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Fundamental properties of conditional value-at-risk (CVaR), as a measure of risk with significant advantages over value-at-risk (VaR), are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. CVaR is… (More)

- R. Tyrrell Rockafellar
- Math. Oper. Res.
- 1976

The theory of the proximal point algorithm for maximal monotone operators is applied to three algorithms for solving convex programs, one of which has not previously been formulated. Rate-of-convergence results for the "method of multipliers," of the strong sort already known, are derived in a generalized form relevant also to problems beyond the compass of… (More)

- R. Tyrrell Rockafellar, Roger J.-B. Wets
- Math. Oper. Res.
- 1991

- R. Tyrrell Rockafellar, Stan Uryasev, Michael Zabarankin
- Finance and Stochastics
- 2006

General deviation measures are introduced and studied systematically for their potential applications to risk management in areas like portfolio optimization and engineering. Such measures include standard deviation as a special case but need not be symmetric with respect to ups and downs. Their properties are explored with a mind to generating a large… (More)

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C. Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of… (More)

- Asen L. Dontchev, R. Tyrrell Rockafellar
- SIAM Journal on Optimization
- 1996

Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson’s notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new “critical face” condition and in other ways. The consequences for… (More)

- Alan J. King, R. Tyrrell Rockafellar
- Math. Oper. Res.
- 1993

New techniques of local sensitivity analysis for nonsmooth generalized equations are applied to the study of sequences of statistical estimates and empirical approximations to solutions of stochastic programs. Consistency is shown to follow from a certain local invertibility property, and asymptotic distributions are derived from a generalized implicit… (More)

Decisions often need to be made before all the facts are in. A facility must be built to withstand storms, floods, or earthquakes of magnitudes that can only be guessed from historical records. A portfolio must be purchased in the face of only statistical knowledge, at best, about how markets will perform. In optimization, this implies that constraints may… (More)

- R. Tyrrell Rockafellar
- SIAM Review
- 1993

Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the… (More)

- R. Tyrrell Rockafellar
- Math. Program.
- 1973