664 research outputs found
Approximations of countably-infinite linear programs over bounded measure spaces
We study a class of countably-infinite-dimensional linear programs (CILPs)
whose feasible sets are bounded subsets of appropriately defined weighted
spaces of measures. We show how to approximate the optimal value, optimal
points, and minimal points of these CILPs by solving finite-dimensional linear
programs. The errors of our approximations converge to zero as the size of the
finite-dimensional program approaches that of the original problem and are easy
to bound in practice. We discuss the use of our methods in the computation of
the stationary distributions, occupation measures, and exit distributions of
Markov~chains
Inverse Optimization of Convex Risk Functions
The theory of convex risk functions has now been well established as the
basis for identifying the families of risk functions that should be used in
risk averse optimization problems. Despite its theoretical appeal, the
implementation of a convex risk function remains difficult, as there is little
guidance regarding how a convex risk function should be chosen so that it also
well represents one's own risk preferences. In this paper, we address this
issue through the lens of inverse optimization. Specifically, given solution
data from some (forward) risk-averse optimization problems we develop an
inverse optimization framework that generates a risk function that renders the
solutions optimal for the forward problems. The framework incorporates the
well-known properties of convex risk functions, namely, monotonicity,
convexity, translation invariance, and law invariance, as the general
information about candidate risk functions, and also the feedbacks from
individuals, which include an initial estimate of the risk function and
pairwise comparisons among random losses, as the more specific information. Our
framework is particularly novel in that unlike classical inverse optimization,
no parametric assumption is made about the risk function, i.e. it is
non-parametric. We show how the resulting inverse optimization problems can be
reformulated as convex programs and are polynomially solvable if the
corresponding forward problems are polynomially solvable. We illustrate the
imputed risk functions in a portfolio selection problem and demonstrate their
practical value using real-life data
Proper efficiency and tradeoffs in multiple criteria and stochastic optimization
The mathematical equivalence between linear scalarizations in multiobjective programming and expected-value functions in stochastic optimization suggests to investigate and establish further conceptual analogies between these two areas. In this paper, we focus on the notion of proper efficiency that allows us to provide a first comprehensive analysis of solution and scenario tradeoffs in stochastic optimization. In generalization of two standard characterizations of properly efficient solutions using weighted sums and augmented weighted Tchebycheff norms for finitely many criteria, we show that these results are generally false for infinitely many criteria. In particular, these observations motivate a slightly modified definition to prove that expected-value optimization over continuous random variables still yields bounded tradeoffs almost everywhere in general. Further consequences and practical implications of these results for decision-making under uncertainty and its related theory and methodology of multiple criteria, stochastic and robust optimization are discussed
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