TY - JOUR
T1 - Global and superlinear convergence of a restricted class of self-scaling methods with inexact line searches, for convex functions
AU - Al-Baali, M.
N1 - Funding Information:
I acknowledge the research support of an Italian Ministry of University Grant for working at the Department of Electronics, Informatics and Systems, University of Calabria, Italy. I would like to thank the anonymous referees for carefully reading a draft of this paper and making a number of valuable comments. I would also like to thank R.H. Byrd, D. Conforti, R. Fletcher, J.C. Gilbert, L. Grandinetti, J.J. Moré, J. Nocedal, M.J.D. Powell and R.B. Schnabel for valuable comments.
PY - 1998/2
Y1 - 1998/2
N2 - This paper studies the convergence properties of algorithms belonging to the class of self-scaling (SS) quasi-Newton methods for unconstrained optimization. This class depends on two parameters, say θk and τk, for which the choice τk = 1 gives the Broyden family of unsealed methods, where θk = 1 corresponds to the well known DFP method. We propose simple conditions on these parameters that give rise to global convergence with inexact line searches, for convex objective functions. The q-superlinear convergence is achieved if further restrictions on the scaling parameter are introduced. These convergence results are an extension of the known results for the unsealed methods. Because the scaling parameter is heavily restricted, we consider a subclass of SS methods which satisfies the required conditions. Although convergence for the unsealed methods with θk ≥ 1 is still an open question, we show that the global and superlinear convergence for SS methods is possible and present, in particular, a new SS-DFP method.
AB - This paper studies the convergence properties of algorithms belonging to the class of self-scaling (SS) quasi-Newton methods for unconstrained optimization. This class depends on two parameters, say θk and τk, for which the choice τk = 1 gives the Broyden family of unsealed methods, where θk = 1 corresponds to the well known DFP method. We propose simple conditions on these parameters that give rise to global convergence with inexact line searches, for convex objective functions. The q-superlinear convergence is achieved if further restrictions on the scaling parameter are introduced. These convergence results are an extension of the known results for the unsealed methods. Because the scaling parameter is heavily restricted, we consider a subclass of SS methods which satisfies the required conditions. Although convergence for the unsealed methods with θk ≥ 1 is still an open question, we show that the global and superlinear convergence for SS methods is possible and present, in particular, a new SS-DFP method.
KW - Broyden's class
KW - Global and superlinear convergence
KW - Inexact line searches
KW - Quasi-Newton methods
KW - Self-scaling
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U2 - 10.1023/A:1018315205474
DO - 10.1023/A:1018315205474
M3 - Article
AN - SCOPUS:0031999308
SN - 0926-6003
VL - 9
SP - 191
EP - 203
JO - Computational Optimization and Applications
JF - Computational Optimization and Applications
IS - 2
ER -