16,751 research outputs found
Conditional Risk Mappings
We introduce an axiomatic definition of a conditional convex risk mapping. By employing the techniques of conjugate duality we derive properties of conditional risk mappings. In particular, we prove a representation theorem for conditional risk mappings in terms of conditional expectations. We also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings.Risk, Convex Analysis, Conjugate Duality, Stochastic Optimization, Dynamic Programming, Multi-Stage Programming
Markov risk mappings and risk-sensitive optimal stopping
In contrast to the analytic approach to risk for Markov chains based on
transition risk mappings, we introduce a probabilistic setting based on a novel
concept of regular conditional risk mapping with Markov update rule. We confirm
that the Markov property holds for the standard measures of risk used in
practice such as Value at Risk and Average Value at Risk. We analyse the dual
representation for convex Markovian risk mappings and a representation in terms
of their acceptance sets. The Markov property is formulated in several
equivalent versions including a strong version, opening up additional
risk-sensitive optimisation problems such as optimal stopping with exercise lag
and optimal prediction. We demonstrate how such problems can be reduced to a
risk-sensitive optimal stopping problem with intermediate costs, and derive the
dynamic programming equations for the latter. Finally, we show how our results
can be extended to partially observable Markov processes.Comment: 29 pages. New: extension of one-step ahead Markov property to entire
"future", Markov property in terms of acceptance sets, VaR and AVaR examples,
convex Markov risk mappings, application to optimal stopping with exercise
lag. Notable changes: Stopping cost in the partially observable optimal
stopping problem can depend on the unobservable stat
Semi-proximal Mirror-Prox for Nonsmooth Composite Minimization
We propose a new first-order optimisation algorithm to solve high-dimensional
non-smooth composite minimisation problems. Typical examples of such problems
have an objective that decomposes into a non-smooth empirical risk part and a
non-smooth regularisation penalty. The proposed algorithm, called Semi-Proximal
Mirror-Prox, leverages the Fenchel-type representation of one part of the
objective while handling the other part of the objective via linear
minimization over the domain. The algorithm stands in contrast with more
classical proximal gradient algorithms with smoothing, which require the
computation of proximal operators at each iteration and can therefore be
impractical for high-dimensional problems. We establish the theoretical
convergence rate of Semi-Proximal Mirror-Prox, which exhibits the optimal
complexity bounds, i.e. , for the number of calls to linear
minimization oracle. We present promising experimental results showing the
interest of the approach in comparison to competing methods
Optimization of Risk Measures
We consider optimization problems involving coherent risk measures. We derive necessary and sufficient conditions of optimality for these problems, and we discuss the nature of the nonanticipativity constraints. Next, we introdice dynamic risk measures, and we formulate multistage optimization problems involving these measures. Conditions similar to dynamic programming equations are developed. The theoretical considerations are illustrated with many examples of mean-risk models applied in practice.risk measures, mean-risk models, duality, optimization, dynamic programming
- …