A Potential Reduction Method for a Class of Smooth Convex Programming

DELFT UNIVERSITY OF TECHNOLOGY
REPORT 90{01 A POTENTIAL REDUCTION METHOD FOR A CLASS OF SMOOTH CONVEX PROGRAMMING PROBLEMS D. den Hertog, C. Roos and T. Terlaky
ISSN 0922{5641 Reports of the Faculty of Technical Mathematics and Informatics no. 90{01 Delft, 1990
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Address of the authors:
Faculty of Technical Mathematics and Computer Science Delft University of Technology Mekelweg 4 2628 CD Delft Netherlands. This work is completed under the support of a research grant of SHELL. The third author is on leave from the Eotvos University, Budapest, and partially supported by OTKA no. 2116.
ISSN 0922-5641 Copyright c 1990 by the Faculty of Technical Mathematics and Informatics, Delft, The Netherlands. No part of this Journal may be reproduced in any form, by print, photoprint, microfilm, or any other means without permission from the Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands. Copies of these reports may be obtained from the bureau of the Faculty of Technical Mathematics and Informatics, Julianalaan 132, 2628 BL Delft, phone +3115784568. A selection of these reports is available in PostScript form at the Faculty’s anonymous ftp-site. They are located in the directory /pub/publications/tech-reports at ftp.twi.tudelft.nl2 2Βιβλιοθήκη 1 Introduction
Since Karmarkar 6] presented his projective method for the solution of the linear programming problem in 1984, many other variants have been developed by researchers. Among them are the potential reduction methods, such as proposed by Ye 13], Freund 2], Roos and Vial 10], Gonzaga 3], 4] and Den Hertog, Roos and Terlaky 1]. The advantages of these methods are that they don't use projective transformations as the projective methods do, and that they don't need to follow the socalled central path closely, contrary to the path{following methods. In Jarre 5] and Mehrota and Sun 9] path{following algorithms are proposed for smooth convex programming problems. Again, the great disadvantage of these methods is that they are based on very small stepsizes to remain in the vicinity of the central trajectory. This characteristic of these methods make them unattractable for practice. In this paper we propose a potential reduction method for smooth convex programming problems, which ful l the socalled Relative Lipschitz Condition. Jarre 5] also uses this condition. Our method is a generalization of our potential reduction method for linear programming in 1] and is also based on Jarre's paper. In our potential reduction method we do a linesearch along the Newton direction with respect to a certain strictly convex potential function. If we are close to the current analytic center we update the lower bound somehow, whereafter we do linesearches aiming at getting close to the analytic center associated with the new lower bound. We proof that after a linesearch the potential value reduces at least with a certain constant. Using this result, we proof that the number of iterations required by the algorithm to converge to an {optimal solution is polynomial in , the dimension of the problem and the Lipschitz constant. We note that Liu and Goldfarb 8] already proposed a potential reduction algorithm for convex quadratic programming and Kojima, Mizuno and Yoshise 7] for linear complementarity problems. To our knowledge our algorithm is the rst potential reduction algorithm for (a class of) smooth convex programming. To extend the method to all smooth convex programs (without the Relative Lipschitz Condition) is currently under research. This paper is organized as follows. In Section 2 we will do some preliminary work. In Section 3 we describe our algorithm. Then, in Section 4 we prove some lemmas, needed for the convergence analysis in Section 5. Notation. Throughout the paper we use the following notation. If s denotes a vector, then the corresponding capital letter S will denote the diagonal matrix with the components of s on the diagonal. The vector e will always denote an all one vector of appropriate length. The identity matrix will be denoted by I . Superscripts will be used to denote di erent iterates, e.g. y and y .
Copyright c 1990 by Faculty of Technical Mathematics and Informatics, Delft, The Netherlands. No part of this Journal may be reproduced in any form, by print, photoprint, micro lm or any other means without written permission from Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands.