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2nd Symposium on Inverse Problems, Design and Optimization

作者： Wang, Yanfei Du, Yonghua Hu, Tiandou Chinese Acad Sci Inst Geol & Geophys Beijing Peoples R China Chinese Acad Sci Inst High Energy Phys Beijing Peoples R China Chinese Acad Sci Grad Univ Beijing Peoples R China

The theory of synchrotron radiation (SR) has been well understood and published. We study the numerical methods for the reconstruction of the spectral distribution function of SR by measurement of the attenuation of the SR energy spectrum. The reconstruction of the spectral distribution function of SR is an ill-posed integral operator equation of the first kind. Therefore, how to overcome the ill-posedness is a major task in numerical computation. We study the projected gradient methods and propose a non-monotone decreasing algorithm, which is called the projected Barzilai-Borwein (PBB) method. The feasibility of the method is studied in detail by using a hypothetical SR spectrum. The applied results of the spectrum of 4W1B beamline (an unfocused X-ray monochromator beamline with 4 mrad (milliradian) of horizontal acceptance which is extracted from the wavelength shifter 4W1) in BSRF (Beijing Synchrotron Radiation Facility) are shown.

关键词： synchrotron radiation (SR) projected gradient methods numerical inversion X-ray attenuation

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projected gradient methods for synchrotron radiation spectra distribution function reconstruction

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 2009年第2期17卷 175-186页

关键词： synchrotron radiation (SR) projected gradient methods numerical inversion X-ray attenuation

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Universal nonmonotone line search method for nonconvex multiobjective optimization problems with convex constraints

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COMPUTATIONAL & APPLIED MATHEMATICS 2025年第1期44卷 1-22页

作者： Pinheiro, Maria Eduarda Grapiglia, Geovani Nunes Trier Univ Dept Math Univ ring 15 D-54296 Trier Germany Catholic Univ Louvain ICTEAM INMA Ave Georges Lemaitre 4 B-1348 Louvain La Neuve Belgium

In this work we propose a general nonmonotone line-search method for nonconvex multiobjective optimization problems with convex constraints. At the kth iteration, the degree of nonmonotonicity is controlled by a vector nu k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _{k}$$\end{document} with nonnegative components. Different choices for nu k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _{k}$$\end{document} lead to different nonmonotone step-size rules. Assuming that the sequence nu kk >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \nu _{k}\right\} _{k\ge 0}$$\end{document} is summable, and that the ith objective function has H & ouml;lder continuous gradient with smoothness parameter theta i is an element of(0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _i \in (0,1]$$\end{document}, we show that the proposed method takes no more than O & varepsilon;-1+1 theta min\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\left( \epsilon <^>{-\left( 1+\frac{1}{\theta _{\min }}\right) }\right) $$\end{document} iterations to find a & var

关键词： Multiobjective optimization projected gradient methods Nonmonotone line searches Worst-case complexity

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Globally convergent trust-region methods for self-consistent field electronic structure calculations

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JOURNAL OF CHEMICAL PHYSICS 2004年第22期121卷 10863-10878页

作者： Francisco, JB Martínez, JM Martínez, L State Univ Campinas UNICAMP IMECC Dept Appl Math BR-13081970 Campinas SP Brazil State Univ Campinas UNICAMP IQ Dept Phys Chem BR-13081970 Campinas SP Brazil

As far as more complex systems are being accessible for quantum chemical calculations, the reliability of the algorithms used becomes increasingly important. Trust-region strategies comprise a large family of optimization algorithms that incorporates both robustness and applicability for a great variety of problems. The objective of this work is to provide a basic algorithm and an adequate theoretical framework for the application of globally convergent trust-region methods to electronic structure calculations. Closed shell restricted Hartree-Fock calculations are addressed as finite-dimensional nonlinear programming problems with weighted orthogonality constraints. A Levenberg-Marquardt-like modification of a trust-region algorithm for constrained optimization is developed for solving this problem. It is proved that this algorithm is globally convergent. The subproblems that ensure global convergence are easy-to-compute projections and are dependent only on the structure of the constraints, thus being extendable to other problems. Numerical experiments are presented, which confirm the theoretical predictions. The structure of the algorithm is such that accelerations can be easily associated without affecting the convergence properties. (C) 2004 American Institute of Physics.

关键词： projected gradient methods HARTREE-FOCK EQUATIONS CONVEX-SETS MINIMIZATION ALGORITHM BARZILAI SYSTEMS

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A projected gradient method for vector optimization problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2004年第1期28卷 5-29页

作者： Drummond, LMG Iusem, AN Univ Fed Rio de Janeiro COPPE Programa Engn sistemas Comp BR-21945970 Rio De Janeiro Brazil Inst Matematica Pura & Aplicada BR-22460320 Rio De Janeiro Brazil

Vector optimization problems are a significant extension of multiobjective optimization, which has a large number of real life applications. In vector optimization the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative orthant. We consider extensions of the projected gradient gradient method to vector optimization, which work directly with vector-valued functions, without using scalar-valued objectives. We provide a direction which adequately substitutes for the projected gradient, and establish results which mirror those available for the scalar-valued case, namely stationarity of the cluster points ( if any) without convexity assumptions, and convergence of the full sequence generated by the algorithm to a weakly efficient optimum in the convex case, under mild assumptions. We also prove that our results still hold when the search direction is only approximately computed.

关键词： Pareto optimality weak efficiency projected gradient methods convexity with respect to cones quasi-Fejer convergence

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A cyclic projected gradient method

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2013年第2期54卷 417-440页

作者： Setzer, Simon Steidl, Gabriele Morgenthaler, Jan Univ Saarland Dept Math & Comp Sci D-66041 Saarbrucken Germany Univ Kaiserslautern Dept Math Felix Klein Ctr D-67653 Kaiserslautern Germany

In recent years, convex optimization methods were successfully applied for various image processing tasks and a large number of first-order methods were designed to minimize the corresponding functionals. Interestingly, it was shown recently in Grewenig et al. (2010) that the simple idea of so-called "superstep cycles" leads to very efficient schemes for time-dependent (parabolic) image enhancement problems as well as for steady state (elliptic) image compression tasks. The "superstep cycles" approach is similar to the nonstationary (cyclic) Richardson method which has been around for over sixty years. In this paper, we investigate the incorporation of superstep cycles into the projected gradient method. We show for two problems in compressive sensing and image processing, namely the LASSO approach and the Rudin-Osher-Fatemi model that the resulting simple cyclic projected gradient algorithm can numerically compare with various state-of-the-art first-order algorithms. However, due to the nonlinear projection within the algorithm convergence proofs even under restrictive assumptions on the linear operators appear to be hard. We demonstrate the difficulties by studying the simplest case of a two-cycle algorithm in a"e(2) with projections onto the Euclidean ball.

关键词： Constrained optimization projected gradient methods Fast explicit diffusion Image denoising Sparse recovery

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Convergence analysis of a nonmonotone projected gradient method for multiobjective optimization problems

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OPTIMIZATION LETTERS 2019年第6期13卷 1365-1379页

作者： Fazzio, N. S. Schuverdt, M. L. Univ La Plata CONICET Dept Math FCE La Plata Buenos Aires Argentina

In this work we consider an extension of the projected gradient method (PGM) to constrained multiobjective problems. The projected gradient scheme for multiobjective optimization proposed by Grana Drummond and Iusem and analyzed by Fukuda and Grana Drummond is extended to include a nonmonotone line search based on the average of the successive previous functions values instead of the traditional Armijo-like rules. Under standard assumptions, stationarity of the accumulation points is established. Moreover, under standard convexity assumptions, we prove full convergence to weakly Pareto optimal solutions of any sequence produced by the proposed algorithm.

关键词： Multiobjective optimization projected gradient methods Nonmonotone line search Global convergence

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Interior gradient and epsilon-subgradient descent methods for constrained convex minimization

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MATHEMATICS OF OPERATIONS RESEARCH 2004年第1期29卷 1-26页

作者： Auslender, A Teboulle, M Univ Lyon 1 Dept Math F-69365 Lyon France Tel Aviv Univ Sch Math Sci IL-69978 Tel Aviv Israel

We extend epsilon-subgradient descent methods for unconstrained nonsmooth convex minimization to constrained problems over polyhedral sets, in particular over R-+(p). This is done by replacing the usual squared quadratic regularization term used in subgradient schemes by the logarithmic-quadratic distancelike function recently introduced by the authors. We then obtain interior epsilon-subgradient descent methods, which allow us to provide a natural extension of bundle methods and Polyak's subgradient projection methods for nonsmooth convex minimization. Furthermore, similar extensions are considered for smooth constrained minimization to produce interior gradient descent methods. Global convergence as well as an improved global efficiency estimate are established for these methods within a unifying principle and minimal assumptions on the problem's data.

关键词： nondifferentiable convex optimization subgradient algorithms bundle methods logarithmic-quadratic proximal methods projected gradient methods convex programming

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Two-phase model algorithm with global convergence for nonlinear programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1998年第2期96卷 397-436页

作者： Martinez, JM Univ Campinas UNICAMP IMECC Dept Math Appl Campinas SP Brazil

The family of feasible methods for minimization with nonlinear constraints includes the nonlinear projected gradient method, the generalized reduced gradient method (GRG), and many variants of the sequential gradient restoration algorithm (SGRA). Generally speaking, a particular iteration of any of these methods proceeds in two phases. In the restoration phase, feasibility is restored by means of the resolution of an auxiliary nonlinear problem, generally a nonlinear system of equations. In the minimization phase, optimality is improved by means of the consideration of the objective function, or its Lagrangian, on the tangent subspace to the constraints. In this paper, minimal assumptions are stated on the restoration phase and the minimization phase that ensure that the resulting algorithm is globally convergent. The key point is the possibility of comparing two successive nonfeasible iterates by means of a suitable merit function that combines feasibility and optimality. The merit function allows one to work with a high degree of infeasibility at the first iterations of the algorithm. Global convergence is proved and a particular implementation of the model algorithm is described.

关键词： nonlinear programming trust regions GRG methods SGRA methods projected gradient methods SQP methods global convergence

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A Modification of Karmarkar's Linear Programming Algorithm

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ALGORITHMICA 1986年第1-4期1卷 395-407页

作者： Vanderbei, Robert J. Meketon, Marc S. Freedman, Barry A. AT&T Bell Labs Holmdel NJ 07733 USA

We present a modification of Karmarkar's linear programming algorithm. Our algorithm uses a recentered projected gradient approach thereby obviating a priori knowledge of the optimal objective function value. Assuming primal and dual nondegeneracy, we prove that our algorithm converges. We present computational comparisons between our algorithm and the revised simplex method. For small, dense constraint matrices we saw little difference between the two methods.

关键词： Linear programming Karmarkar's algorithm projected gradient methods Least squares

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