Simultaneous Perturbation Stochastic Approximation Algorithm Combined with Neural Network and Fuzzy Simulation

Many optimization problems contain fuzzy information. Possibility theory [1] has been well developed and applied to this kind of optimization problems [2—5]. Fuzzy programming is an important tool to handle the optimization problems, which usually includes three types of models: fuzzy expected value model [6] , fuzzy chance-constrained programming model [7,8] , and fuzzy dependent-chance programming model [9]. Fuzzy programming models can be converted into crisp equivalents in some special cases, and then solved by the classical algorithms for crisp programming models. However, it is difficult for complex fuzzy programming models. To solve fuzzy programming models, hybrid intelligent algorithms (HIAs) were designed [6 ü 9]. The HIAs first generate a set of training input-output data for the functions with fuzzy variables by fuzzy simulation, then train a neural network (NN) to approximate the functions according to the generated training data, and finally embed the trained NN in genetic algorithm (GA). GA is used to search the optimal solution in the entire space. The HIAs are feasible in solving fuzzy programming models. However, GA is time-consuming in finding the global optimum, especially in solving large-dimensional optimization problems. Therefore, it is necessary to seek out other optimization methods to solve fuzzy programming models. Usually, one would always need the global optimal solution. However, in practice this solution is not often available and one must be satisfied with obtaining a local optimal solution. In a lot of practical optimization problems, a local optimal solution is a fully acceptable solution for the resources available (human time, money, material, computer time, etc.) to be spent on the optimization. In fuzzy environments, it is difficult to directly obtain the value of objective function, especially the gradient of the function with respect to the decision vector. Stochastic approximation (SA) algorithm uses the (possibly noisy) measurements of objective function to approximate the gradient, which was first applied to finding extrema of functions