The Role of the Off-diagonal Elements of the Hessian Matrix in the Construction of Tight Convex Underestimators for Nonconvex Functions

In this paper, we propose a new method that produces new forms of tight convex underestimators for twice continuously differentiable nonconvex functions. The algorithm generalizes the main ideas used in aBB. The key idea is to determine a new convexification function that is able to handle the off-diagonal elements of the Hessian matrix of the original nonconvex function. The new convexification function is based on that used in aBB, but it is enhanced with extra convex parametric terms. The Hessian matrix of the new convexification function is a constant non-diagonal matrix. The values of the parameters are determined in such a way that the effect of the off-diagonal elements in the overall underestimating function is minimized. As a result, the new underestimator is tighter than that produced by aBB. We discuss the theoretical properties of the new underestimator and we present several illustrative examples where we demonstrate the improvements of the new convex underestimators over those used in the original aBB method.