By substituting an L-p loss function for the L-1 loss function in the optimization problem defining quantiles, one obtains L-p-quantiles that, as shown recently, dominate their classical L-1-counterparts in financial ...
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By substituting an L-p loss function for the L-1 loss function in the optimization problem defining quantiles, one obtains L-p-quantiles that, as shown recently, dominate their classical L-1-counterparts in financial risk assessment. In this work, we propose a concept of multivariate L-p-quantiles generalizing the spatial (L-1-)quantiles introduced by Probal Chaudhuri (J. Amer. Statist. Assoc. 91 (1996) 862-872). Rather than restricting to power loss functions, we actually allow for a large class of convex loss functions rho. We carefully study existence and uniqueness of the resulting rho-quantiles, both for a general probability measure over R-d and for a spherically symmetric one. Interestingly, the results crucially depend on rho and on the nature of the underlying probability measure. Building on an investigation of the differentiability properties of the objective function defining rho-quantiles, we introduce a companion concept of spatial rho-depth, that generalizes the classical spatial depth. We study extreme rho-quantiles and show in particular that extreme L-p-quantiles behave in fundamentally different ways for p <= 2 and p > 2. Finally, we establish Bahadur representation results for sample rho-quantiles and derive their asymptotic distributions. Throughout, we impose only very mild assumptions on the underlying probability measure, and in particular we never assume absolute continuity with respect to the Lebesgue measure.
We consider a primal-dual potential reduction algorithm for nonlinear convex optimization problems over symmetric cones. The same complexity estimates as in the case of the linear objective function are obtained provi...
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We consider a primal-dual potential reduction algorithm for nonlinear convex optimization problems over symmetric cones. The same complexity estimates as in the case of the linear objective function are obtained provided a certain nonlinear system of equations can be solved with a given accuracy. This generalizes the result of K. Kortanek, F. Potra and Y. Ye [7]. We further introduce a generalized Nesterov-Todd direction and show how it can be used to achieve a required accuracy (by solving the linearization of above mentioned nonlinear system) for a class of nonlinear convexfunctions satisfying scaling Lipschitz condition. This result is a far-reaching generalization of results of F. Potra, Y. Ye and J. Zhu [8], [9]. Finally, we show that a class of functions (which contains quantum entropy function) satisfies scaling Lipschitz condition. (C) 2017 Elsevier Inc. All rights reserved.
In Part I of this paper, we proposed and analyzed a novel algorithmic framework for the minimization of a nonconvexobjective function, subject to nonconvex constraints, based on inner convex approximations. This Part...
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A standard approach to the minimization of a state constrained objective function in Control/Shape Optimization problems is to consider the minimax of the associated Lagrangian. In this paper, this construction is use...
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This paper considers a downlink heterogeneous network, where different types of multiantenna base stations (BSs) communicate with a number of single-antenna users. Multiple BSs can serve the users by spatial multiflow...
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