 
Summary: TwoTimescale Simultaneous Perturbation
Stochastic Approximation Using
Deterministic Perturbation Sequences
SHALABH BHATNAGAR
Indian Institute of Science, Bangalore
MICHAEL C. FU and STEVEN I. MARCUS
University of Maryland, College Park, Maryland
and
IJENG WANG
Johns Hopkins University, Applied Physics Laboratory, Laurel, Maryland
Simultaneous perturbation stochastic approximation (SPSA) algorithms have been found to be very
effective for highdimensional simulation optimization problems. The main idea is to estimate the
gradient using simulation output performance measures at only two settings of the Ndimensional
parameter vector being optimized rather than at the N + 1 or 2N settings required by the usual
onesided or symmetric difference estimates, respectively. The two settings of the parameter vec
tor are obtained by simultaneously changing the parameter vector in each component direction
using random perturbations. In this article, in order to enhance the convergence of these algo
rithms, we consider deterministic sequences of perturbations for twotimescale SPSA algorithms.
Two constructions for the perturbation sequences are considered: complete lexicographical cycles
and much shorter sequences based on normalized Hadamard matrices. Recently, onesimulation
