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This paper considers localization of a source or a sensor from distance measurements. We argue that linear algorithms proposed for this purpose are susceptible to poor noise performance. Instead given a set of sensors/anchors of known positions and measured distances of the source/sensor to be localized from them we propose a potentially nonconvex weighted cost function whose global minimum estimates the location of the source/sensor one seeks. The contribution of this paper is to provide nontrivial ellipsoidal and polytopic regions surrounding these sensors/anchors of known positions, such that if the object to be localized is in this region, localization occurs by globally exponentially convergent gradient descent in the noise free case. Exponential convergence in the noise free case represents practical convergence as it ensures graceful performance degradation in the presence of noise. These results guide the deployment of sensors/anchors so that small subsets can be made responsible for practical localization in geographical areas determined by our approach. [ABSTRACT FROM AUTHOR]

Iterative shrinkage/thresholding (1ST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic regularizer (e.g., total variation or wavelet-based regularization). It happens that the convergence rate of these 1ST algorithms depends heavily on the linear observation operator, becoming very slow when this operator is ill-conditioned or ill-posed. In this paper, we introduce two-step 1ST (TwIST) algorithms, exhibiting much faster convergence rate than 1ST for ill-conditioned problems. For a vast class of nonquadratic convex regularizers (lP norms, some Besov norms, and total variation), we show that TwIST converges to a minimizer of the objective function, for a given range of values of its parameters. For noninvertible observation operators, we introduce a monotonic version of TwIST (MTwIST); although the convergence proof does not apply to this scenario, we give experimental evidence that MTwIST exhibits similar speed gains over 1ST. The effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples. [ABSTRACT FROM AUTHOR]

We apply nonsmooth optimization techniques to classification problems, with particular reference to the Transductive Support Vector Machine (TSVM) approach, where the considered decision function is nonconvex and nondifferentiable, hence difficult to minimize. We present some numerical results obtained by running the proposed method on some standard test problems drawn from the binary classification literature. [ABSTRACT FROM AUTHOR]