Abstract. We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For ddimensional Qk-type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the fin...

This paper analyzes sequences generated by infeasible interior point methods. In convex and nonconvex settings, we prove that moving the primal feasibility at the same rate as complementarity will ensure that the Lagrange multiplier sequence will remain bounded, provided the limit point of the primal sequence has a Lagrange multiplier, without constraint qualification assumptions. We also show ...

A Lagrange multiplierrules that uses small generalized gradients is introduced. It includes both inequality and set constraints. The generalized gradient is the linear generalized gradient. It is smaller than the generalized gradients of Clarke and Mordukhovich but retains much of their nice calculus. Its convex hull is the generalized gradient of Michel and Penot if a function is Lipschitz. Th...

In this paper, cross entropy method is used for solving dual Lagrange support vector machine (SVM). Cross entropy (CE) method is a new practical approach which is widely used in some applications such as combinatorial optimization, learning algorithm and simulation. Our approach refers to Kernel Adatron which is solving dual Lagrange SVM using gradient ascent method. Hereby, the cross entropy m...

Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.

It is proposed how to impose a general type of “noncommutativity” within classical mechanics from first principles. Formulation is performed in completely alternative way, i.e. without any resort to fuzzy and/or star product philosophy, which are extensively applied within noncommutative quantum theories. Newton–Lagrange noncommutative equations of motion are formulated and their properties are...

Most algorithms for reinforcement learning work by estimating action-value functions. Here we present a method that uses Lagrange multipliers, the costate equation, and multilayer neural networks to compute policy gradients. We show that this method can find solutions to time-optimal control problems, driving linear mechanical systems quickly to a target configuration. On these tasks its perfor...

Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formul...

In [3] we described a technique for solving certain linear operator equations by studying the operator power series de…ned by the system. Essential for obtaining explicit solutions is a Lagrange inversion formula for power series with coe¢ cients in an integral domain K. Such a formula can be found in “Recursive Matrices and Umbral Calculus”by Barnabei, Brini and Nicoletti [1]. J. F. Freeman’s ...