where: [e.sub.0] is the rigid configuration of the beam, [lambda] is the vector of the Lagrange multipliers
, and [[PI].sub.e] is defined as:
Instead of imposing the perfect match of the solution with each constraints separately--this would generate false structures specific to noise and would proliferate the Lagrange multipliers
, Gull and Daniell melted down them into a single expression:
The first one is based on separation and support of convex sets theorems (theorems of the Hausdorff type), the second is based on penalty functions and the third and final approach is based on the classical theory of Lagrange multipliers
. These optimal conditions, assuming differentiability, are the Karush-Kuhn-Tucker conditions.
The LRMR algorithms include APG (accelerated proximal gradient) and ALM (augmented Lagrange multipliers
) ; ALM is divided into two algorithms, EALM (exact augmented Lagrange multipliers
) and IALM (inexact augmented Lagrange multipliers
), and the three algorithms are described below.
We shall adopt the Batalin-Vilkovisky (BV) prescription  combined with the Blau-Thompson minimal action gauge-fixing [12, 24], which fix the Lagrange multipliers
associated with the gauge conditions (we propose the following notation for the superfield charges: [sup.s][[OMEGA].sup.g.sub.p], where s stands for the SUSY number, g denotes the ghost number, and p indicates the form degree).
At the same time, it is worth pointing out that the NIM does not need to approximately identify the general Lagrange multipliers
via the variational theory.
According to the Lagrange multipliers
method, Lagrange function is as follows:
To simplify quadratic equations, Lagrange multipliers
ai are applied with the constraint and subtracted from the condition for maximal margin, which is 1/2 <W,W>.
The obtained coefficients were then optimized using Augmented Lagrange multipliers
The Lagrangian method for contact modeling, where nonpenetration condition is strictly and exactly satisfied and the contact force is represented by constraint matrix and Lagrange multipliers
, has also attracted attentions from a lot of researchers [10-13].
where [N.sub.s] is the number of nonzero Lagrange multipliers
. The pseudocode of the SELM training algorithm is summarized in Algorithm 1.
So, we presented an inexact augmented Lagrange multipliers
(IALM) algorithm to solve the RPCA problem .