quantization methods andapplications to numerical problems in finance” University of Paris.,2003.

Georg Pflug, Alois Pichler, “A distance for multistage stochasticoptimization models,” SIAM Journal on Optimization, 22., 2011.

Cedric Villani, “Topics in Optimal Transportation, volume 58 ofGraduate Studies in Mathematics,” American Mathematical Society.,2003.

Georg Pflug, Alois Pichler, “Approximations for ProbabilityDistributions and Stochastic Optimization Problems,” Springer NewYork., 2011.

R.S. Lipster, A.N. Shiryayev, “Statistics of Random Processes,Applications” Springer., 1978, 339 pp.


Thank you for your attention!


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Multi-stage stochastic optimization and approximations ...homepage.univie.ac.at/anna.timonina/website/docs/ISMP2012.pdf · ug)Multi-stage stochastic optimization and approximations