This paper follows the recent discussion on the sparse solution recovery with quasi-norms Lq; q\in(0,1) when the sensing matrix possesses a Restricted Isometry Constant \delta_{2k} (RIC). Our key tool is an improvement on a version of “the converse of a generalized Cauchy-Schwarz inequality” extended to the setting of quasi-norm. We show that, if \delta_{2k}\le 1/2, … Read more
We develop a new modeling and exact solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the … Read more
We consider the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. We propose … Read more
This paper provides a theoretical and experimental comparison between conjugate-gradients and multigrid, two iterative schemes for solving linear systems, in the context of applying diffusion-based correlation models in data assimilation. In this context, a large number of such systems has to be (approximately) solved if the implicit mode is chosen for integrating the involved diffusion … Read more
The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations $Ax=b$, where $A\in\Re^{n\times n}$ is symmetric positive definite. Let $x_k$ denote the $k$–th iterate of CG. In this paper we obtain an expression for $J_k$, the Jacobian matrix of $x_k$ with respect to $b$. We use … Read more
We introduce a flexible optimization framework for nuclear norm minimization of matrices with linear structure, including Hankel, Toeplitz and moment structures, and catalog applications from diverse fields under this framework. We discuss various first-order methods for solving the resulting optimization problem, including alternating direction methods of multipliers, proximal point algorithms and gradient projection methods. We … Read more
The continuous adjoint sensitivity analysis for a class of optimal design problem, formerly studied in (Turteltaub, 2005), is revisited in this note. Full details of derivation is presented. It is shown that the adjoint PDE derived in this study is not identical to one derived in (Turteltaub, 2005). Article Download View Note: Optimal non-homogeneous composites … Read more
We consider a class of optimization problems for sparse signal reconstruction which arise in the field of Compressed Sensing (CS). A plethora of approaches and solvers exist for such problems, for example GPSR, FPC AS, SPGL1, NestA, l1 ls, PDCO to mention a few. CS applications lead to very well conditioned optimization problems and therefore … Read more
Many statistical $M$-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension $\pdim$ to grow with (and possibly exceed) … Read more
We present an approach for nonlinear programming (NLP) based on the direct minimization of an exact di erentiable penalty function using trust-region Newton techniques. As opposed to existing algorithmic approaches to NLP, the approach provides all the features required for scalability: it can eciently detect and exploit directions of negative curvature, it is superlinearly convergent, and … Read more