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Finding a model with simple bounds

In this section, I present the problem of finding simple bounds and the method that solves it.

The goal of bound-constrained optimization is to find a vector of model parameters ${\bf m}$ such that we minimize  
\mbox{ min } f({\bf m})\mbox{ subject to }{\bf m}\in\Omega,\end{displaymath} (1)
{\bf m}\in\Omega = \{ {\bf m} \in \Re^N\mid l_i\leq m_i\leq u_i\},\end{displaymath} (2)
with li and ui being the lower and upper bounds for the model mi, respectively. In this case, li and ui are called simple bounds. They can be different for each point of the model space. The model vector that obeys equation (1) is called ${\bf m^*}$.

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