Abstract
This paper presents an algorithm for the minimization of a nonlinear objective function subject to nonlinear inequality and equality constraints. The proposed method has the two distinguishing properties that, under weak assumptions, it converges to a Kuhn-Tucker point for the problem and under somewhat stronger assumptions, the rate of convergence is quadratic. The method is similar to a recent method proposed by Rosen in that it begins by using a penalty function approach to generate a point in a neighborhood of the optimum and then switches to Robinson's method. The new method has two new features not shared by Rosen's method. First, a correct choice of penalty function parameters is constructed automatically, thus guaranteeing global convergence to a stationary point. Second, the linearly constrained subproblems solved by the Robinson method normally contain linear inequality constraints while for the method presented here, only linear equality constraints are required. That is, in a certain sense, the new method “knows” which of the linear inequality constraints will be active in the subproblems. The subproblems may thus be solved in an especially efficient manner.
Preliminary computational results are presented.
Zusammenfassung
Diese Arbeit beschreibt einen Algorithmus zur Minimierung einer nichtlinearen Funktion mit nichtlinearen Ungleichungen und Gleichungen als Nebenbedingungen. Die vorgeschlagene Methode hat die Eigenschaft, daß sie unter schwachen Voraussetzungen gegen einen Kuhn-Tucker-Punkt des betrachteten Optimierungsproblems konvergiert und unter stärkeren Voraussetzungen eine quadratische Konvergenzgeschwindigkeit aufweist. Ähnlich wie eine vor kurzem von Rosen vorgeschlagene Methode benutzt der Algorithmus eine Straffunktion, um einen Punkt in der Nähe der optimalen Lösung zu berechnen und schaltet dann auf Robinsons Methode um. Die neue Methode hat gegenüber dem Verfahren von Rosen zwei neue Eigenschaften. Erstens wird der richtige Wert des Parameters in der Straffunktion automatisch gefunden. Zweitens enthalten die mit der Methode von Robinson gelösten Teilprobleme nur lineare Gleichungen als Nebenbedingungen. Die Teilprobleme können daher besonders leicht gelöst werden.
Vorläufige numerische Ergebnisse werden berichtet.
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Sponsored by the United States Army under Contract No. DAAG29-75-C-0024, by the National Research Council of Canada under Research Grant A8189, and by the National Science Foundation under Grant No. MCS74-20584 A02.
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Best, M.J., Bräuninger, J., Ritter, K. et al. A globally and quadratically convergent algorithm for general nonlinear programming problems. Computing 26, 141–153 (1981). https://doi.org/10.1007/BF02241780
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DOI: https://doi.org/10.1007/BF02241780