Abstract
For solving linear programming problems, we describe and evaluate descent directions for projective methods in the setting of the method of de Ghellinck and Vial (1986). It is shown that choosing search directions from the projected simplex centered at 1 in transformed space guarantees a polynomial time bound. In particular, the projection of 1 coincides with the direction chosen by de Ghellinck and Vial (1986) which is well known to be equivalent to Karmarkar's search direction when the initial solution is feasible. Close to that search direction the complexity of the method is unchanged [0(nL) iterations] while in general the bound is 0(n2 ·L), whereL is the size of the linear programming problem.
The investigations were not made in order to improve worst case bounds, but rather to enable a free choice of search directions from a quite large class of “polynomial” directions. We show how different directions can be computed in one iteration using the projection of 1 with relatively small extra computational effort. In projective methods linear programming is reformulated as one-parametric feasibility problem where the parameter is the currently known upper bound on the optimal value. It is therefore very important to improve that bound whenever possible. Generalizing bounds from de Ghellinck and Vial (1986) we prove some infeasibility criteria. In particular, any computed search direction yields some upper bound which eventually improves that currently used. All bounds can be computed by solving sets of simple linear and quadratic equations.
Zusammenfassung
Zur Lösung linearer Optimierungsaufgaben beschreiben und bewerten wir Abstiegsrichtungen bei projektiven Methoden in der Formulierung von de Ghellinck and Vial (1986). Es zeigt sich, daß bei Wahl von Suchrichtungen aus dem projizierten Simplex um 1 im transformierten Raum stets polynomiale Laufzeitabschätzungen gelten. Insbesondere die Projektion der 1 stimmt mit der von de Ghellinck and Vial (1986) gewählten Suchrichtung überein, die bekanntlich bei zulässiger Startlösung äquivalent zu Karmarkar's Suchrichtung ist. Nahe bei dieser Suchrichtung bleibt die Komplexität der Methode unverändert [0(nL) Iterationen], während im allgemeinen die Schranke 0(n2 ·L) gilt, wobeiL die Größe des Linearen Optimierungsproblemes beschreibt.
Die Untersuchungen zielen nicht darauf ab, Schranken für den ungünstigsten Fall zu verbessern, sondern sollen die freie Wahl von Suchrichtungen aus einer großen Klasse „polynomialer“ Richtungen zulassen. Wir zeigen, wie man unterschiedliche Suchrichtungen innerhalb einer Iteration aus der Projektion der 1 mit relativ kleinem zusätzlichen Aufwand berechnen kann. In projektiven Methoden wird die Lineare Optimierungsaufgabe als einparametrisches Zulässigkeitsproblem reformuliert, wobei als Parameter die jeweils bekannte obere Schranke des Optimalwertes dient. Daher ist es sehr wichtig, diese Schranke bei jeder Gelegenheit zu verbessern. Durch Verallgemeinerung von Schranken aus de Ghellinck and Vial (1986) leiten wir einige Unzulässigkeitstests her. Insbesondere liefert jede berechnete Suchrichtung eine obere Schranke, die möglicherweise die bekannte obere Schranke verbessert. Alle Schranken können durch Lösung einer Reihe einfacher linearer und quadratischer Gleichungen berechnet werden.
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Zimmermann, U. Search directions for a class of projective methods. ZOR - Methods and Models of Operations Research 34, 353–379 (1990). https://doi.org/10.1007/BF01416226
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DOI: https://doi.org/10.1007/BF01416226