133 research outputs found
Primal-Dual Stability in Local Optimality
Much is known about when a locally optimal solution depends in a
single-valued Lipschitz continuous way on the problem's parameters, including
tilt perturbations. Much less is known, however, about when that solution and a
uniquely determined multiplier vector associated with it exhibit that
dependence as a primal-dual pair. In classical nonlinear programming, such
advantageous behavior is tied to the combination of the standard strong
second-order sufficient condition (SSOC) for local optimality and the linear
independent gradient condition (LIGC) on the active constraint gradients. But
although second-order sufficient conditions have successfully been extended far
beyond nonlinear programming, insights into what should replace constraint
gradient independence as the extended dual counterpart have been lacking.
The exact answer is provided here for a wide range of optimization problems
in finite dimensions. Behind it are advances in how coderivatives and strict
graphical derivatives can be deployed. New results about strong metric
regularity in solving variational inequalities and generalized equations are
obtained from that as well
Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging
The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation
Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise
The Primitive Equations are a basic model in the study of large scale Oceanic
and Atmospheric dynamics. These systems form the analytical core of the most
advanced General Circulation Models. For this reason and due to their
challenging nonlinear and anisotropic structure the Primitive Equations have
recently received considerable attention from the mathematical community.
In view of the complex multi-scale nature of the earth's climate system, many
uncertainties appear that should be accounted for in the basic dynamical models
of atmospheric and oceanic processes. In the climate community stochastic
methods have come into extensive use in this connection. For this reason there
has appeared a need to further develop the foundations of nonlinear stochastic
partial differential equations in connection with the Primitive Equations and
more generally.
In this work we study a stochastic version of the Primitive Equations. We
establish the global existence of strong, pathwise solutions for these
equations in dimension 3 for the case of a nonlinear multiplicative noise. The
proof makes use of anisotropic estimates, estimates on the
pressure and stopping time arguments.Comment: To appear in Nonlinearit
Structure of shocks in Burgers turbulence with L\'evy noise initial data
We study the structure of the shocks for the inviscid Burgers equation in
dimension 1 when the initial velocity is given by L\'evy noise, or equivalently
when the initial potential is a two-sided L\'evy process . When
is abrupt in the sense of Vigon or has bounded variation with
, we prove that the set
of points with zero velocity is regenerative, and that in the latter case this
set is equal to the set of Lagrangian regular points, which is non-empty. When
is abrupt we show that the shock structure is discrete. When
is eroded we show that there are no rarefaction intervals.Comment: 22 page
On polyhedral projection and parametric programming
Submitted versio
Numerical approach to a model for quasistatic damage with spatial BV-regularization
We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rate-independent system. Moreover, we present our numerical results for two benchmark problems
Superquantile/CVaR Risk Measures: Second-Order Theory
Superquantiles, which refer to conditional value-at-risk (CVaR) in the same way that
quantiles refer to value-at-risk (VaR), have many advantages in the modeling of risk in finance and engineering. However, some applications may benefit from a further step, from superquantiles to second-
order superquantiles. Measures of risk based on second-order superquantiles have recently been explored
in some settings, but key parts of the theory have been lacking: descriptions of the associated risk envelopes and risk identifiers. Those missing ingredients are supplied in this paper, and moreover not just
for second-order superquantiles, but also for a much broader class of mixed superquantile measures of
risk. Such dualizing expressions facilitate the development of dual methods for mixed and second-order
superquantile risk minimization as well as superquantile regression, a proposed second-order version of
quantile regression.U.S. Air Force Office of Scientific Research grant FA9550-11-1-0206U.S. Air Force Office of Scientific Research grant F1ATAO1194GOO1DARPA grant HR0011517798U.S. Air Force Office of Scientific Research grant FA9550-11-1-0206U.S. Air Force Office of Scientific Research grant F1ATAO1194GOO1DARPA grant HR001151779
Engineering decisions under risk-averseness
Engineering decisions are invariably made under substantial uncertainty about current
and future system cost and response, including cost and response associated with low-probability, high- consequence events. A risk-neutral decision maker would rely on expected values when comparing designs, while a risk-averse decision maker might adopt nonlinear utility functions or failure probability criteria. The paper shows that these models for making decisions are related to a framework of risk measures that includes many possibilities. We describe how risk measures provide an expanded set of models for handling risk-averse decision makers. General recommendations for selecting risk measures lead to decision models for risk-averse decision making that comprehensively represent risks in engi- neering systems, avoid paradoxes, and accrue substantial benefits in subsequent risk, reliability, and cost optimization. The paper provides an overview of the framework of decision making based on risk measures.This material is based upon work supported in part by the U. S. Air Force Office of Scientific Research under grants FA9550-11-1-0206 and F1ATAO1194GOO1
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