We develop the idea of using Monte Carlo sampling of random portfolios to
solve portfolio investment problems. In this first paper we explore the need
for more general optimization tools, and consider the means by which
constrained random portfolios may be generated. A practical scheme for the
long-only fully-invested problem is developed and tested for the classic QP
application. The advantage of Monte Carlo methods is that they may be extended
to risk functions that are more complicated functions of the return
distribution, and that the underlying return distribution may be computed
without the classical Gaussian limitations. The optimization of quadratic
risk-return functions, VaR, CVaR, may be handled in a similar manner to
variability ratios such as Sortino and Omega, or mathematical constructions
such as expected utility and its behavioural finance extensions.
Robustification is also possible. Grid computing technology is an excellent
platform for the development of such computations due to the intrinsically
parallel nature of the computation, coupled to the requirement to transmit only
small packets of data over the grid. We give some examples deploying
GridMathematica, in which various investor risk preferences are optimized with
differing multivariate distributions. Good comparisons with established results
in Mean-Variance and CVaR optimization are obtained when ``edge-vertex-biased''
sampling methods are employed to create random portfolios. We also give an
application to Omega optimization.