Economic inexact restoration for derivative-free expensive function minimization and applications

The Inexact Restoration approach has proved to be an adequate tool for handling the problem of minimizing an expensive function within an arbitrary feasible set by using different degrees of precision in the objective function. The Inexact Restoration framework allows one to obtain suitable convergence and complexity results for an approach that rationally combines low- and high-precision evaluations. In the present research, it is recognized that many problems with expensive objective functions are nonsmooth and, sometimes, even discontinuous. Having this in mind, the Inexact Restoration approach is extended to the nonsmooth or discontinuous case. Although optimization phases that rely on smoothness cannot be used in this case, basic convergence and complexity results are recovered. A derivative-free optimization phase is defined and the subproblems that arise at this phase are solved using a regularization approach that take advantage of different notions of stationarity. The new methodology is applied to the problem of reproducing a controlled experiment that mimics the failure of a dam

On complexity and convergence of high-order coordinate descent algorithms

Coordinate descent methods with high-order regularized models for box-constrained minimization are introduced. High-order stationarity asymptotic convergence and first-order stationarity worst-case evaluation complexity bounds are established. The computer work that is necessary for obtaining first-order $\varepsilon$-stationarity with respect to the variables of each coordinate-descent block is $O(\varepsilon^{-(p+1)/p})$ whereas the computer work for getting first-order $\varepsilon$-stationarity with respect to all the variables simultaneously is $O(\varepsilon^{-(p+1)})$. Numerical examples involving multidimensional scaling problems are presented. The numerical performance of the methods is enhanced by means of coordinate-descent strategies for choosing initial points

SPECTRAL PROJECTED GRADIENT METHOD WITH INEXACT RESTORATION FOR MINIMIZATION WITH NONCONVEX CONSTRAINTS

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)This work takes advantage of the spectral projected gradient direction within the inexact restoration framework to address nonlinear optimization problems with nonconvex constraints. The proposed strategy includes a convenient handling of the constraints, together with nonmonotonic features to speed up convergence. The numerical performance is assessed by experiments with hard-spheres problems, pointing out that the inexact restoration framework provides an adequate environment for the extension of the spectral projected gradient method for general nonlinearly constrained optimization.31316281652Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [E-26/171.164/2003 - APQ1]FAPESP [01/04597-4, 06/53768-0

BB: An R Package for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function

We discuss <code>R</code> package <b>BB</b>, in particular, its capabilities for solving a nonlinear system of equations. The function <code>BBsolve</code> in <b>BB</b> can be used for this purpose. We demonstrate the utility of these functions for solving: (a) large systems of nonlinear equations, (b) smooth, nonlinear estimating equations in statistical modeling, and (c) non-smooth estimating equations arising in rank-based regression modeling of censored failure time data. The function <code>BBoptim</code> can be used to solve smooth, box-constrained optimization problems. A main strength of <b>BB</b> is that, due to its low memory and storage requirements, it is ideally suited for solving high-dimensional problems with thousands of variables

Trading book and credit risk: How fundamental is the Basel review?

AbstractWithin the new Basel regulatory framework for market risks, non-securitization credit positions in the trading book are subject to a separate default risk charge (formally incremental default risk charge). Banks using the internal model approach are required to use a two-factor model and a 99.9% VaR capital charge. This model prescription is intended to reduce risk-weighted asset variability, a known feature of internal models, and improve their comparability among financial institutions. In this paper, we analyze the theoretical foundations and relevance of these proposals. We investigate the practical implications of the two-factor and correlation calibration constraints through numerical applications. We introduce the Hoeffding decomposition of the aggregate unconditional loss to provide a systematic-idiosyncratic representation. In particular, we examine the impacts of a J-factor correlation structure on risk measures and risk factor contributions for long-only and long-short credit-sensitive portfolios