Nevertheless, the relation between the penalty formulation with a partial regularizer and the original problem was not much studied yet. Under some suitable assumptions, we show that the penalty formulation based on a partial regularization is an exact reformulation of the original problem in the ...
The identified equation is typically an ill-conditioned inverse problem. To improve this problem and ensure that the obtained results have high robustness, the regularization method - Tikhonov (ℓ2-norm) and sparse (ℓ1-norm) regularization is used. Due to the possibility of obtaining a ...
As mentioned in Section II, the oldest idea for determining the regularization parameter is to consider it as a Lagrange multiplier of an equivalent constrained problem (Andrews and Hunt, 1977; Luenberger, 1984). This approach leads to the two following criteria (Bertero et al., 1988): 1. Am...
A central problem in machine learning is how to make an algorithm that will perform wellnot just on the training data, but also on new inputs. Many strategies used in machine learning are explicitly designed to reduce the test error, possibly at the expense of increased training error. These...
Under validity of a constraint qualification, we show that the stationary points of the regularized problem converge to a stationary point of the relaxed reformulation and under additional condition it is even a stationary point of the original problem. We conclude the paper by a numerical example....
It is an extension of the steepest descent method for solving smooth unconstrained optimization problem. The feasible steepest descent direction has an explicit expression and the method is easy to implement. Under very mild conditions, we show that the proposed method is globally convergent. We ...
Suppose that for a known matrixAand vectorb, we wish to find a vectorxsuch that: The standard approach isordinary least squareslinear regression. However, if noxsatisfies the equation or more than onexdoes—that is, the solution is not unique—the problem is said to beill posed. In such...
First, we give results of existence and uniqueness and prove the link between the constrained minimization problem and the minimization of an associated Lagrangian functional. Then we describe a relaxation method for computing the solution, and give a proof of convergence. After this, we explain ...
On the other hand, the second approach requires only sufficient knowledge of the system under study to permit a reasonable selection of dictionary functions; nonlinearity is introduced through the positivity constraint. This problem becomes increasingly ill-posed with increasing M, and will in general ...
It is worth to remark that both the problems (10) and (18) can be cast as in (22) with \(\Phi (\varvec{x},\varvec{y})=K(\varvec{x},\varvec{y})\), where, for the first problem, \(\varvec{y}=\varvec{\mu }_{\varvec{v}}\). The regularizer h is not present neit...