The Effect of Hessian Evaluations in the Global Optimization {\alpha}BB Method

We consider convex underestimators that are used in the global optimization {\alpha}BB method and its variants. The method is based by augmenting the original nonconvex function by a relaxation term that is derived from an interval enclosure of the Hessian matrix. In this paper, we discuss the advantages of symbolic computation of the Hessian matrix. Symbolic computation often allows simplifications of the resulting expressions, which in turn means less conservative underestimators. We show by examples that even a small manipulation with the symbolic expressions, which can be processed automatically by computers, can have a large effect on the quality of underestimators.Comment: 11 pages, 6 figure

Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

In this paper, we propose two algorithms for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs). However, it is often too demanding to find exact solutions of SDPs. Our first approach does not rely on solving SDPs but on approximately following {a path leading to a solution, which is formed on the intersection of the semi-infinite region and the interior of the semi-definite region. We show weak* convergence of this method to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions and further provide with sufficient conditions for strong convergence. Moreover, as the second method, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with Monteiro-Zhang scaling technique into the first method. We particularly prove two-step superlinear convergence of the second method using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs obtained by finite relaxation of the SISDP