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Zbl 1028.90060
Forsgren, Anders; Gill, Philip E.; Wright, Margaret H.
Interior methods for nonlinear optimization. (English)
[J] SIAM Rev. 44, No.4, 525-597 (2002). ISSN 0036-1445; ISSN 1095-7200/e

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. \par 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.

MSC 2000:: *90C30 Nonlinear programming
49M37 Methods of nonlinear programming type
49M30 Methods of successive approximation, not based on necessary cond.
65F05 Direct methods for linear systems
65K05 Mathematical programming (numerical methods)
90C51 Interior-point methods

Keywords: nonlinear programming; constrained minimization; nonlinear constraints; primal-dual methods; interior methods; penalty methods; barrier methods

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