Interior Methods for Nonlinear Optimization

Abstract

Abstract.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.

Entwurf robuster modellbasierter Fehlerisolationsfilter

“…Neben der Anwendung spezieller Lösungsverfahren (s. zum Beispiel [73]) bietet sich hierfür eine konvexe Approximation an. Das durch p T k 2 ≥ 1 beschriebene nichtkonvexe Gebiet wird dazu durch mehrere konvexe Gebiete angenähert.…”

“…Die Lösung kann zum Beispiel auf Basis von zufällig gewählten Startwerten mithilfe iterativer Lösungsstrategien erfolgen (s. zum Beispiel [73]). Aufgrund der geringen Anzahl von freien Parametern lassen sich die Optimierungsprobleme effizient lösen, die Konvergenz auf das globale Optimum ist für dieses Teilproblem jedoch nicht sichergestellt.…”

The interior-point revolution in optimization: History, recent developments, and lasting consequences

“…(This behavior constitutes an obvious rationale for the descriptors "barrier" and "interior".) Numerous properties of B(x, µ) are known; see, for example, the classic reference [9] or [36,12]. For small µ, unconstrained minimizers of B(x, µ) are related in an intuitively appealing way to the solution x * of (4).…”

“…For small µ, unconstrained minimizers of B(x, µ) are related in an intuitively appealing way to the solution x * of (4). Given that x * satisfies the sufficient optimality conditions given in Section 2.2.1, then, for a sequence of monotonically decreasing and sufficiently small values of µ, there is an associated sequence {x µ } of isolated local unconstrained minimizers of the barrier function (11) such that (12) lim µ→0 x µ = x * and lim…”

“…For proofs and additional details, see, for example, [36,12]. To illustrate the behavior of the log barrier function, we consider the two-variable inequality-constrained problem:…”