Global Convergence of Damped Newton's Method for Nonsmooth Equations, via the Path Search

A natural damping of Newton's method for nonsmooth equations is presented. This damping, via the path search instead of the traditional line search, enlarges the domain of convergence of Newton's method and therefore is said to be globally convergent. Convergence behavior is like that of line search damped Newton's method for smooth equations, including Q-quadratic convergence rates under appropriate conditions. Applications of the path search include damping Robinson-Newton's method for nonsmooth normal equations corresponding to nonlinear complementarity problems and variational inequalities, hence damping both Wilson's method (sequential quadratic programming) for nonlinear programming and Josephy-Newton's method for generalized equations. Computational examples from nonlinear programming are given

A Newton Collocation Method for Solving Dynamic Bargaining Games

We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.

Mathematical programs with equilibrium constraints: automatic reformulation and solution via constrained optimization

Constrained optimization has been extensively used to solve many large scale deterministic problems arising in economics, including, for example, square systems of equations and nonlinear programs. A separate set of models have been generated more recently, using complementarity to model various phenomenon, particularly in general equilibria. The unifying framework of mathematical programs with equilibrium constraints (MPEC) has been postulated for problems that combine facets of optimization and complementarity. This paper briefly reviews some methods available to solve these problems and described a new suite of tools for working with MPEC models. Computational results demonstrating the potential of this tool are given that automatically construct and solve a variety of different nonlinear programming reformulations of MPEC problems.\ud \ud This material is based on research partially supported by the National Science Foundation Grant CCR-9972372, the Air Force Office of Scientific Research Grant F49620-01-1-0040, Microsoft Corporation and the Guggenheim Foundation

Deflation for semismooth equations

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure

Nonsmooth Nonconvex Variational Reconstruction for Electrical Impedance Tomography

Electrical Impedance Tomography is an imaging technique that aims to reconstruct the inner conductivity distribution of a medium starting from a set of measured voltages registered by a series of electrodes that are positioned on the surface of the medium. Such technique was used for the first time in geological studies in 1930 and then applied to industrial procedures. The first clinical use of EIT dates back to 1987. In 2018 EIT was validated in tissue engineering as a noninvasive and label-free imaging and monitoring tool for cell distribution (cell growth, differentiation and tissue formation) in 3D scaffolds. EIT problem can be split into a Forward and an Inverse problem. The aim of Forward EIT is to define the set of measured voltages starting from a known conductivity distribution. If the forward problem is characterized by a nonlinear mapping, called Forward Operator, from the conductivity distribution to the measured voltages, inverse EIT consists of inverting the Forward Operator. This leads to an ill-posed problem which requires regularization, either in the model or in the numerical method that is applied to define the solution. The inverse problem is modelled as a Nonlinear Least Squares problem, where one seeks to minimize the mismatch beetween the measured voltages and the ones generated by the reconstructed conductivity. Reconstruction techniques require the introduction of a regularization term which forces the reconstructed data to stick to certain prior information. In this dissertation, some state-of-the-art regularization methods are analyzed and compared via EIDORS, a specific software for EIT problems. The aim is to reconstruct the variation in conductivity within a 2D section of a 3D scaffold. Furthermore a variational formulation on a 2D mesh for a space-variant regularization is proposed, based on a combination of high order and nonconvex operators, which respectively seek to recover piecewise inhomogeneous and piecewise linear regions

Nonsmooth Optimization; Proceedings of an IIASA Workshop, March 28 - April 8, 1977

Optimization, a central methodological tool of systems analysis, is used in many of IIASA's research areas, including the Energy Systems and Food and Agriculture Programs. IIASA's activity in the field of optimization is strongly connected with nonsmooth or nondifferentiable extreme problems, which consist of searching for conditional or unconditional minima of functions that, due to their complicated internal structure, have no continuous derivatives. Particularly significant for these kinds of extreme problems in systems analysis is the strong link between nonsmooth or nondifferentiable optimization and the decomposition approach to large-scale programming. This volume contains the report of the IIASA workshop held from March 28 to April 8, 1977, entitled Nondifferentiable Optimization. However, the title was changed to Nonsmooth Optimization for publication of this volume as we are concerned not only with optimization without derivatives, but also with problems having functions for which gradients exist almost everywhere but are not continous, so that the usual gradient-based methods fail. Because of the small number of participants and the unusual length of the workshop, a substantial exchange of information was possible. As a result, details of the main developments in nonsmooth optimization are summarized in this volume, which might also be considered a guide for inexperienced users. Eight papers are presented: three on subgradient optimization, four on descent methods, and one on applicability. The report also includes a set of nonsmooth optimization test problems and a comprehensive bibliography