We consider a proximal algorithm with quasi distance applied to nonconvex and nonsmooth functions involving analytic properties for an unconstrained minimization problem. We show the behavioral importance of this proximal point model for habit’s formation in Decision and Making Sciences. ArticleDownload View PDF
The possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the Barzilai-Borwein choice of step-size, and other related ideas. These methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems. This paper suggests a method which is able to take advantage … Read more
The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second-order behaviour of the objective function. Following earlier work by Gratton and Toint (2009), we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take … Read more
This view of the development of algorithms for nonlinear optimization is based on the research that has been of particular interest to the author since 1959, including several of his own contributions. After a brief survey of classical methods, which may require good starting points in order to converge successfully, the huge impact of variable … Read more
Newton’s method is a classical method for solving a nonlinear equation $F(z)=0$. We derive inexact Newton steps that lead to an inexact Newton method, applicable near a solution. The method is based on solving for a particular $F'(z_{k’})$ during $p$ consecutive iterations $k=k’,k’+1,\dots,k’+p-1$. One such $p$-cycle requires $2^p-1$ solves with the matrix $F'(z_{k’})$. If matrix … Read more
We describe a collection of test problems which have been used to develop and test data fitting software for identifying parameters in explicit model functions, dynamical systems of equations, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time-dependent partial differential equations with or without algebraic equations. The test cases … Read more
Mathematica is an advanced software system that enables symbolic computing, numerics, program code development, model visualization and professional documentation in a unified framework. Our MathOptimizer software package serves to solve global and local optimization models developed using Mathematica. We introduce MathOptimizer’s key features and discuss its usage options that support a range of operational modes. … Read more
Image restoration and reconstruction from blurry and noisy observation is known to be ill-posed. To stabilize the recovery, total variation (TV) regularization was introduced by Rudin, Osher and Fatemi in \cite{LIR92}, which has demonstrated superiority in preserving image edges. However, the nondifferentiability of TV makes the underlying optimization problems difficult to solve. In this paper, … Read more
EASY-FIT is an interactive software system to identify parameters and compute optimal designs in explicit model functions, steady-state systems, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time dependent partial differential equations with or without algebraic equations. Proceeding from given experimental data, i.e. observation times and measurements, the minimum … Read more
The availability of nonlinear programming test problems is extremely important to test optimization codes or to develop new algorithms. We describe the usage of the Fortran subroutines for all 306 test problems of two previous collections of the author, see Hock and Schittkowski (1981) and Schittkowski (1987). For each test example, we provide an optimal … Read more