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An Adaptive Radial Basis Algorithm (ARBF) for Mixed-Integer Expensive Constrained Global Optimization137Solve this problem formulating the global non-convex optimization problem:IS2. The maximal distance ∆ from any point in the feasible region Ω to the closest samplepointminx,∆−∆x L ≤ x ≤ x Us/t 0 ≤ ∆ ≤ ∞0 ≤ ||x − x i || 2 − ∆ 2 ≤ ∞, i = 1, ..., n.(10)3.2 The Global Search (G-step) in ARBFTo find a good point first a sequence of global optimization problems are solved. If this phasefails, the global solution x Globs nof the RBF surface is considered. If rejected, the point mostdistance to all the sampled points, as well as to the boundary, is computed by solving a globaloptimization problem.G-step Phase 1. For a sequence of decreasing target values f ∗ n k, solve the global optimizationproblemg n (x k g n) =minx∈Ωgiving the global minimum g n (x k g n) for each f ∗ n k. Findf ˆkn =µ n (x) [ s n (x) − f ∗ n k] 2(11)minα(x k gn ))=0 f ∗ n kIf f ˆkn is non-empty, pick the global minimum corresponding to this target value, xˆkgn, as theG-step. Otherwise turn to G-step Phase 2. If α(x k g n)) = d during the search, there is no pointin decreasing f ∗ n kfurther, and the sequence is interrupted. If a minimum with α(x k g n)) = 0 isfound, a refined set of target values are tried in order to try to decrease the target values f ∗ n kfurther.G-step Phase 2. Select the global RBF solution x Globs nvalue and sufficiently far from currently best point x Min . If α(x Globs n0.01 max(1, f Min ) and ‖x GlobG-step Phase 3.s n− x Min ‖ > 0.2∆ pick x Globs nif it is interior, with sufficiently low) = 0, f Min − s n (xs Globn) ≥as the G-step point. Otherwise turn toG-step Phase 3. Find the point which have the maximal distance to any sample point as wellas the boundary, solving the global optimization problemδ IP =s/tminx ∈ Ω, δ−δx Li ≤ x i − δ/d , i = 1, ..., n.x i + δ/d ≤ x Ui , i = 1, ..., n.0 ≤ δ ≤ ∆0 ≤ ||x − x i || 2 − δ 2 ≤ ∞ , i = 1, ..., n.(12)3.3 The Global-Local Search (GL-step) in ARBFTo find a good point first a sequence of global optimization problems are solved. If this phasefails, the global solution x Globs nof the RBF surface is considered. If rejected, the InfStep point= −∞) is used.(f ∗ n