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$$\alpha BB$$: A global optimization method for general constrained nonconvex problems. (English) Zbl 0846.90087
Summary: A branch-and-bound global optimization method, $$\alpha BB$$, for general continuous optimization problems involving nonconvexities in the objective function and/or constraints is presented. The nonconvexities are categorized as being either of special structure or generic. A convex relaxation of the original nonconvex problem is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, fractional, signomial) with customized tight convex lower bounding functions and (ii) by utilizing the $$\alpha$$ parameter as defined by the second and the third author [J. Global Optim. 4, No. 2, 135-170 (1994; Zbl 0797.90114)] to underestimate nonconvex terms of generic structure. The proposed branch-and-bound type algorithm attains finite $$\varepsilon$$-convergence to the global minimum through the successive subdivision of the original region and the subsequent solution of a series of nonlinear convex minimization problems. The global optimization method, $$\alpha BB$$, is implemented in $$C$$ and tested on a variety of example problems.

##### MSC:
 90C26 Nonconvex programming, global optimization 90C30 Nonlinear programming
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