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 l...

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Bibliographic Details
Published in:SIAM review 2002-12, Vol.44 (4), p.525-597
Main Authors: Anders Forsgren, Gill, Philip E., Wright, Margaret H.
Format: Article
Language:English
Subjects:
Algorithms
Applied sciences
Calculus of variations and optimal control
Constrained optimization
Exact sciences and technology
Hessian matrices
Inequality constraints
Lagrangian function
Linear programming
Mathematical analysis
Mathematical functions
Mathematical programming
Mathematics
Nonlinear programming
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Operational research and scientific management
Operational research. Management science
Optimization
Quadratic programming
Sciences and techniques of general use
Survey and Review
Trajectories
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Summary: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.
ISSN:0036-1445
1095-7200
DOI:10.1137/s0036144502414942