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In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters. When the parameters are known only within certain bounds, one approach to tackling such problems is called robust optimization. Here the goal is to find a solution which is feasible for all such data and optimal in some sense. Stochastic programming models are similar in style but take advantage of the fact that probability distributions governing the data are known or can be estimated. The goal here is to find some policy that is feasible for all (or almost all) the possible data instances and maximizes the expectation of some function of the decisions and the random variables. More generally, such models are formulated, solved analytically or numerically, and analyzed in order to provide useful information to a decision-maker.[1]

As an example, consider two-stage linear programs. Here the decision maker takes some action in the first stage, after which a random event occurs affecting the outcome of the first-stage decision. A recourse decision can then be made in the second stage that compensates for any bad effects that might have been experienced as a result of the first-stage decision. The optimal policy from such a model is a single first-stage policy and a collection of recourse decisions (a decision rule) defining which second-stage action should be taken in response to each random outcome.

Stochastic programming has applications in a broad range of areas ranging from finance to transportation to energy optimization.[2][3] This article includes an example of optimizing an investment portfolio over time.

Contents

Two-stage problemsEdit

The basic idea of two-stage stochastic programming is that (optimal) decisions should be based on data available at the time the decisions are made and cannot depend on future observations. The two-stage formulation is widely used in stochastic programming. The general formulation of a two-stage stochastic programming problem is given by:

 
where   is the optimal value of the second-stage problem
 

The classical two-stage linear stochastic programming problems can be formulated as

 

where   is the optimal value of the second-stage problem

 

In such formulation   is the first-stage decision variable vector,   is the second-stage decision variable vector, and   contains the data of the second-stage problem. In this formulation, at the first stage we have to make a "here-and-now" decision   before the realization of the uncertain data  , viewed as a random vector, is known. At the second stage, after a realization of   becomes available, we optimize our behavior by solving an appropriate optimization problem.

At the first stage we optimize (minimize in the above formulation) the cost   of the first-stage decision plus the expected cost of the (optimal) second-stage decision. We can view the second-stage problem simply as an optimization problem which describes our supposedly optimal behavior when the uncertain data is revealed, or we can consider its solution as a recourse action where the term   compensates for a possible inconsistency of the system   and   is the cost of this recourse action.

The considered two-stage problem is linear because the objective functions and the constraints are linear. Conceptually this is not essential and one can consider more general two-stage stochastic programs. For example, if the first-stage problem is integer, one could add integrality constraints to the first-stage problem so that the feasible set is discrete. Non-linear objectives and constraints could also be incorporated if needed.[4]

Distributional assumptionEdit

The formulation of the above two-stage problem assumes that the second-stage data   can be modeled as a random vector with a known probability distribution (not just uncertain). This would be justified in many situations. For example,   could be information derived from historical data and the distribution does not significantly change over the considered period of time. In such situations one may reliably estimate the required probability distribution and the optimization on average could be justified by the law of large numbers. Another example is that   could be realizations of a simulation model whose outputs are stochastic. The empirical distribution of the sample could be used as an approximation to the true but unknown output distribution.

DiscretizationEdit

To solve the two-stage stochastic problem numerically, one often needs to assume that the random vector   has a finite number of possible realizations, called scenarios, say  , with respective probability masses  . Then the expectation in the first-stage problem's objective function can be written as the summation:

 
and, moreover, the two-stage problem can be formulated as one large linear programming problem (this is called the deterministic equivalent of the original problem, see section § Deterministic equivalent of a stochastic problem).

When   has an infinite (or very large) number of possible realizations the standard approach is then to represent this distribution by scenarios. This approach raises three questions, namely:

  1. How to construct scenarios, see § Scenario construction;
  2. How to solve the deterministic equivalent. Optimizers such as CPLEX, GLPK and Gurobi can solve large linear/nonlinear problems. The NEOS Server,[5] hosted at the University of Wisconsin, Madison, allows free access to many modern solvers. The structure of a deterministic equivalent is particularly amenable to apply decomposition methods,[6] such as Benders' decomposition or scenario decomposition;
  3. How to measure quality of the obtained solution with respect to the "true" optimum.

These questions are not independent. For example, the number of scenarios constructed will affect both the tractability of the deterministic equivalent and the quality of the obtained solutions.

Stochastic linear programEdit

A stochastic linear program is a specific instance of the classical two-stage stochastic program. A stochastic LP is built from a collection of multi-period linear programs (LPs), each having the same structure but somewhat different data. The   two-period LP, representing the   scenario, may be regarded as having the following form:

 

The vectors   and   contain the first-period variables, whose values must be chosen immediately. The vector   contains all of the variables for subsequent periods. The constraints   involve only first-period variables and are the same in every scenario. The other constraints involve variables of later periods and differ in some respects from scenario to scenario, reflecting uncertainty about the future.

Note that solving the   two-period LP is equivalent to assuming the   scenario in the second period with no uncertainty. In order to incorporate uncertainties in the second stage, one should assign probabilities to different scenarios and solve the corresponding deterministic equivalent.

Deterministic equivalent of a stochastic problemEdit

With a finite number of scenarios, two-stage stochastic linear programs can be modelled as large linear programming problems. This formulation is often called the deterministic equivalent linear program, or abbreviated to deterministic equivalent. (Strictly speaking a deterministic equivalent is any mathematical program that can be used to compute the optimal first-stage decision, so these will exist for continuous probability distributions as well, when one can represent the second-stage cost in some closed form.) For example, to form the deterministic equivalent to the above stochastic linear program, we assign a probability   to each scenario  . Then we can minimize the expected value of the objective, subject to the constraints from all scenarios:

 

We have a different vector   of later-period variables for each scenario  . The first-period variables   and   are the same in every scenario, however, because we must make a decision for the first period before we know which scenario will be realized. As a result, the constraints involving just   and   need only be specified once, while the remaining constraints must be given separately for each scenario.

Scenario constructionEdit

In practice it might be possible to construct scenarios by eliciting experts' opinions on the future. The number of constructed scenarios should be relatively modest so that the obtained deterministic equivalent can be solved with reasonable computational effort. It is often claimed that a solution that is optimal using only a few scenarios provides more adaptable plans than one that assumes a single scenario only. In some cases such a claim could be verified by a simulation. In theory some measures of guarantee that an obtained solution solves the original problem with reasonable accuracy. Typically in applications only the first stage optimal solution   has a practical value since almost always a "true" realization of the random data will be different from the set of constructed (generated) scenarios.

Suppose   contains   independent random components, each of which has three possible realizations (for example, future realizations of each random parameters are classified as low, medium and high), then the total number of scenarios is  . Such exponential growth of the number of scenarios makes model development using expert opinion very difficult even for reasonable size  . The situation becomes even worse if some random components of   have continuous distributions.

Monte Carlo sampling and Sample Average Approximation (SAA) MethodEdit

A common approach to reduce the scenario set to a manageable size is by using Monte Carlo simulation. Suppose the total number of scenarios is very large or even infinite. Suppose further that we can generate a sample   of   replications of the random vector  . Usually the sample is assumed to be independent and identically distributed (i.i.d sample). Given a sample, the expectation function   is approximated by the sample average

 

and consequently the first-stage problem is given by

 

This formulation is known as the Sample Average Approximation method. The SAA problem is a function of the considered sample and in that sense is random. For a given sample   the SAA problem is of the same form as a two-stage stochastic linear programming problem with the scenarios  .,  , each taken with the same probability  .

Statistical inferenceEdit

Consider the following stochastic programming problem

 

Here   is a nonempty closed subset of  ,   is a random vector whose probability distribution   is supported on a set  , and  . In the framework of two-stage stochastic programming,   is given by the optimal value of the corresponding second-stage problem.

Assume that   is well defined and finite valued for all  . This implies that for every   the value   is finite almost surely.

Suppose that we have a sample   of  realizations of the random vector  . This random sample can be viewed as historical data of   observations of  , or it can be generated by Monte Carlo sampling techniques. Then we can formulate a corresponding sample average approximation

 

By the Law of Large Numbers we have that, under some regularity conditions   converges pointwise with probability 1 to   as  . Moreover, under mild additional conditions the convergence is uniform. We also have  , i.e.,   is an unbiased estimator of  . Therefore, it is natural to expect that the optimal value and optimal solutions of the SAA problem converge to their counterparts of the true problem as  .

Consistency of SAA estimatorsEdit

Suppose the feasible set   of the SAA problem is fixed, i.e., it is independent of the sample. Let   and   be the optimal value and the set of optimal solutions, respectively, of the true problem and let   and   be the optimal value and the set of optimal solutions, respectively, of the SAA problem.

  1. Let   and   be a sequence of (deterministic) real valued functions. The following two properties are equivalent:
    • for any   and any sequence   converging to   it follows that   converges to  
    • the function   is continuous on   and   converges to   uniformly on any compact subset of  
  2. If the objective of the SAA problem   converges to the true problem's objective   with probability 1, as  , uniformly on the feasible set  . Then   converges to   with probability 1 as  .
  3. Suppose that there exists a compact set   such that
    • the set   of optimal solutions of the true problem is nonempty and is contained in  
    • the function   is finite valued and continuous on  
    • the sequence of functions   converges to   with probability 1, as  , uniformly in  
    • for   large enough the set   is nonempty and   with probability 1
then   and   with probability 1 as  . Note that   denotes the deviation of set   from set  , defined as
 

In some situations the feasible set   of the SAA problem is estimated, then the corresponding SAA problem takes the form

 

where   is a subset of   depending on the sample and therefore is random. Nevertheless, consistency results for SAA estimators can still be derived under some additional assumptions:

  1. Suppose that there exists a compact set   such that
    • the set   of optimal solutions of the true problem is nonempty and is contained in  
    • the function   is finite valued and continuous on  
    • the sequence of functions   converges to   with probability 1, as  , uniformly in  
    • for   large enough the set   is nonempty and   with probability 1
    • if   and   converges with probability 1 to a point  , then  
    • for some point   there exists a sequence   such that   with probability 1.
then   and   with probability 1 as  .

Asymptotics of the SAA optimal valueEdit

Suppose the sample   is i.i.d. and fix a point  . Then the sample average estimator  , of  , is unbiased and has variance  , where   is supposed to be finite. Moreover, by the central limit theorem we have that

 

where   denotes convergence in distribution and   has a normal distribution with mean   and variance  , written as  .

In other words,   has asymptotically normal distribution, i.e., for large  ,   has approximately normal distribution with mean   and variance  . This leads to the following (approximate)  % confidence interval for  :

 

where   (here   denotes the cdf of the standard normal distribution) and

 

is the sample variance estimate of  . That is, the error of estimation of   is (stochastically) of order  .

Multistage portfolio optimizationEdit

The following is an example from finance of multi-stage stochastic programming. Suppose that at time   we have initial capital   to invest in   assets. Suppose further that we are allowed to rebalance our portfolio at times   but without injecting additional cash into it. At each period   we make a decision about redistributing the current wealth   among the   assets. Let   be the initial amounts invested in the n assets. We require that each   is nonnegative and that the balance equation   should hold.

Consider the total returns   for each period  . This forms a vector-valued random process  . At time period  , we can rebalance the portfolio by specifying the amounts   invested in the respective assets. At that time the returns in the first period have been realized so it is reasonable to use this information in the rebalancing decision. Thus, the second-stage decisions, at time  , are actually functions of realization of the random vector  , i.e.,  . Similarly, at time   the decision   is a function   of the available information given by   the history of the random process up to time  . A sequence of functions  ,  , with   being constant, defines an implementable policy of the decision process. It is said that such a policy is feasible if it satisfies the model constraints with probability 1, i.e., the nonnegativity constraints  ,  ,  , and the balance of wealth constraints,

 

where in period   the wealth   is given by

 

which depends on the realization of the random process and the decisions up to time  .

Suppose the objective is to maximize the expected utility of this wealth at the last period, that is, to consider the problem

 

This is a multistage stochastic programming problem, where stages are numbered from   to  . Optimization is performed over all implementable and feasible policies. To complete the problem description one also needs to define the probability distribution of the random process  . This can be done in various ways. For example, one can construct a particular scenario tree defining time evolution of the process. If at every stage the random return of each asset is allowed to have two continuations, independent of other assets, then the total number of scenarios is  

In order to write dynamic programming equations, consider the above multistage problem backward in time. At the last stage  , a realization   of the random process is known and   has been chosen. Therefore, one needs to solve the following problem

 

where   denotes the conditional expectation of   given  . The optimal value of the above problem depends on   and   and is denoted  .

Similarly, at stages  , one should solve the problem

 

whose optimal value is denoted by  . Finally, at stage  , one solves the problem

 

Stagewise independent random processEdit

For a general distribution of the process  , it may be hard to solve these dynamic programming equations. The situation simplifies dramatically if the process   is stagewise independent, i.e.,   is (stochastically) independent of   for  . In this case, the corresponding conditional expectations become unconditional expectations, and the function  ,   does not depend on  . That is,   is the optimal value of the problem

 

and   is the optimal value of

 

for  .

Biological applicationsEdit

Stochastic dynamic programming is frequently used to model animal behaviour in such fields as behavioural ecology.[7][8] Empirical tests of models of optimal foraging, life-history transitions such as fledging in birds and egg laying in parasitoid wasps have shown the value of this modelling technique in explaining the evolution of behavioural decision making. These models are typically many-staged, rather than two-staged.

Economic applicationsEdit

Stochastic dynamic programming is a useful tool in understanding decision making under uncertainty. The accumulation of capital stock under uncertainty is one example; often it is used by resource economists to analyze bioeconomic problems[9] where the uncertainty enters in such as weather, etc.

Software toolsEdit

Modelling languagesEdit

All discrete stochastic programming problems can be represented with any algebraic modeling language, manually implementing explicit or implicit non-anticipativity to make sure the resulting model respects the structure of the information made available at each stage. An instance of an SP problem generated by a general modelling language tends to grow quite large (linearly in the number of scenarios), and its matrix loses the structure that is intrinsic to this class of problems, which could otherwise be exploited at solution time by specific decomposition algorithms. Extensions to modelling languages specifically designed for SP are starting to appear, see:

  • AIMMS - supports the definition of SP problems
  • EMP SP (Extended Mathematical Programming for Stochastic Programming) - a module of GAMS created to facilitate stochastic programming (includes keywords for parametric distributions, chance constraints and risk measures such as Value at risk and Expected shortfall).
  • SAMPL - a set of extensions to AMPL specifically designed to express stochastic programs (includes syntax for chance constraints, integrated chance constraints and Robust Optimization problems)

They both can generate SMPS instance level format, which conveys in a non-redundant form the structure of the problem to the solver.

See alsoEdit

ReferencesEdit

  1. ^ Shapiro, Alexander; Dentcheva, Darinka; Ruszczyński, Andrzej (2009). Lectures on stochastic programming: Modeling and theory (PDF). MPS/SIAM Series on Optimization. 9. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Mathematical Programming Society (MPS). pp. xvi+436. ISBN 978-0-89871-687-0. MR 2562798.
  2. ^ Stein W. Wallace and William T. Ziemba (eds.). Applications of Stochastic Programming. MPS-SIAM Book Series on Optimization 5, 2005.
  3. ^ Applications of stochastic programming are described at the following website, Stochastic Programming Community.
  4. ^ Shapiro, Alexander; Philpott, Andy. A tutorial on Stochastic Programming (PDF).
  5. ^ http://www.neos-server.org/neos/
  6. ^ Ruszczyński, Andrzej; Shapiro, Alexander (2003). Stochastic Programming. Handbooks in Operations Research and Management Science. 10. Philadelphia: Elsevier. p. 700. ISBN 978-0444508546.
  7. ^ Mangel, M. & Clark, C. W. 1988. Dynamic modeling in behavioral ecology. Princeton University Press ISBN 0-691-08506-4
  8. ^ Houston, A. I & McNamara, J. M. 1999. Models of adaptive behaviour: an approach based on state. Cambridge University Press ISBN 0-521-65539-0
  9. ^ Howitt, R., Msangi, S., Reynaud, A and K. Knapp. 2002. "Using Polynomial Approximations to Solve Stochastic Dynamic Programming Problems: or A "Betty Crocker " Approach to SDP." University of California, Davis, Department of Agricultural and Resource Economics Working Paper.

Further readingEdit

External linksEdit