Related papers: Nonlinear approximation of functions based on non-…
We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the…
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…
We propose an algorithm for reduction of the problem of maximization of fraction of two functionals to the equivalent procedure including maximization of difference between the functionals and the solution of an equation of scalar unknown.…
Minimization methods that search along a curvilinear path composed of a non-ascent nega- tive curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a…
We introduce a new model of linear regression for random functional inputs taking into account the first order derivative of the data. We propose an estimation method which comes down to solving a special linear inverse problem. Our…
Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…
The real-life merit functions have an unimaginable complexity of an M-dimensional topography, where M is the number of the parameters. It is shown that there is an underlying noise-free merit function, called follower merit function which…
In this paper we focus on the linear functionals defining an approximate version of the gradient of a function. These functionals are often used when dealing with optimization problems where the computation of the gradient of the objective…
Sequential quadratic optimization algorithms are proposed for solving smooth nonlinear optimization problems with equality constraints. The main focus is an algorithm proposed for the case when the constraint functions are deterministic,…
We investigate the nonlinear regression problem under L2 loss (square loss) functions. Traditional nonlinear regression models often result in non-convex optimization problems with respect to the parameter set. We show that a convex…
It is shown that whenever the multiplicative normalization of a fitting function is not known, least square fitting by $\chi^2$ minimization can be performed with one parameter less than usual by converting the normalization parameter into…
Having a function $f$ and a set of functionals $\{\mathcal{C}_{n}\}$, $c_n^f \equiv \mathcal{C}_n \left(f\right)$, one can interpret function approximation very generally as a construction of some function $\mathcal{A}_{N}^{f}$ such that…
We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive…
We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…
We develop and analyze stochastic inexact Gauss-Newton methods for nonlinear least-squares problems and for nonlinear systems ofequations. Random models are formed using suitable sampling strategies for the matrices involved in the…
Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…
We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were…
We study the optimization of functions with $n>2$ arguments that have a representation as a sum of several functions that have only $2$ of the $n$ arguments each, termed sums of bivariates, on finite domains. The complexity of optimizing…
It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…
Linear Least Squares is a very well known technique for parameter estimation, which is used even when sub-optimal, because of its very low computational requirements and the fact that exact knowledge of the noise statistics is not required.…