Stochastic optimization programs

Stochastic optimization methods described previously, such as simulated annealing, can also be used to solve the general nonlinear programming problem. These have the advantage that the search is sometimes allowed to move uphill in a minimization problem, rather than always searching for a downhill move. Or, in a maximization problem, the search is sometimes allowed to move downhill, rather than always searching for an uphill move. In this way, the technique is less vulnerable to the problems associated with local optima. 

One of the approaches that can be used in design is to carry out structural and parameter optimization of a superstructure. The structural optimization required can be carried out using mixed integer linear programming in the case of a linear problem or mixed integer nonlinear programming in the case of a nonlinear problem. Stochastic optimization can also be very effective for structural optimization problems. 

Gorelick, S. M. (1990). Large scale nonlinear deterministic and stochastic optimization Formulations involving simulation of subsurface contamination. Mathematical Programming, 48, 19-39. 

Cheema, J.J.S., Sankpal, N.V., Tambe, S.S. and Kulkami, B.D. 2002. Genetic Programming Assisted Stochastic Optimization Strategies for Optimization of Glucose to Gluconic Acid Fermentation. Biotechnol. Progr., 18, 1356-1365. 

Dynamic programming (DP) is an approach for the modeling of dynamic and stochastic decision problems, the analysis of the structural properties of these problems, and the solution of these problems. Dynamic programs are also referred to as Markov decision processes (MDP). Slight distinctions can be made between DP and MDP, such as that in the case of some deterministic problems the term dynamic programming is used rather than Markov decision processes. The term stochastic optimal control is also often used for these types of problems. We shall use these terms synonymously. 

A basic assumption of stochastic programming is that the probability distribution of the random variable is known. The target then is to find an optimal solution that makes the expected value of the system to be minimum (or maximum). According to the type of the objective function and constraints, the stochastic programming problem can be divided into stochastic linear programming problems and stochastic nonlinear programming problems. 

Evolutionary programming (EP), created by Lawrence J. Fogel in 1960, is a stochastic optimization strategy similar to genetic algorithms and evolution... 

The deterministic equivalent program of a multistage stochastic optimization can be formulated in a similar manner to the program (A.22). However, special care should be taken to preserve the so called non-anticipativity principle in such cases. 

Whatever model is used to describe an operations research problem, be it a differential equation, a mathematical program, or a stochastic process, there is a natural tendency to seek a maximum or a minimum with a certain purpose in mind. Thus, one often finds optimization problems imbedded in the models of operations research. 

---