This book discusses an important area of numerical optimization,
called interior-point method. This topic has been popular since the
1980s when people gradually realized that all simplex algorithms
were not convergent in polynomial time and many interior-point
algorithms could be proved to converge in polynomial time. However,
for a long time, there was a noticeable gap between theoretical
polynomial bounds of the interior-point algorithms and efficiency
of these algorithms. Strategies that were important to the
computational efficiency became barriers in the proof of good
polynomial bounds. The more the strategies were used in algorithms,
the worse the polynomial bounds became. To further exacerbate the
problem, Mehrotra's predictor-corrector (MPC) algorithm (the most
popular and efficient interior-point algorithm until recently) uses
all good strategies and fails to prove the convergence. Therefore,
MPC does not have polynomiality, a critical issue with the simplex
method. This book discusses recent developments that resolves the
dilemma. It has three major parts. The first, including Chapters 1,
2, 3, and 4, presents some of the most important algorithms during
the development of the interior-point method around the 1990s, most
of them are widely known. The main purpose of this part is to
explain the dilemma described above by analyzing these algorithms'
polynomial bounds and summarizing the computational experience
associated with them. The second part, including Chapters 5, 6, 7,
and 8, describes how to solve the dilemma step-by-step using
arc-search techniques. At the end of this part, a very efficient
algorithm with the lowest polynomial bound is presented. The last
part, including Chapters 9, 10, 11, and 12, extends arc-search
techniques to some more general problems, such as convex quadratic
programming, linear complementarity problem, and semi-definite
programming.
General
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