Enabling gradient-based optimization methods in problems with unreliable or absent derivatives

In this thesis, we focus on problems in which the derivative of the objective function is either unavailable or unreliable, which can occur in a variety of situations including the presence of legacy codes (codes written in the past but not maintained), problems of parameter tuning for simulation or optimization algorithms and engineering problems where the objective functions are the output of black-box simulation software. Despite the absence or the unreliability of the derivatives, our interest is in the resolution of the optimization problem using gradient-based methods, which take advantage of the rich and relevant information normally included in the gradient of the objective function. We address the lack of derivatives considering two different scenarios. In the first one, we consider smooth problems with additive noise affecting objective function evaluations. We assume that objective function evaluations can be obtained in a cheap and fast way and we focus on gradient approximation methods that use objective function evaluations to somehow filter the noise and build an estimate of the gradient. In the second scenario, we consider potentially non-smooth simulation-based optimization problems in which neither the objective function nor its (eventual) derivative have an explicit expression. Assuming the expensiveness of the evaluations of objective functions, we enable the usage of gradient-based methods by following an approach that is based on the creation of a neural network model that replaces the simulation software used for computing the objective function. In this way, the smooth function obtained with the neural network model and its gradient are considered in the optimization procedure.