Global optimization has found widespread applications in chemical engineering . Deterministic global optimization techniques attempt to determine an approximate optimal solution within a specified tolerance and terminate with a certificate of its optimality in finite time . The performance of branch-and-bound algorithms for deterministic global optimization is strongly dependent on the ability to construct tight and rapidly convergent convex relaxations of nonconvex functions.
Recently, Bompadre and coworkers [3, 4] analyzed the convergence order of convex relaxations based on McCormick, Taylor, and McCormick-Taylor models. They showed that if a function is in C2, the scheme of relaxations corresponding to its convex and concave envelopes is at least second-order pointwise convergent which, in turn, implies Hausdorff convergence of at least second-order. Establishing that a scheme of relaxations is at least second-order Hausdorff convergent is important from many viewpoints, notably in mitigating the so-called cluster problem in unconstrained problems [5-7]. A similar analysis of clustering and convergence order for constrained optimization algorithms is lacking.
In this work, we investigate the convergence order of branch-and-bound algorithms in global optimization by extending the convergence analysis of Bompadre and coworkers to constrained problems. Specifically, we propose a definition of the convergence order for constrained optimization problems, and analyze the convergence order of commonly used branch-and-bound schemes.
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