Communications Project

Document Type:Master's Thesis
Name:Zafar A. Ansari
Title:Limited Memory Space Dilation and Reduction Algorithms
Degree:Master of Science
Department:Industrial and Systems Engineering
Committee Chair: Dr. Hanif D. Sherali
Committee Members:Dr. S C. Sarin
Dr. G. V. Loganathan
Keywords:Nondifferentiable Optimization, Space Dilation, r-algorithm, Subgradient Optimization
Date of defense:July 13, 1998
Availability:Release the entire work immediately worldwide.


abstract                    Limited Memory Space Dilation and Reduction Algorithms

                                                       Zafar A. Ansari

                                            Dr. Hanif D. Sherali, Chairman

                                         Industrial and Systems Engineering


In this thesis, we present variants of Shor and Zhurbenko’s r-algorithm, motivated by the
memoryless and limited memory updates for differentiable quasi-Newton methods. This well
known r-algorithm, which employs a space dilation strategy in the direction of the difference
between two successive subgradients, is recognized as being one of the most effective procedures
for solving nondifferentiable optimization problems. However, the method needs to store the
space dilation matrix and update it at every iteration, resulting in a substantial computational
burden for large-sized problems. To circumvent this difficulty, we first develop a memoryless
update scheme. In the space transformation sense, the new update scheme can be viewed as a
combination of space dilation and reduction operations. We prove convergence of this new
algorithm, and demonstrate how it can be used in conjunction with a variable target value method
that allows a practical, convergent implementation of the method.  For performance comparisons
we examine other memoryless and limited memory variants, and also prove a modification of a
related algorithm due to Polyak that employs a projection on a pair of Kelley’s cutting planes.
These variants are tested along with Shor’s r-algorithm on a set of standard test problems from
the literature as well as on randomly generated dual transportation and assignment problems. Our
computational experiments reveal that the proposed memoryless space dilation and reduction
algorithm (VT-MSDR) and the proposed modification of the Polyak-Kelly cutting plane method
(VT-PKC) provide an overall competitive performance relative to the other methods tested with
respect to solution quality and computational effort. The r-Algorithm becomes increasingly more
expensive with an increase in problem size, while not providing any gain in solution quality. The
fixed dilation (with no reduction) strategy (VT-MSD) provides a comparable, though second-
choice, alternative to VT-MSDR. Employing a two-step limited memory extension over VT-
MSD sometimes helps in improving the solution quality, although it adds to computational
effort, and is not as robust a procedure.

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