NOZZLE and MULSTREG: numerical optimization tools for energy industry - black-box optimization for clean energy technologies

In this work, we study how to exploit Derivative-Free Optimization and Black-Box Optimization in the design and validation phases of a cooling system in a gas turbine. Firstly, we define NOZZLE, a numerical model of a section of the cooling system, and we use an optimization method to obtain an efficient design; secondly, we develop MUℓSTREG: an optimization method to enhance the validation procedures of an entire cooling system. NOZZLE is a Black-Box function that simulates an impingement cooling system for a turbine nozzle starting from a well-known model that correlates the design features of the cooling system with efficiency parameters. The optimization model is defined as a mixed-variable constrained BBO problem and we numerically illustrate how to use DFO algorithms to find a reference solution that is useful for practitioners. MULSTREG is a new multilevel stochastic framework for the solution of optimization problems where the value of the objective function is noisy. We focus on data-fitting problems with random uncertainty as in the validation phase of a complete cooling system in a turbine. The proposed approach uses random regularized first-order models that exploit a hierarchical description of the problem, being either in the variable space or in the function space, allowing different levels of accuracy for the objective function. The convergence analysis of the method is conducted and its numerical behavior is tested on finite-sum minimization problems. The multilevel framework is tailored to the solution of such problems resulting in fact in a nontrivial variance reduction technique with adaptive step-size that outperforms standard approaches when solving nonconvex problems. Our stochastic method does not require the finest approximation to coincide with the original objective function. This allows us to avoid the evaluation of the full sum in finite-sum minimization problems, opening to the solution of large classification problems.