Stochastic Nonlinear Programming with Linear Constraints by MonteCarlo Estimators

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Stochastic Nonlinear Programming with Linear
Constraints by Monte-Carlo Estimators
Leonidas Sakalauskas Institute of Mathematics and
Informatics Akademijos st 4, 08663 Vilnius,
Lithuania Fax[phone] E-mail
ltsakal_at_ktl.mii.lt
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OUTLINE

• Introduction
• Unconstrained nonlinear stochastic optimization
  by Monte-Carlo method
• Optimization with nonlinear stochastic
  constraints
• - feasible solutions for linearly constrained
  stochastic problems
• Application for portfolio optimization
• Conclusions

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1. Introduction

• We consider nonlinear stochastic programming
  problem
• with linear constraints
• 
  (1)
• ,
  ,
• P is absolute continuous with density
  ,
• , is
  a bounded set
• , is the
  matrix

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• The methods of stochastic approximation were
  first proposed to solve stochastic optimization
  problems
• Robins-Monro (1951), Mikhalevitch et al (1987),
  Kushner (1997), Han-Fu-Chen (2002), Ermoliev et
  al (2003), etc.
• Problems
• the rate of convergence of stochastic
  approximation slows down for constrained problems
  (Polyak (1987), Uriasyev (1990))
• the gradient - type projection method, usually
  applied here, can no converge when constraints
  are linear due to zigzagging or jamming (see
  Bertsekas (1982), Polyak (1987), etc.)
• procedures for optimality testing arent
  developed, etc.

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• Application of Monte-Carlo method, frequently
  applied to stochastic problems (Prekopa (1999),
  Ermoliev et al (2003), etc,), is based on
  replacement of the objective function, being
  mathematical expectation, by Monte-Carlo
  estimators (see, e.g. Shapiro (1989), etc.)
• Problems
• choice of the sample size N
• numerical complexity
• optimality testing and accuracy of solution
• zigzagging or jamming

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2. Unconstrained nonlinear stochastic
Optimization by Monte-Carlo method
2.1. Stochastic differentiation
(Rubinstein (1983), Prekopa (1999), Ermolyev et
al (2003), Uriasyev (1994), etc.)
,
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Assume possibilities I) to simulate
Monte-Carlo samples
II) to compute vectors
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III) Denote sampling statistics
Remark. Statistics are asymptotically normal !!!
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2.2. Optimality testing

• if the hypothesis of equality of gradient to
  zero is rejected

• or the confidence interval of the objective
  function
• is longer than admissible value

there is no a reason to accept the optimality of
decision
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Testing functions
subject to
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Fig 1. Probability to accept the solution on
distance r,
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2.3. Optimization by Monte-Carlo estimators
Convergence a.s. of such procedure to the optimal
solution is proved with linear rate by martingale
approach (Sakalauskas, Informatica, 2001) Since
the sample size can be taken small at first
iterations (about 20-100) adjustment of sample
size enables us essentially to decrease the
volume of computations needed to get the optimal
decision with admissible accuracy.
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Due to linear rate of convergence
Thus, if we have a certain resource to compute
one value of the objective or constraint function
with an admissible accuracy, then the
optimization requires in fact only several times
more computations.
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2.4. The General Algorithm.
and initial step size
be given,
Step 0. Let initial decision
of size
Step 1. Simulate the sample
Step 2. To test the hypothesis of equality of
gradient to zero
Step 3. To test the confidence interval of the
objective function
Step 4.
Step 5. Otherwise, terminate the algorithm
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3. Optimization with nonlinear stochastic
constraints3.1. Problem statement
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3.2. Lagrange method by Monte-Carlo estimators
Sakalauskas, EJOR (2002)
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3.3. Optimality testing
Together with an hypothesis of zero Lagrange
function gradient and admissible confidence
interval of the objective function
a validity and the confidence interval of the
constraint are tested
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Fig. 2. Change of the objective, constraint,
sample size, solution
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4. - feasible solutions for linearly
constrained stochastic problems
4.1. The feasible set
The necessary optimality condition
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4.2. The - feasible set
where
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4.2. Optimization by - feasible solutions
and Monte-Carlo method
(2)
is
- feasible gradient projection
Sakalauskas, Informatica (2004)
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Theorem 1. Let the function
be differentiable,
the gradient of this function be Lipshitzian with
the constant
Assume, the set
to be bounded and having more than one element,
- matrix.
,
is the
Then, starting from any initial approximation
,
formulae (2) define the sequence
so that
and there exist values
such that
a.s.
for
.
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4.3. Application to portfolio problem
Financial planning in the case of uncertainty is
frequently can be reduced to stochastic nonlinear
optimization with linear Constraints (D.Duffie
and J.Pan (1997), R.Mansini et al (2003)). Let
us to consider an application of developed
approach to optimization of portfolio of the
Lithuanian Stock Market with n4 securities
(Table 1). We make the analysis for daily
returns of the following assets ENRG joint
stock company Lietuvos energija (power
industry) MAZN - joint stock company Mazeikiu
Nafta (oil refinery) ROKS - joint stock company
Rokiskio suris (dairy products) RST - joint
stock company Rytu skirstomieji tinklai

(power industry)
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Table 1
The problem is to maximize a probability of
portfolio return to exceed the desired threshold
R under the assumption of lognormal distribution
of assets
subject to a simple set of constitutional
constraints
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Table 2
results of optimization







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5. Conclussion

• The numerical methodology for stochastic
  nonlinear
• programming by Monte-Carlo estimators has been
• developed, based on gradient-type algorithms.
• The methodology distinguishes itself by
  peculiarities
• the optimality of the solution is tested with
  respect to
• statistical criteria
• the Monte-Carlo sample size is adjusted in an
  iterative way so as to guarantee the convergence
  to optimal solution with linear rate and to
  estimate the objective function with an
  admissible confidence after a finite number of
  series.
• The method of -feasible approximations for
  stochastic
• nonlinear problems with linear constraints has
  been
• developed as a framework of general method.

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THANK YOU !