Ignace Loris

Sparse and stable Markowitz portfolios

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.

Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints

Regularization of ill-posed linear inverse problems via ℓ1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ℓ1 penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to ℓ1-constraints, using a gradient method, with projection on ℓ1-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.

Accelerating gradient projection methods for l1-constrained signal recovery by steplength selection rules

Abstract We propose a new gradient projection algorithm that compares favorably with the fastest algorithms available to date for l 1 -constrained sparse recovery from noisy data, both in the compressed sensing and inverse problem frameworks. The method exploits a line-search along the feasible direction and an adaptive steplength selection based on recent strategies for the alternation of the well-known Barzilai–Borwein rules. The convergence of the proposed approach is discussed and a computational study on both well conditioned and ill-conditioned problems is carried out for performance evaluations in comparison with five other algorithms proposed in the literature.

On a direct bilinearization method: Kaup's higher-order water wave equation as a modified nonlocal Boussinesq equation

A systematic procedure for the bilinearization of classes of soliton equations is developed with the help of a generalization of Bells exponential polynomials (1934). Application of this procedure to Kaups higher-order wave equation (1975) discloses several links with other soliton systems. In particular, it is found that the Kaup equation is the modified version of a sech square soliton system which constitutes an alternative to the good Boussinesq equation.

On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty

An explicit algorithm for the minimization of an l1-penalized least-squares functional, with non-separable l1 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 l1 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 non-separable penalties than those based on an l1-norm can be handled as well.

BILINEAR FORM AND SOLUTIONS OF THE K-CONSTRAINED KADOMTSEV-PETVIASHVILI HIERARCHY

We show how to derive an alternative bilinear formulation for the k-constrained Kadomtsev - Petviashvili hierarchy. This Hirota form allows for the easy identification of a broad class of solutions to these equations.

Symmetry reductions of the BKP hierarchy

A general symmetry of the bilinear BKP hierarchy is studied in terms of tau functions. We use this symmetry to define reductions of the BKP hierarchy, among which new integrable systems can be found. The reductions are connected to constraints on the Lax operator as well as on the bilinear formulation. A class of solutions for the reduced equations is derived.

On reduced CKP equations

Symmetry reductions of the CKP hierarchy are discussed in a pseudodifferential and tau-function context. Solutions of the resulting nonlinear partial differential equations are obtained via the methods of gauge transformations and of tau functions. Reductions of other (2 + 1)-dimensional hierarchies related to KP are also briefly investigated.

Variable Metric Inexact Line-Search-Based Methods for Nonsmooth Optimization

We develop a new proximal-gradient method for minimizing the sum of a differentiable, possibly nonconvex, function plus a convex, possibly nondifferentiable, 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 stepsize 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...

Binary Darboux transformations for constrained KP hierarchies

We describe how Darboux transformations and binary Darboux transformations can be constructed for (vector-) constrained KP hierarchies. These transformations are then used to obtain explicit classes of Wronskian and Grammian solutions for these hierarchies. The relationship between these two types of solutions is also discussed.