The problem

The goal of the proposed algorithm is to find a vector of model parameters ${\bf x}$ such that we minimize Kelley (1999)  

$$\mbox{ min } f({\bf x})\mbox{ subject to }{\bf x}\in\Omega,$$ (1)

where

$${\bf x}\in\Omega = \{ {\bf x} \in \Re^N\mid l_i\leq x_i\leq u_i\},$$ (2)

with l_i and u_i being the lower and upper bounds for the model x_i, respectively. In this case, l_i and u_i are called simple bounds. They can be different for each point of the model space. The model vector that obeys equation (1) is called ${\bf x^*}$.

The sets of indices i for which the ith constraint are active/inactive are called the active/inactive sets A(x)/I(x). Most of the algorithms used to solve bound constrained problems first identify A(x) and then solve the minimization problem for the free variables of I(x).