(183n) Global Optimization of a Class of Black-Box Problems with Bounded Hessian
Constructing a valid underestimator is a major impediment for applying branch and bound algorithm on black-box problems. Note that the lower bound at a node is calculated by minimizing the relaxed problem. Therefore, an underestimator whose minima can be found by performing finite number of evaluations is desirable for it to be used in global optimization of black-box problems.
To this end, edge-concave underestimator is an attractive alternative  because its minima lie at one of the corner points. To determine the underestimator, the upper bound of the diagonal elements of the hessian of the original function are needed. Therefore, if we assume to know the maximum of the upper bound of the diagonal elements of the hessian matrix, we can evaluate the underestimator at different x values. When we evaluate the edge-concave underestimator at all the corner points, then the minimum of these points is a valid lower bound of the original function.
In this work, we present a global optimization scheme based on a branch and bound framework. At each node, a valid lower bound is found using the scheme from previous paragraph and BOBYQA  is used to find the upper bound. The algorithm is implemented on problems listed in [8 (Table 3)] and all the problems converge to Îµ-global optimality. For many problems, the number of function evaluations and the nodes explored are economical. For instance, for problem 1 (n=5), the number of nodes explored and function evaluations to obtain global minima are 8 and 240 respectively. For problem 4 (n=5), the number of nodes explored and required function evaluations are 44 and 970 respectively. The algorithm will be implemented on several problems and compared with other DFO solvers to illustrate its effectiveness.
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