133 research outputs found

    Primal-Dual Stability in Local Optimality

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    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

    Social Choice Scoring Functions

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    Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging

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    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

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    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, LtpLxqL^{p}_{t}L^{q}_{x} 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

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    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 ψ0\psi_0. When ψ0\psi_0 is abrupt in the sense of Vigon or has bounded variation with lim suph0h2ψ0(h)=\limsup_{|h| \downarrow 0} h^{-2} \psi_0(h) = \infty, 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 ψ0\psi_0 is abrupt we show that the shock structure is discrete. When ψ0\psi_0 is eroded we show that there are no rarefaction intervals.Comment: 22 page

    Numerical approach to a model for quasistatic damage with spatial BV-regularization

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    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

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    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

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    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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