ΑΒΒ

αΒΒ is a second-order deterministic global optimization algorithm for finding the optima of general, twice continuously differentiable functions. The algorithm is based around creating a relaxation for nonlinear functions of general form by superposing them with a quadratic of sufficient magnitude, called α, such that the resulting superposition is enough to overcome the worst-case scenario of non-convexity of the original function. Since a quadratic has a diagonal Hessian matrix, this superposition essentially adds a number to all diagonal elements of the original Hessian, such that the resulting Hessian is positive-semidefinite. Thus, the resulting relaxation is a convex function.

Theory
Let a function $${f(\boldsymbol{x}) \in C^2}$$ be a function of general non-linear non-convex structure, defined in a finite box $$X=\{\boldsymbol{x}\in \mathbb{R}^n:\boldsymbol{x}^L\leq\boldsymbol{x}\leq\boldsymbol{x}^U\}$$. Then, a convex underestimation (relaxation) $$L(\boldsymbol{x})$$ of this function can be constructed over $$X$$ by superposing a sum of univariate quadratics, each of sufficient magnitude to overcome the non-convexity of $${f(\boldsymbol{x})}$$ everywhere in $$X$$, as follows:


 * $$L(\boldsymbol{x})=f(\boldsymbol{x})+\sum_{i=1}^{i=n}\alpha_i(x_i^L - x_i)(x_i^U - x_i)$$

$$L(\boldsymbol{x})$$ is called the $$\alpha \text{BB}$$ underestimator for general functional forms. If all $$\alpha_i$$ are sufficiently large, the new function $$L(\boldsymbol{x})$$ is convex everywhere in $$X$$. Thus, local minimization of $$L(\boldsymbol{x})$$ yields a rigorous lower bound on the value of $${f(\boldsymbol{x})}$$ in that domain.

Calculation of $$\boldsymbol{\alpha}$$
There are numerous methods to calculate the values of the $$\boldsymbol{\alpha}$$ vector. It is proven that when $$\alpha_i=\max\{0,-\frac{1}{2}\lambda_i^{\min}\}$$, where $$\lambda_i^{\min}$$ is a valid lower bound on the $$i$$-th eigenvalue of the Hessian matrix of $${f(\boldsymbol{x})}$$, the $$L(\boldsymbol{x})$$ underestimator is guaranteed to be convex.

One of the most popular methods to get these valid bounds on eigenvalues is by use of the Scaled Gerschgorin theorem. Let $$H(X)$$ be the interval Hessian matrix of $${f(X)}$$ over the interval $$X$$. Then, $$\forall d_i>0$$ a valid lower bound on eigenvalue $$\lambda_i$$ may be derived from the $$i$$-th row of $$H(X)$$ as follows:


 * $$\lambda_i^{\min}=\underline{h_{ii}}-\sum_{i\neq j}(\max(|\underline{h_{ij}}|,|\overline{h_{ij}}|\frac{d_j}{d_i})$$