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A Global Optimization Method, αBB, for General Twice-Differentiable Constrained NLPs: I - Theoretical Advances
, 1997
"... In this paper, the deterministic global optimization algorithm, αBB, (α-based Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twice-differentiable NLPs. The key idea is the constru ..."
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Cited by 41 (2 self)
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In this paper, the deterministic global optimization algorithm, αBB, (α-based Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twice-differentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...
A Global Optimization Method, alphaBB, for General Twice-Differentiable Constrained NLPs: II - Implementation and Computational Results
"... Part I of this paper (Adjiman et al., 1997) described the theoretical foundations of a global optimization algorithm, the ffBB algorithm, which can be used to solve problems belonging to the broad class of twice-differentiable NPLs. For any such problem, the ability to automatically generate progres ..."
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Cited by 6 (2 self)
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Part I of this paper (Adjiman et al., 1997) described the theoretical foundations of a global optimization algorithm, the ffBB algorithm, which can be used to solve problems belonging to the broad class of twice-differentiable NPLs. For any such problem, the ability to automatically generate progressively tighter convex lower bounding problems at each iteration guarantees the convergence of the branchand -bound ffBB algorithm to within ffl of the global optimum solution. Several methods were presented for the construction of convex valid underestimators for general nonconvex functions. In this second part, the performance of the proposed algorithm and its alternative underestimators is studied through their application to a variety of problems. An implementation of the ffBB is described and a number of rules for branching variable selection and variable bound updates are shown to enhance convergence rates. A user-friendly parser facilitates problem input and provides flexibility in the...
