Related papers: An efficient family of optimal eight-order iterati…
In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain…
We study the efficiency of the multisection method for univariate nonlinear equations, relative to that for the well-known bisection method. We show that there is a minimal effort algorithm that uses more sections than the bisection method,…
We introduce a new form of Lagrangian and propose a simple first-order algorithm for nonconvex optimization with nonlinear equality constraints. We show the algorithm generates bounded dual iterates, and establish the convergence to KKT…
In this paper, the Newton-Anderson method, which results from applying an extrapolation technique known as Anderson acceleration to Newton's method, is shown both analytically and numerically to provide superlinear convergence to non-simple…
This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…
This paper proposes an implicit family of sub-step integration algorithms grounded in the explicit singly diagonally implicit Runge-Kutta (ESDIRK) method. The proposed methods achieve third-order consistency per sub-step and thus the…
This work proposes a higher-order iterative framework for solving matrix equations, inspired by the structure and functionality of neural networks. A modification of the classical Jacobi iterative method is introduced to compute…
Conic optimization has recently emerged as a powerful tool for designing tractable and guaranteed algorithms for non-convex polynomial optimization problems. On the one hand, tractability is crucial for efficiently solving large-scale…
Given a full column rank matrix $A \in \mathbb{R}^{m\times n}$ ($m\geq n$), we consider a special class of linear systems of the form $A^\top Ax=A^\top b+c$ with $x, c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. The occurrence of $c$ in…
We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$…
In this paper, we offer a comparison in terms of computational efficiency between two techniques to avoid order reduction when using Strang method to integrate nonlinear initial boundary value problems with time-dependent boundary…
This work wishes to support various mathematical issues concerning the iterative methods with the help of new programming languages. We consider a way to show how problems in math have an answer by using different academic resources and…
This paper proposes new derivations of three well-known sorting algorithms, in their functional formulation. The approach we use is based on three main ingredients: first, the algorithms are derived from a simpler algorithm, i.e. the…
A simple iteration methodology for the solution of a set of a linear algebraic equations is presented. The explanation of this method is based on a pure geometrical interpretation and pictorial representation. Convergence using this method…
This paper presents an inverse optimality method to solve the Hamilton-Jacobi-Bellman equation for a class of nonlinear problems for which the cost is quadratic and the dynamics are affine in the input. The method is inverse optimal because…
A sequential quadratic optimization algorithm is proposed for solving smooth nonlinear equality constrained optimization problems in which the objective function is defined by an expectation of a stochastic function. The algorithmic…
While the theory of operator approximation with any given accuracy is well elaborated, the theory of {best constrained} constructive operator approximation is still not so well developed. Despite increasing demands from applications this…
A new effective method for factorization of a class of nonrational $n\times n$ matrix-functions with \emph{stable partial indices} is proposed. The method is a generalization of the one recently proposed by the authors which was valid for…
We aim at computing the derivative of the solution to a parametric optimization problem with respect to the involved parameters. For a class broader than that of strongly convex functions, this can be achieved by automatic differentiation…
A new pattern search method for bound constrained optimization is introduced. The proposed algorithm employs the coordinate directions, in a suitable way, with a nonmonotone line search for accepting the new iterate, without using…