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promote the selection of isolated individuals along the front and thus to ensure a maximum of diversity. Once each individual has been given a pseudo-fitness value, it is selected for reproduction using a proportional selection operator; finally, the individuals retained in the mating pool are submitted to crossover and mutation in order to explore the research space and yield a front of rank 1 closer and closer to the global Pareto front of the problem. The main tuning parameters of the algorithm are the population size, the number of generations, the crossover and mutation probabilities pc, pm and the so-called sharing parameter σ used to take into account the relative isolation of an individual along a dominance front. Typical values for pc, pm are respectively 0.9 and 0.1; values of σ are retained following a formula given in [78] which takes into account the population size and the number of objectives. Theoretically, the population size and the number of generations should be chosen according to the number of parameters and objectives of the optimization problem under study; in practice however, the population size and number of generations are fixed by the global amount of CPU time devoted to the computation : taking too small a population may rapidly lead to a local optimum and make useless the iteration process on the generation number; reversely, a large population will impose a limited number of generations which could not allow the population to evolve sufficiently towards the optimum. 3.2. Gradient Based Methods For computing the BFGS algorithm, the multi-objective design environment (mode)FRONTIER software (produced by ESTECO s.r.l.) has been used. The classical Broyden-Fletcher-Goldfar-Shanno (BFGS) optimization algorithm is a variable metric method, i.e., a so-called quasi-Newton method. The function is locally approximated with a quadratic form, function of the gradient and of a suitable approximation of the Hessian. For a quadratic function the Newton Method gives the solution for a minimum. The Hessian matrix is not used, but instead an approximation of it is built up iteratively, making use of the information about the function and the gradient values in the successive iterated points: so the “quasi” in the name of the method. Once the Newton step is computed, the BFGS algorithm performs a linear search along that direction, starting from the full step, and then if the decrease in the objective function is not satisfactory, trying iteratively a shorter and shorter step until a satisfactory point is found. In this way, if the 42PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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