Nonmonotone spectral projected gradient methods on convex sets

Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo-Lampariello-Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the one-dimensional search process. Convergence properties and extensive numerical results are presented.1041196121

Optical constants and thickness determination of very thin amorphous semiconductor films

This contribution addresses the relevant question of retrieving, from transmittance data, the optical constants, and thickness of very thin semiconductor and dielectric films. The retrieval process looks for a thickness that, subject to the physical input of the problem, minimizes the difference between the measured and the theoretical spectra. This is a highly underdetermined problem but, the use of approximate-though simple-functional dependencies of the index of refraction and of the absorption coefficient on photon energy, used as an a priori information, allows surmounting the ill posedness of the problem. The method is illustrated with the analysis of transmittance data of very thin amorphous silicon films. The method enables retrieval of physically meaningful solutions for films as thin as 300 A. The estimated parameters agree well with known data or with optical parameters measured by independent methods. The limitations of the adopted model and the shortcomings of the optimization algorithm are presented and discussed. (C) 2002 American Institute of Physics.9263093310

Parameter Estimation and Quantitative Parametric Linkage Analysis with GENEHUNTER-QMOD

Objective: We present a parametric method for linkage analysis of quantitative phenotypes. The method provides a test for linkage as well as an estimate of different phenotype parameters. We have implemented our new method in the program GENEHUNTER-QMOD and evaluated its properties by performing simulations. Methods: The phenotype is modeled as a normally distributed variable, with a separate distribution for each genotype. Parameter estimates are obtained by maximizing the LOD score over the normal distribution parameters with a gradient-based optimization called PGRAD method. Results: The PGRAD method has lower power to detect linkage than the variance components analysis (VCA) in case of a normal distribution and small pedigrees. However, it outperforms the VCA and Haseman-Elston regression for extended pedigrees, nonrandomly ascertained data and non-normally distributed phenotypes. Here, the higher power even goes along with conservativeness, while the VCA has an inflated type I error. Parameter estimation tends to underestimate residual variances but performs better for expectation values of the phenotype distributions. Conclusion: With GENEHUNTER-QMOD, a powerful new tool is provided to explicitly model quantitative phenotypes in the context of linkage analysis. It is freely available at http://www.helmholtz-muenchen.de/genepi/downloads. Copyright (C) 2012 S. Karger AG, Base

Filter-based DIRECT method for constrained global optimization

This paper presents a DIRECT-type method that uses a filter methodology to assure convergence to a feasible and optimal solution of nonsmooth and nonconvex constrained global optimization problems. The filter methodology aims to give priority to the selection of hyperrectangles with feasible center points, followed by those with infeasible and non-dominated center points and finally by those that have infeasible and dominated center points. The convergence properties of the algorithm are analyzed. Preliminary numerical experiments show that the proposed filter-based DIRECT algorithm gives competitive results when compared with other DIRECT-type methods.The authors would like to thank two anonymous referees and the Associate Editor for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ciência e Tecnologia within the projects UID/CEC/00319/2013 and ˆ UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Combining filter method and dynamically dimensioned search for constrained global optimization

In this work we present an algorithm that combines the filter technique and the dynamically dimensioned search (DDS) for solving nonlinear and nonconvex constrained global optimization problems. The DDS is a stochastic global algorithm for solving bound constrained problems that in each iteration generates a randomly trial point perturbing some coordinates of the current best point. The filter technique controls the progress related to optimality and feasibility defining a forbidden region of points refused by the algorithm. This region can be given by the flat or slanting filter rule. The proposed algorithm does not compute or approximate any derivatives of the objective and constraint functions. Preliminary experiments show that the proposed algorithm gives competitive results when compared with other methods.The first author thanks a scholarship supported by the International Cooperation Program CAPES/ COFECUB at the University of Minho. The second and third authors thanks the support given by FCT (Funda¸c˜ao para Ciˆencia e Tecnologia, Portugal) in the scope of the projects: UID/MAT/00013/2013 and UID/CEC/00319/2013. The fourth author was partially supported by CNPq-Brazil grants 308957/2014-8 and 401288/2014-5.info:eu-repo/semantics/publishedVersio

On a smoothed penalty-based algorithm for global optimization

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an ε -global minimizer is proved. At each iteration k, the framework requires the ε(k) -global minimizer of a subproblem, where ε(k)→ε . We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an ε(k) -global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the ε(k)-neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.The authors would like to thank two anonymous referees for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ci ˜ encia e Tecnologia within the projects UID/CEC/00319/2013 and ˆ UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Filter-based stochastic algorithm for global optimization

We propose the general Filter-based Stochastic Algorithm (FbSA) for the global optimization of nonconvex and nonsmooth constrained problems. Under certain conditions on the probability distributions that generate the sample points, almost sure convergence is proved. In order to optimize problems with computationally expensive black-box objective functions, we develop the FbSA-RBF algorithm based on the general FbSA and assisted by Radial Basis Function (RBF) surrogate models to approximate the objective function. At each iteration, the resulting algorithm constructs/updates a surrogate model of the objective function and generates trial points using a dynamic coordinate search strategy similar to the one used in the Dynamically Dimensioned Search method. To identify a promising best trial point, a non-dominance concept based on the values of the surrogate model and the constraint violation at the trial points is used. Theoretical results concerning the sufficient conditions for the almost surely convergence of the algorithm are presented. Preliminary numerical experiments show that the FbSA-RBF is competitive when compared with other known methods in the literature.The authors are grateful to the anonymous referees for their fruitful comments and suggestions.The first and second authors were partially supported by Brazilian Funds through CAPES andCNPq by Grants PDSE 99999.009400/2014-01 and 309303/2017-6. The research of the thirdand fourth authors were partially financed by Portuguese Funds through FCT (Fundação para Ciência e Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM and UIDB/00319/2020

Nonmonotone Barzilai-Borwein Gradient Algorithm for ℓ1\ell_1-Regularized Nonsmooth Minimization in Compressive Sensing

This paper is devoted to minimizing the sum of a smooth function and a nonsmooth $\ell_1$-regularized term. This problem as a special cases includes the $\ell_1$-regularized convex minimization problem in signal processing, compressive sensing, machine learning, data mining, etc. However, the non-differentiability of the $\ell_1$-norm causes more challenging especially in large problems encountered in many practical applications. This paper proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the $\ell_1$-norm. Moreover, a nonmonotone line search technique is incorporated to find a suitable stepsize along this direction. The algorithm is easily performed, where the values of the objective function and the gradient of the smooth term are required at per-iteration. Under some conditions, the proposed algorithm is shown to be globally convergent. The limited experiments by using some nonconvex unconstrained problems from CUTEr library with additive $\ell_1$-regularization illustrate that the proposed algorithm performs quite well. Extensive experiments for $\ell_1$-regularized least squares problems in compressive sensing verify that our algorithm compares favorably with several state-of-the-art algorithms which are specifically designed in recent years.Comment: 20 page