We consider the problem of portfolio selection within the classical Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the absolute values of the portfolio weights. This penalty regularizes (stabilizes) the optimization problem, encourages sparse portfolios (i.e., portfolios with only few active positions), and allows accounting for transaction costs. Our approach recovers as special cases the no-short-positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets constructed by Fama and French. Using only a modest amount of training data, we construct portfolios whose out-of-sample performance, as measured by Sharpe ratio, is consistently and significantly better than that of the naïve evenly weighted portfolio.penalized regression | portfolio choice | sparsity I n 1951, Harry Markowitz ushered in the modern era of portfolio theory by applying simple mathematical ideas to the problem of formulating optimal investment portfolios (1). He argued that single-minded pursuit of high returns constitutes a poor strategy, and suggested that rational investors must, instead, balance their desires for high returns and for low risk, as measured by variability of returns.It is not trivial, however, to translate Markowitz's conceptual framework into a portfolio selection algorithm in a real-world context. The recent survey (2) examined several portfolio construction algorithms inspired by the Markowitz framework. Given a reasonable amount of training data, the authors found none of the surveyed algorithms able to significantly or consistently outperform the naïve strategy where each available asset is given an equal weight in the portfolio. This disappointing performance is partly due to the structure of Markowitz's optimization framework. Specifically, the optimization at the core of the Markowitz scheme is empirically unstable: small changes in assumed asset returns, volatilities, or correlations can have large effects on the output of the optimization procedure. In this sense, the classic Markowitz portfolio optimization is an ill-posed (or ill-conditioned) inverse problem. Such problems are frequently encountered in other fields; a variety of regularization procedures have been proposed to tame the troublesome instabilities (3).In this article, we discuss a regularization of Markowitz's portfolio construction. We will restrict ourselves to the traditional Markowitz mean-variance approach. (Similar ideas could also be applied to different portfolio construction frameworks considered in the literature.) Moreover, we focus on one particular regularization method, and highlight some very special properties of the regularized portfolios obtained through its use.Our proposal consists of augmenting the original Markowitz objective function by adding a penalty term proportional to the sum of the absolute values of the portfolio weigh...

Regularization of ill-posed linear inverse problems via 1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an 1 penalized functional is via an iterative softthresholding algorithm. We propose an alternative implementation to 1 -constraints, using a gradient method, with projection on 1 -balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.

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We develop a new proximal-gradient method for minimizing the sum of a differentiable, possibly nonconvex, function plus a convex, possibly non differentiable, function. The key features of the proposed method are the definition of a suitable descent direction, based on the proximal operator associated to the convex part of the objective function, and an Armijo-like rule to determine the step size along this direction ensuring the sufficient decrease of the objective function. In this frame, we especially address the possibility of adopting a metric which may change at each iteration and an inexact computation of the proximal point defining the descent direction. For the more general nonconvex case, we prove that all limit points of the iterates sequence are stationary, while for convex objective functions we prove the convergence of the whole sequence to a minimizer, under the assumption that a minimizer exists. In the latter case, assuming also that the gradient of the smooth part of the objective function is Lipschitz, we also give a convergence rate estimate, showing the O( 1 k ) complexity with respect to the function values. We also discuss verifiable sufficient conditions for the inexact proximal point and we present the results of a numerical experience on a convex total variation based image restoration problem, showing that the proposed approach is competitive with another state-of-the-art method.

S U M M A R YWe propose the use of 1 regularization in a wavelet basis for the solution of linearized seismic tomography problems Am = d, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used to find a sparse solution m that contains no more fine-scale structure than is necessary to fit the data d to within its assigned errors.

An explicit algorithm for the minimization of an ℓ 1 penalized least squares functional, with non-separable ℓ 1 term, is proposed. Each step in the iterative algorithm requires four matrix vector multiplications and a single simple projection on a convex set (or equivalently thresholding). Convergence is proven and a 1/N convergence rate is derived for the functional. In the special case where the matrix in the ℓ 1 term is the identity (or orthogonal), the algorithm reduces to the traditional iterative soft-thresholding algorithm. In the special case where the matrix in the quadratic term is the identity (or orthogonal), the algorithm reduces to a gradient projection algorithm for the dual problem.By replacing the projection with a simple proximity operator, other convex nonseparable penalties than those based on an ℓ 1 -norm can be handled as well.

International audienceWe propose a class of spherical wavelet bases for the analysis of geophysical models and for the tomographic inversion of global seismic data. Its multiresolution character allows for modelling with an effective spatial resolution that varies with position within the Earth. Our procedure is numerically efficient and can be implemented with parallel computing. We discuss two possible types of discrete wavelet transforms in the angular dimension of the cubed sphere. We describe benefits and drawbacks of these constructions and apply them to analyse the information in two published seismic wave speed models of the mantle, using the statistics of wavelet coefficients across scales. The localization and sparsity properties of wavelet bases allow finding a sparse solution to inverse problems by iterative minimization of a combination of the ℓ2 norm of the data residuals and the ℓ1 norm of the model wavelet coefficients. By validation with realistic synthetic experiments we illustrate the likely gains from our new approach in future inversions of finite-frequency seismic dat

We consider a variable metric linesearch based proximal gradient method for the minimization of the sum of a smooth, possibly nonconvex function plus a convex, possibly nonsmooth term. We prove convergence of this iterative algorithm to a critical point if the objective function satisfies the Kurdyka-Lojasiewicz property at each point of its domain, under the assumption that a limit point exists. The proposed method is applied to a wide collection of image processing problems and our numerical tests show that our algorithm results to be flexible, robust and competitive when compared to recently proposed approaches able to address the optimization problems arising in the considered applications.