An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices

Abstract

The classical αBB method is a global optimization method the important step of which is to determine a convex underestimator of an objective function on an interval domain. Its particular point is to enclose the range of a Hessian matrix in an interval matrix. To have a tighter estimation of the Hessian matrices, we investigate a linear parametric form enclosure in this paper. One way to obtain this form is by using a slope extension of the Hessian entries. Numerical examples indicate that our approach can sometimes significantly reduce overestimation on the objective function. However, the slope extensions highly depend on a choice of the center of linearization. We compare some naive choices and also propose a heuristic one, which performs well in executed examples, but it seems there is no one global winner.

A modification piecewise convexification method with a classification strategy for box-constrained non-convex optimization programs

“…Up to now, most global approximation algorithms have been developed such as convex relaxation-based method; see [5,6,7,8,9,10,11]. In particular, the αBB method, which is one of the global optimization methods and is based on the idea of the convex relaxation, plays a substantial role in the design of efficient and computationally tractable numerical algorithms for non-convex optimization problems; see, e.g., [12,13,14,15]. It is worth remarking that these algorithms generally aim at finding a single globally optimal solution, but most application problems may exist with many or even an infinite number of globally optimal solutions; see, e.g., [16,17].…”

A Modification Piecewise Convexification Method for Box-Constrained Non-Convex Optimization Programs

“…So far, the majority of global approximation algorithms are designed, see, [9,10,14,20,11,16,18,19]. In particular, the αBB method has become an increasingly important global optimization method in the design of efficient and computationally tractable numerical algorithms for non-convex optimization problems, see, [3,12,6,8]. It is worth noticing that these algorithms aim in general at determining a single globally optimal solution, and the majority of application problems may be existed with many or even infinite of many globally optimal solutions.…”