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An Adaptive Radial Basis Algorithm (ARBF) for Mixed-Integer Expensive Constrained Global Optimization135Table 1.Different choices of Radial Basis Functions.RBF φ(r) > 0 p(x) mcubic r 3 a T · x + b 1thin plate spline r 2 log r a T · x + b 1linear rpb 0multiquadric(r 2 + γ 2 ) -Gaussian exp(−γr 2 ) -2.1 The Radial Basis Algorithm (RBF)Find initial set of n ≥ d + 1 sample points x using experimental design.Compute costly f(x) for initial set of n points, best point (x Min , f Min ).Use the n sampled points to build a smooth RBF interpolation model (surrogate model,response surface model) as an approximation of the f(x) surface.Iterate until f Goal , known goal for f(x), achieved, n > n Max , MaxCycle iteration cycleswith no progress or maximal CPU time used.– Find minimum of RBF surface, s n (x sn ) = minx∈Ω s n(x).– In every iteration in sequence pick one of the N + 2 step choices.1. Cycle step −1 (InfStep). Set target value fn ∗ = −∞, i.e. solve the globalproblemgn∞ = min µ n(x), (7)x∈Ωwhere the coefficient µ n (x) is an estimate of the new λ-coefficient if the trial xis included in the RBF interpolation2. Cycle step k = 0, 1, ..., N − 1 (Global Search).(Define target)value fn ∗ ∈(−∞, s n (x sn )] as fn(k) ∗ = s n (x sn ) − w k · max f(x i ) − s n (x sn ) , with w k =i(1 − k/N) 2 or w k = 1 − k/N. Solve the global optimization problemg n (x k g n) =minx∈Ωµ n (x) [s n (x) − f ∗ n (k)]2 . (8)3. Cycle step N (Local search).If s n (x sn ) < f Min − 10 −6 |f Min |, accept x sn as the new trial point.Otherwise set f ∗ n(k) = f Min − 10 −2 |f Min | and solve (8).– Compute and validate new (x, f(x)), increase n.– Update (x Min , f Min ) and compute RBF surface.2.2 Properties of the basic RBF algorithmGutmann had the view that the global optimum of the surface, x sn would be close to thecurrent best point x Min . However, as seen in practice, this is not case. In cycle step N, x snshould give local convergence. If the point is far from x Min , this is not true. In rbfSolve alocal optimization solver (SNOPT or NPSOL) has been used, however in almost all cases ithas found the global minimum, not the closest local minimum.Define the number of active variables α(x) as the number of coefficients in the point x thathas components close to the bounds in the box, i.e