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PRECONDITIONED spectral PROJECTED gradient method ON CONVEX SETS

Journal of Computational Mathematics 2005年第3期23卷 225-232页

作者： LenysBello MarcosRaydan Dpto.deMatemdtica FacultaddeCiencias(FACYT)UniversidaddeCaraboboValenciaVenezuela Dpto.deComputación UniversidadCentraldeVenezuelaAp.47002Caracas1041-AVenezuela

The spectral gradient method has proved to be effective for solving large-scale unconstrained optimization problems. It has been recently extended and combined with the projected gradient method for solving optimization problems on convex sets. This combination includes the use of nonmonotone line search techniques to preserve the fast local convergence. In this work we further extend the spectral choice of steplength to accept preconditioned directions when a good preconditioner is available. We present an algorithmthat combines the spectral projected gradient method with preconditioning strategies toincrease the local speed of convergence while keeping the global properties. We discuss implementation details for solving large-scale problems.

关键词： spectral gradient method Projected gradient method Preconditioning tech-niques Nonmonotone line search

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A LOW-COST OPTIMIZATION APPROACH FOR SOLVING MINIMUM NORM LINEAR SYSTEMS AND LINEAR LEAST-SQUARES PROBLEMS

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Journal of Computational Mathematics 2024年第4期42卷 932-954页

作者： Debora Cores Johanna Figueroa Defense University Center at the Spanish Naval Academy Plaza de Espana36920 MarinPontevedraSpain Departamento de Matematica Facultad de MatematicaPontificia Universidad Catolica de ChileAv.Vicuna Mackenna 4860San JoaquinSantiago 894000Chile

Recently,the authors proposed a low-cost approach,named Optimization Approach for Linear Systems(OPALS)for solving any kind of a consistent linear system regarding the structure,characteristics,and dimension of the coefficient matrix *** results obtained by this approach for matrices with no structure and with indefinite symmetric part were encouraging when compare with other recent and well-known *** this work,we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem(LLSP)and the Minimum Norm Linear System Problem(MNLSP)using any iterative low-cost gradient-type method,avoiding the construction of the matrices AT A or AAT,and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient *** combination of those conditions together with the choice of the initial iterate allow us to produce a novel and efficient low-cost numerical scheme for solving both ***,the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares *** include encouraging numerical results to illustrate the practical behavior of the proposed schemes.

关键词： Nonlinear convex optimization gradient-type methods spectral gradient method Minimum norm solution linear systems Linear least-squares solution

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spectral DY-TYPE PROJECTION method FOR NONLINEAR MONOTONE SYSTEM OF EQUATIONS

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Journal of Computational Mathematics 2015年第4期33卷 341-355页

作者： Jinkui Liu Shengjie Li College of Mathematics and Statistics Chongqing University Chongqing 401331 China School of Mathematics and Statistics Chongqing Three Gorges University Wanzhou 404100 China College of Mathematics and Statistics Chongqing University Chongqing 401331 China.

In this paper, we propose a spectral DY-type projection method for nonlinear mono- tone system of equations, which is a reasonable combination of DY conjugate gradient method, the spectral gradient method and the projection technique. Without the differen- tiability assumption on the system of equations, we establish the global convergence of the proposed method, which does not rely on any merit function. Furthermore, this method is derivative-free and so is very suitable to solve large-scale nonlinear monotone systems. The preliminary numerical results show the feasibility and effectiveness of the proposed method.

关键词： Nonlinear monotone system of equations spectral gradient method DY conjugate gradient method Projection method Global convergence.

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