Please provide us with some information on the structure of your stochastic optimization problem. Otherwise, it is difficult to give a quality answer to your question.

Dear Hermann,

I do not know why the structure of the problem is important for answering. However, all parameters of the problem have a probability distribution function. Moreover, the decision variables are also stochastic and random. The time horizon of the problem is not multi period.

Structure: Is the model linear or nonlinear in the variables and parameters, are there constraints? Is the objective function an expected value, or does it include terms depending on the variation, what is its dimension, are the variables continuous or discrete, etc.

Thanks Hermann,

The model is Mixed Integer Programming (MIP). In addition, the objective function can contain expected value and variance together, or only expected value. I do not know this feature exactly now.

I also edited the question.

decomposition with branch and cut (Sherali and Sen, Math. Programming, 2006), http://link.springer.com/content/pdf/10.1007/s10107-005-0592-5.pdf

there are branch and price approaches (Lulli and Sen, 2004), http://www.jstor.org/stable/30046118

what is the dimension of your problem?

The problem has O(n4 ) variables and O(n4) constraints.

Your description of the problem

" Moreover, the decision variables are also stochastic and random. The time horizon of the problem is not multi period."

is somehow confusing.

- What do you mean by stochastic decision variables? Does that mean deciding one variable in your problem is choosing a distribution function? Or can the decision variables be chosen dependinmg on the outcome of the data? This would mean a wait-and-see-approach.

- If it is not just wait-and-see: The problem being not multi-period, does that mean your model does not include corrective second-stage (or multi-stage) variables (rcourse actions)?

In this case - what does feasibility mean, if all your variables do have to be chosen fixed beforehand? Optimality in solme sense requires feasibility as far as I understand the subject (that's why the presupposition of complete recourse).

Dear Ralf

I mean the optimal results depends on the particular realization of the random event. I need to consider variables for each possible realization.

My problem time horizon is not multiple. It means the optimized solution is valid for just one time horizon such as one year or month.

Dear Hossein,

from your explanation I understand that all your variables could be chosen in dependence of the realization of the data, which I regard a wait-and-see problem. This means in principle that you could solve the problem after knowing the data.

I cannot see how expected value or variance this way can enter the objective. Your solutions, as you describe it, should be dependent on the current realization, not on all possible ones. This way decomposition of a two- or multi-stage stochastic problem does not help wrt. the thing you are interested in.

A possible question might be: what is the distribution of the optimal value / optimal solution set depending on the distributions of the data. I am not aware whether there are known answers to this type of question.

If you want to solve yout problem for all possible realizations of the data means the same as being able to solve the full-parametric optimization problem. To solve the abovementioned distribution problem the best would be availability of an expression for the solution in an analytic way, but I do not know a possibility of doing this. at least for nonlinear problems.

Addendum: The notion of a two- or multistage stochastic problem is not the time horizon of a dynamic problem, though in many cases the choice of the stages is influenced by the time, since it is assumed that the data realizations reveal over time. In these problems it is assumed that a part of the variables has to be decided without any knowledge of the realizazion and afterwards, as part of the data realizations is known, more variables have to be decided - every time assuming the knowledge of the data realiszations already revealed, but in a nonanticipative way (i.e. without sipposed knowledge of the other data) just based on knowledge of the possible scenarios.

Thus your problem does not belong to that class of problems and none of the methods for two- or multistage stochastic optimization is applicable.