Interior Methods for Nonlinear Optimization
Interior methods are an omnipresent, conspicuous feature of the constrained
optimization landscape today, but it was not always so. Primarily in
the form of barrier methods, interior-point techniques were popular
during the 1960s for solving nonlinearly constrained problems.
However, their use for linear programming was not even contemplated
because of the total dominance of the simplex method. Vague but
continuing anxiety about barrier methods eventually led to their
abandonment in favor of newly emerging, apparently
more efficient alternatives such as augmented Lagrangian and
sequential quadratic programming methods. By the early 1980s, barrier
methods were almost without exception regarded as a closed chapter in the
history of optimization.
This picture changed dramatically with Karmarkar's widely publicized
announcement in 1984 of a fast polynomial-time interior method for linear
programming; in 1985, a formal connection was established between his
method and classical barrier methods. Since then,
interior methods have advanced so far, so fast, that
their influence has transformed both the theory and practice of
constrained optimization. This article provides a
condensed, selective look at classical
material and recent research about interior methods for nonlinearly
constrained optimization.
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