  We present novel algorithms for simulation optimization using random
directions stochastic approximation (RDSA). These include first-order
(gradient) as well as second-order (Newton) schemes. We incorporate both
continuous-valued as well as discrete-valued perturbations into both our
algorithms. The former are chosen to be independent and identically distributed
(i.i.d.) symmetric, uniformly distributed random variables (r.v.), while the
latter are i.i.d., asymmetric, Bernoulli r.v.s. Our Newton algorithm, with a
novel Hessian estimation scheme, requires N-dimensional perturbations and three
loss measurements per iteration, whereas the simultaneous perturbation Newton
search algorithm of [1] requires 2N-dimensional perturbations and four loss
measurements per iteration. We prove the unbiasedness of both gradient and
Hessian estimates and asymptotic (strong) convergence for both first-order and
second-order schemes. We also provide asymptotic normality results, which in
particular establish that the asymmetric Bernoulli variant of Newton RDSA
method is better than 2SPSA of [1]. Numerical experiments are used to validate
the theoretical results.
