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A reformulation framework for global optimization. (English) Zbl 1277.90102
Summary: We present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions, two versions of the so-called $$\alpha$$BB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem forms a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.

MSC:
 90C26 Nonconvex programming, global optimization 90C11 Mixed integer programming
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