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    The basic elements of the method (presented in  [6]) are easily summarized. The shape design problem  (before starting the optimization phase)  is considered affected by uncertainty:  i.e.  the optimal  design in,  obviously,  unknown  and a  uniform probability of  occurrence in the design space  is assumed. The design space  is  then sampled randomly: a number S of geometries or shapes 1 {}Skjs  are generated (but no objective function evaluation  is performed!) by using any arbitrary geometry deformation techniques (we have been using a Free Form Deformation technique, preferred since it allows high design flexibility and  it is independent of grid topology) to sample the space.  KLE is then used to find the so called  principal  direction  of this design space.  The  average geometry  or shape, s,  is defined as: 11/Sj js S s   and  any geometry in the space can be representeed as: 1Kkk ks s z    (the  Karhunen–Lòeve  expansion). kz  are the principal directions, defined as the solutions of the eigenproblem: k k kRz z  , where ( ) /TR GG S , with 1 [ ,..., ] S G s s s s   . 3 G KN is the actual dimension of the design space, and G N  is the number of grid points used to discretize the hull of the sampled geometries.  The eigenvalues k represent the geometric variance associated with the corresponding eigenvector kz and are used to assess  the total geometric variance and build  a reduced-dimensionality space.  The generic geometry or shape is then defined as:    defining the geometrical variance of the design space.  In conclusion, by taking a random set of designs, KLE provides us  the principal directions of the design space (eigenvectors) with  the associated geometric variance (eigenvalues). New designs are generated by linear combination of principal geometries. According to KLE theory, no greater geometric variance can be retained by any other linear expansion of order n. 3.2 New Development in Global Optimization, Derivative-Free Algorithms Here is briefly described an evolutionary type, derivative free, global optimization method, the Deterministic Particle Swarm Optimization  (DPSO)  [7], where substantial modifications to the basic algorithm  were introduced with respect to the original  PSO  version. The  fundamental DPSO steps are here briefly and can be summarized as follows:  Step 0. (Initialization) Distribute a set of particles inside the design space with deterministic distribution and given initial velocities. Set  the iteration index 0 i .  Step 1  (Analysis) Set 1 ii . For each particle of the swarm evaluate the objective function, identify the minimum value in the swarm ijg, and the minimum value encountered by the single particle in its own history ijp. Step 2.(Velocity vector) For each particle calculate a velocity vector jv  using the particle's memory and the knowledge gained by the swarm:     112i i i i i ij j j j jv wv c p x c g x                   (8) Here, χ  is a  limit  factor introduced to limit the maximum speed of the particles, w  is the inertia weight,  c1  and  c2  are positive parameters (cognitive  and  social  respectively), and ijx  is the position of the j-th  particle at the step i. Step 3. (Updating) Update the position of each particle xj using the velocity vector and previous position: 11 i i ij j jx x v; Step 4. (Stopping criterion) Go to Step 1 and repeat until some convergence criterion is satisfied.  This basic algorithm has been very recently modified [8], integrating the global phase with a line search-based derivative-free method  (local  phase), providing  a robust method to force the convergence of  a  subsequence  of points toward a stationary point, which satisfies first order optimality conditions for the objective function.  The method  (LS-DF  PSO, [8]),  extends  the DPSO scheme with a line search-based method. Specifically, a Positively Spanning Set (PSS) is used, where the set of search directions () D is defined by  the unit vectors ke , 1, , kn , as shown in the following equation and in Figure 1. 0 1 0 1,,,1 0 1 0D                                
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