Nonextensive Statistical Mechanics

7.4 Computer Sciences 269Fig. 7.64 Quantitative comparison between the skewed implied volatilities obtained from a set ofMicrosoft options traded on November 19, 2003, and the theoretical model with q = 1.4, whichfits well the returns distribution of the underlying stock (from [594]).7.4 Computer Sciences7.4.1 Optimization AlgorithmsGlobal optimization consists in numerically finding a global minimum of a given(not necessarily convex) cost/energy function, defined in a continuous D-dimensionalspace. Such algorithms have a plethora of useful applications. A well-known classicalprocedure, referred to as the Boltzmann machine, is the so-called SimulatedAnnealing, introduced in 1983 [623], which visits phase-space with a Gaussian distribution.A few years later, a faster procedure, referred to as the Cauchy machine becauseitvisitsphase-spaceusingaCauchy–Lorentzdistribution,wasintroduced[624].Finally, inspired by q-statistics, an algorithm was introduced [625,626], named GeneralizedSimulated Annealing (GSA), which recovers the two just mentioned ones asparticular cases.GSA consists, like the Boltzmann and Cauchy machines, of two algorithms thatare to be used with alternation. These are the Visiting algorithm and the Acceptance