Related papers: Set Reduction In Nonlinear Equations
Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…
The convergence of inexact Newton methods is studied for solving generalized equations on Riemannian manifolds by using the metric regularity property, which is also explored. Under appropriate conditions and without any additional…
A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica and Matlab) do not offer a solution, is to find all the real solutions of a system of N nonlinear equations in a certain finite…
In this paper a special piecewise linear system is studied. It is shown that, under a mild assumption, the semi-smooth Newton method applied to this system is well defined and the method generates a sequence that converges linearly to a…
This paper presents a methodology for finding numerically, by means of curve-following, all real solutions of a general system of $n$ nonlinear equations in $n$ unknowns, within a given $n$-dimensional box. The main idea behind our method…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the…
We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for…
Neural networks functions are supposed to be able to encode the desired solution of an inverse problem very efficiently. In this paper, we consider the problem of solving linear inverse problems with neural network coders. First we…
Newton's method is the most widespread high-order method, demanding the gradient and the Hessian of the objective function. However, one of the main disadvantages of Newtons method is its lack of global convergence and high iteration cost.…
Motivated by machine learning problems over large data sets and distributed optimization over networks, we develop and analyze a new method called incremental Newton method for minimizing the sum of a large number of strongly convex…
In this paper, we propose a third-order Newton's method which in each iteration solves a semidefinite program as a subproblem. Our approach is based on moving to the local minimum of the third-order Taylor expansion at each iteration,…
The paper deals with the following system of nonlinear difference equations \begin{equation*} x_{n+1}=ax_{n}^{2}y_{n}+bx_{n}y_{n}^{2},\ y_{n+1}=cx_{n}^{2}y_{n}+dx_{n}y_{n}^{2},\ n\in \mathbb{N}_{0}, \end{equation*} where the initial values…
The problem of finding a point in the intersection of closed sets can be solved by the method of alternating projections and its variants. It was shown in earlier papers that for convex sets, the strategy of using quadratic programming (QP)…
We present a derivative-based algorithm for nonlinearly constrained optimization problems that is tolerant of inaccuracies in the data. The algorithm solves a semi-smooth set of nonlinear equations that are equivalent to the first-order…
Superlinear convergence has been an elusive goal for black-box nonsmooth optimization. Even in the convex case, the subgradient method is very slow, and while some cutting plane algorithms, including traditional bundle methods, are popular…
In this paper we investigate how standard nonlinear programming algorithms can be used to solve constrained optimization problems in a distributed manner. The optimization setup consists of a set of agents interacting through a…
Small-scale plasticity problems are often characterised by different patterning behaviours ranging from macroscopic down to the atomistic scale. In successful models of such complex behaviour, its origin lies within non-convexity of the…
Unconstrained convex optimization problems have enormous applications in various field of science and engineering. Different iterative methods are available in literature to solve such problem, and Newton method is among the oldest and…
This paper proposes several novel optimization algorithms for minimizing a nonlinear objective function. The algorithms are enlightened by the optimal state trajectory of an optimal control problem closely related to the minimized objective…