Consider the problem: given a real number x and an error bound ε, find an interval such that it contains √ x and its width is less than ε. One way to solve the problem is to start with an initial interval and repeatedly to update it by applying an interval refinement map on it until it becomes narrow enough. In this paper, we prove that the well known Secant-Newton map is the optimal among a ce...

In this paper, we presented a logarithmic-quadratic proximal alternating direction method for structured variational inequalities. The method generates the new iterate by searching the optimal step size along the descent direction. Global convergence of the new method is proved under certain assumptions.

This paper presents two methods for the design of FIR lters with arbitrary magnitude and phase responses according to a weighted mean squared error criterion with constraints on the resulting magnitude and phase errors. This constrained least square criterion allows for an arbitrary trade-o between pure L2 lters and Chebyshev lters. The resulting nonlinear optimization problem is either convert...

We prove a nontrivial bound of O(|D|27/56+ ) for the 3-part of the class number of a quadratic field Q( √ D) by using a variant of the square sieve and the q-analogue of van der Corput’s method to count the number of squares of the form 4x3− dz2 for a square-free positive integer d and bounded x, z.

Under the strongly convex assumption, several recent works studied the global linear convergence rate of the proximal incremental aggregated gradient (PIAG) method for minimizing the sum of a large number of smooth component functions and a non-smooth convex function. In this paper, under the quadratic growth condition–a strictly weaker condition than the strongly convex assumption, we derive a...

We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and first-order algorithms. The simplified Lagrangian-CPP relaxation of such QOPs proposed by Arima, Kim, and Kojima in 2012 takes one of the simplest forms, an unconstrained conic linear o...

This paper presents an inexact proximal method for solving monotone variational inequality problems with a given separable structure. The proposed algorithm is natural extension of the Proximal Multiplier Algorithm Distances (PMAPD) by Sarmiento et al . [ Optimization 65 (2016) 501–537], which unified works Chen and Teboulle (PCPM method), Kyono Fukushima (NPCPMM) developed convex programs part...

In this article, we aim to solve high-dimensional convex quadratic programming (QP) problems with a large number of terms, linear equality, and inequality constraints. To the targeted QP problem desired accuracy efficiently, consider restricted-Wolfe dual develop two-phase Proximal Augmented Lagrangian method (QPPAL), Phase I generate reasonably good initial point warm start II obtain an accura...

This paper offers a complete method to find the exact frequency, damping, amplitude, and phase of the exponential molds. A simulated signal is taken to fit the real one. When this simulated signal is equal to the real one, the parameters of the simulated signal are identical to the real values. This method includes three major steps, initial value setting, gradient method, and quadratic interpo...

One approach to computing a square root of a matrix A is to apply Newton's method to the quadratic matrix equation F( X) = X2 A =0. Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simpl...