Deterministic approximations in stochastic programming with applications to a class of portfolio allocation problems
Abstract
Optimal decision making under uncertainty involves modeling stochastic systems
and developing solution methods for such models. The need to incorporate
randomness in many practical decision-making problems is prompted
by the uncertainties associated with today’s fast-paced technological environment.
The complexity of the resulting models often exceeds the capabilities
of commercially available optimization software, and special purpose solution
techniques are required.
Three main categories of solution approaches exist for attacking a particular
stochastic programming instance. These are: large-scale mathematical
programming algorithms, Monte-Carlo sampling-based techniques, and deterministically
valid bound-based approximations. This research contributes to
the last category.
First, second-order lower and upper bounds are developed on the expectation
of a convex function of a random vector. Here, a “second-order bound”
means that only the first and second moments of the underlying random parameters
are needed to compute the bound. The vector’s random components
are assumed to be independent and to have bounded support contained in a
hyper-rectangle. Applications to stochastic programming test problems and
analysis of numerical performance are also presented.
Second, assuming additional relevant moment information is available,
higher-order upper bounds are developed. In this case the underlying random
vector can have support contained in either a hyper-rectangle or a multidimensional
simplex, and the random parameters can be either dependent
or independent. The higher-order upper bounds form a decreasing sequence
converging to the true expectation, and yielding convergence of the optimal
decisions.
Finally, applications of the higher-order upper bounds to a class of portfolio
optimization problems are presented. Mean-variance and mean-varianceskewness
efficient portfolio frontiers are considered in the context of a specific
portfolio allocation model as well as in general and connected with applications
of the higher-order upper bounds in utility theory
Department
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