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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 such that we minimize
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 .
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).