Suchergebnisse

A product warranty is an agreement offered by a producer to a consumer
to replace or repair a faulty item, or to partially or fully reimburse
the consumer in the event of a failure. Warranties are very
widespread and serve many purposes, including protection for producer,
seller, and consumer. They are used as signals of quality and
as elements of marketing strategies. In this study we review the notion
of an online convex optimization algorithm and its variations,
and apply it in warranty context. We introduce a class of profit functions,
which are functions of warranty, and use it to formulate the
problem of maximizing the company's profit over time as an online
convex optimization problem. We use this formulation to present an
approach to setting the warranty based on an online algorithm with
low regret. Under a dynamic environment, this algorithm provides
a warranty strategy for the company that maximises its profit over
time.

"This book presents a methodology for comprehensive treatment of nonsmooth laws in mechanics in accordance with contemporary theory and algorithms of optimization. The author deals with theory and numeiral algorithms comprehensively, providing a new perspective n nonsmooth mechanics based on contemporary optimization. Covering linear programs; semidefinite programs; second-order cone programs; complementarity problems; optimality conditions; Fenchel and Lagrangian dualities; algorithms of operations research, and treating cable networks; membranes; masonry structures; contact problems; plasticity, this is an ideal guide of nonsmooth mechanics for graduate students and researchers in civil and mechanical engineering, and applied mathematics"--

In this paper, we study private optimization problems for non-smooth convex functions $F(x)=\mathbb{E}_i f_i(x)$ on $\mathbb{R}^d$.We show that modifying the exponential mechanism by adding an $\ell_2^2$ regularizer to $F(x)$ and sampling from $\pi(x)\propto \exp(-k(F(x)+\mu\|x\|_2^2/2))$ recovers both the known optimal empirical risk and population loss under $(\eps,\delta)$-DP. Furthermore, we show how to implement this mechanism using $\widetilde{O}(n \min(d, n))$ queries to $f_i(x)$ for the DP-SCO where $n$ is the number of samples/users and $d$ is the ambient dimension.We also give a (nearly) matching lower bound $\widetilde{\Omega}(n \min(d, n))$ on the number of evaluation queries.
Our results utilize the following tools that are of independent interest:\begin{itemize}\item We prove Gaussian Differential Privacy (GDP) of the exponential mechanism if the loss function is strongly convex and the perturbation is Lipschitz. Our privacy bound is \emph{optimal} as it includes the privacy of Gaussian mechanism as a special case and is proved using the isoperimetric inequality for strongly log-concave measures.\item We show how to sample from $\exp(-F(x)-\mu \|x\|^2_2/2)$ for $G$-Lipschitz $F$ with $\eta$ error in total variation (TV) distance using $\widetilde{O}((G^2/\mu) \log^2(d/\eta))$ unbiased queries to $F(x)$. This is the first sampler whose query complexity has \emph{polylogarithmic dependence} on both dimension $d$ and accuracy $\eta$.\end{itemize}