Stability results are given for a class of feedback systems arising from the regulation of time-varying discrete-time systems using optimal infinite-horizon and moving-horizon feedback laws. The class is characterized by joint constraints on the state and the control, a general nonlinear cost function and nonlinear equations of motion possessing two special properties. It is shown that weak conditions on the cost function and the constraints are sufficient to guarantee uniform asymptotic stability of both the optimal infinite-horizon and moving-horizon feedback systems. The infinite-horizon cost associated with the moving-horizon feedback law approaches the optimal infinite-horizon cost as the moving horizon is extended.

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Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: stability and moving-horizon approximations

Imposition of a lasso penalty shrinks parameter estimates toward zero and performs continuous model selection. Lasso penalized regression is capable of handling linear regression problems where the number of predictors far exceeds the number of cases. This paper tests two exceptionally fast algorithms for estimating regression coefficients with a lasso penalty. The previously known l2 algorithm is based on cyclic coordinate descent. Our new l1 algorithm is based on greedy coordinate descent and Edgeworth’s algorithm for ordinary l1 regression. Each algorithm relies on a tuning constant that can be chosen by cross-validation. In some regression problems it is natural to group parameters and penalize parameters group by group rather than separately. If the group penalty is proportional to the Euclidean norm of the parameters of the group, then it is possible to majorize the norm and reduce parameter estimation to l2 regression with a lasso penalty. Thus, the existing algorithm can be extended to novel settings. Each of the algorithms discussed is tested via either simulated or real data or both. The Appendix proves that a greedy form of the l2 algorithm converges to the minimum value of the objective function.

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Coordinate descent algorithms for lasso penalized regression

The “Huber function” (or “Huber norm” ) is one of several robust error measures which interpolates between smooth (l2) treatment of small residuals and robust (l1) treatment of large residuals. Since the Huber function is differentiable, it may be minimized reliably with a standard gradient‐based optimizer. We propose to minimize the Huber function with a quasi‐Newton method that has the potential of being faster and more robust than conjugate‐gradient methods when solving nonlinear problems. Tests with a linear inverse problem for velocity analysis with both synthetic and field data suggest that the Huber function gives far more robust model estimates than does a least‐squares fit with or without damping.

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Robust inversion of seismic data using the Huber norm

A strategy for multiple removal consists of estimating a model of the multiples and then adaptively subtracting this model from the data by estimating shaping filters. A possible and efficient way of computing these filters is by minimizing the difference or misfit between the input data and the filtered multiples in a least-squares sense. Therefore, the signal is assumed to have minimum energy and to be orthogonal to the noise. Some problems arise when these conditions are not met. For instance, for strong primaries with weak multiples, we might fit the multiple model to the signal (primaries) and not to the noise (multiples). Consequently, when the signal does not exhibit minimum energy, we propose using the L 1 -norm, as opposed to the L 2 -norm, for the filter estimation step. This choice comes from the well-known fact that the L 1 -norm is robust to ‘large’ amplitude differences when measuring data misfit. The L 1 -norm is approximated by a hybrid L 1 /L 2 -norm minimized with an iteratively reweighted least-squares (IRLS) method. The hybrid norm is obtained by applying a simple weight to the data residual. This technique is an excellent approximation to the L 1 -norm. We illustrate our method with synthetic and field data where internal multiples are attenuated. We show that the L 1 -norm leads to much improved attenuation of the multiples when the minimum energy assumption is violated. In particular, the multiple model is fitted to the multiples in the data only, while preserving the primaries.

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Adaptive subtraction of multiples using the l1-norm

This paper discusses identification of parameters of generalized ordered weighted averaging (GOWA) operators from empirical data. Similarly to ordinary OWA operators, GOWA are characterized by a vector of weights, as well as the power to which the arguments are raised. We develop optimization techniques which allow one to fit such operators to the observed data. We also generalize these methods for functional defined GOWA and generalized Choquet integral based aggregation operators.

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Learning Weights in the Generalized OWA Operators

subject to the given constraints (2) and (3). (In expression (4), b, denotes the ith component of b and A, denotes the ith row of A.) The method is completely general in the sense that no restrictions are imposed on the ranks of the matrices A, C, and E, or on the signs of the elements of f. Furthermore, if no vector x satisfying (2) and (3) exists, the subroutine detects this and informs the user that the problem is infeasible. The algorithm can be used to solve the constrained 11 approximation problem. Suppose that data consisting of k points (t , y,) are to be approximated by a linear

F. D. K. Roberts

Papers

Algorithm 552: Solution of the Constrained I1 Linear Approximation Problem [F4]