A twostep spectral gradient projection method for system of nonlinear monotone equations and image deblurring problems aliyu muhammed awwal 1,2, lin wang 3, poom kumam 1,4, and hassan mohammad 5 1 kmuttfixed point research laboratory, room scl 802 fixed point laboratory, science laboratory. Recently, the work of solodov and svaiter has motivated further studies on monotone equations. Spectral projected gradient methods journal of statistical software. Adaptive twopoint stepsize gradient algorithm springerlink. They were introduced by barzilai and borwein 1 and have been applied successfully to find local minimizers of large scale problems 1,3,4,17,26. 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. A new nonmonotone line search algorithm is proposed and analyzed. Algorithm spgm spectral projected gradient method step 1. Monotone spectral projected gradient method for the dual problem and its convergence the variant of the non. The original problem is reduced to a nonlinear programming one using general rungekutta integration formulas. More recently, spectral gradient methods were extended to minimize general smooth functions on convex sets.
By combining barzilaiborwein nonmonotone ideas with traditional projected gradient algorithm, birgin et al. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. Here, due to the spectral choice of the steplength, we also expect e. A projected adaptive cyclic barzilaiborwein method. A series of numerical experiments show that the proposed nspg method can greatly improve the efficiency of solving teicps in terms of taking much less computing time for higher. Nonmonotone spectral projected gradient methods on convex sets. In doing so, we remove various boundedness conditions that. Pdf analysis of monotone gradient methods researchgate. In the inexact restoration method of 30 the initial gradient step merely guaranteed theoretical global convergence. The spg is modified in the orthogonal projection of the iterates onto the convex feasible set in order to obtain a cheap computation. The user provides objective function and gradient values, and projections onto the. The barzilai and borwein gradient method for the large. In this case, the spectral choice of steplength was combined with the projected gradient method to obtain a robust and e.
Nonmonotone spectral projected gradient methods on convex. Nonmonotone projected gradient methods based on barrier 309 iv. A nonmonotone gradient algorithm for total variation image. It combines the classical projected gradient method with the spectral gradient choice of steplength. Spectral gradient projection method for monotone nonlinear. The barzilai and borwein gradient method for the large scale. Nonmonotone spectral projected gradient methods on. Inexact spectral projected gradient methods on convex sets. In our scheme, we require that an average of the successive function values decreases, while the traditional nonmonotone approach of grippo, lampariello, and lucidi siam j. In this paper, a novel method based on nonmonotone spectral projected gradient pursuit with l 1norm has been proposed to localize the internal fluorescent sources.
S s symmetry article a twostep spectral gradient projection method for system of nonlinear monotone equations and image deblurring problems aliyu muhammed awwal 1,2, lin wang 3, poom kumam 1,4, and hassan mohammad 5 1 kmuttfixed point research laboratory, room scl 802 fixed point laboratory, science laboratory building, department of mathematics, faculty of science, king. In particular, the nonmonotone strategy is combined with the spectral gradient choice of. A nonmonotone spectral projected gradient method for largescale topology optimization problems rouhollahtavakoli1 department of material science and engineering sharif university of technology tehran, p. In these approaches the steplength is adaptively chosen and they showed very good practical performance. Nonmonotone projected gradient methods based on barrier. Each iterate of this algorithm is obtained by the projection of a barzilaiborwein step onto the feasible set consisting of box and one linear constraints volume constraint. Applications of spectral gradient algorithm for solving. An efficient gradient based method to solve the volume constrained topology optimization problems is presented. Combined with the nonmonotone line search, the twopoint stepsize gradient method has successfully been applied for largescale unconstrained optimization. A smoothing perturbed spectral projected gradient method for. The spectral projected gradient spg method was introduced, analyzed and implemented in 17, 18, 19. The spectral projected gradient spg method 24, 25, 26, for solving convex constrained problems, was born from the marriage of the global barzilaborwein spectral nonmonotone scheme 84 with classical projected gradient strategies 15, 56, 74. Efficient simulation of contacts,friction and constraints using.
On the application of the spectral projected gradient. Citeseerx nonmonotone spectral projected gradient methods. However, tv denoising problem is usually illconditioned that the classical monotone projected gradient method cannot solve the problem e ciently. Now we state their convergence theorems as follows.
A variant of the spectral projected gradient spg method proposed by birgin, martinez and raydan is proposed to solve semidefinite programs with \log. Research article a nonmonotone gradient algorithm for total. In this paper, we combined a new nonmonotone techniques with the spectral conjugate gradient method to obtain a more e. Energy minimization of point charges on a sphere with a. A new method is introduced for large scale convex constrained optimization. At each iteration, a spectral projected gradient method approximately minimizes a leastsquares problem with an explicit onenorm constraint.
However, we also note that the properties of the map u remain valid for an arbitrary closed convex set x, that. The general model algorithm involves, at each iteration. Nonmonotone projected gradient methods based on barrier and. Spectral gradient methods for linearly constrained. If t 1, the scaled projected gradient is the continuous projected gradient whose norm kg1xk. This work introduces a modified version of the spectral projected gradient method that can be used for simulating dynamical systems with complex joints and. Oct 04, 2020 a nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems wanbin tong, hongjin he, chen ling and liqun qi 2020, 104. In this paper, we give a more comprehensive theoretical analysis of the spg method. Due to the high nonlinearity resulted by tensors, the corresponding teicps are often not easy to be solved directly by the algorithms tailored for eicps. Convergence analysis of a nonmonotone projected gradient. Spg2 for the minimization of di erentiable functions on nonempty closed and convex sets. This method combines the basic spectral step ideas 8, 53, 54 with projected gradient strategies 10, 37, 45. A new nonmonotone spectral conjugate gradient method for. Spg is a nonmonotone projected gradient algorithm for solving largescale convexconstrained optimization problems.
In this paper, we examine the projected gradient descent method 2628, especially spgqp spectral projected gradients 29 for qp 5. A primer on monotone operator methods stanford university. In this paper, we introduce a nonmonotone spectral projected gradient nspg method equipped with a positive barzilaiborwein step. Sep 01, 2001 spg is a nonmonotone projected gradient algorithm for solving largescale convexconstrained optimization problems. A twostep spectral gradient projection method for system. Spectral gradient methods, originally proposed in 2 for the solution of unconstrained optimization problems have been widely used and developed see 6,15,23 and references therein. The projection of x onto a nonempty closed convex set c is defined as. However, the numerical performances of the algorithm heavily depend on m, one of the parameters in the nonmonotone line search, even for illconditioned problems.
Spectral gradient methods for linearly constrained optimization. Nonmonotone projected gradient method in linear elasticity. A nonmonotone spectral projected gradient method for largescale. Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method jinzuo ye, ayang du, yu an, b chongwei chi, a, and jie tian a, a chinese academy of sciences, key laboratory of molecular imaging of chinese academy of sciences. By combine the smoothing and perturbed technique, we design a smoothing perturbed spectral projected gradient method to solve problem 1. In this paper, we introduce a nonmonotone spectral projected gradient nspg method equipped with a positive barzilaiborwein step size to find a solution of teicps. Oct 01, 2009 spectral gradient methods are lowcost nonmonotone schemes see 8 for finding local minimizers. Subsampled nonmonotone spectral gradient methods arxiv. Pdf a nonmonotone spectral projected gradient method for. A twostep spectral gradient projection method for system of. In 1997, raydan 19 extended the spectral gradient method to unconstrained optimization and proved its global convergence. When spgm is applied to solve the qp problem 3, it may be stated this way. In 2000, by combining the projected gradient method and the nonmonotone line search technique, birgin, mart nez and raydan 6 extended the. The interest for the spg method is due to its faster convergence with respect to classical gradient.
In a recent paper, a nonmonotone spectral projected gradient spg method was introduced by birgin et al. In this paper, we consider extending the spectral projected gradient method to deal with the semismooth equations 1. Research article a nonmonotone gradient algorithm for. Nov 16, 2018 in this work we consider an extension of the projected gradient method pgm to constrained multiobjective problems. It is based on the raydan9 unconstrained generalization of the barzilaiborwein method for quadratics10. The extension of the spectral gradient method to smooth convex. Symmetry free fulltext a twostep spectral gradient projection. Pdf nonmonotone projected gradient techniques are considered for the minimization of dieren tiable functions on closed convex sets. Nonmonotone spectral projected gradient method for.
Dc, the resulting projectionlike map uv,xremains in c. In the paper 4, nonmonotone spectral projected gradient algorithms 4, algorithms 2. Pdf nonmonotone spectral projected gradient methods on. In this paper, by combining with the projection method of solodov and. Automatic differentiation and spectral projected gradient. A smoothing perturbed spectral projected gradient method. It combines the classical projected gradient method with the spectral gradient choice of steplength and a nonmonotone linesearch strategy. Fortran 77 software implementing the spg method is introduced. Introduction to spectral projected gradient method. On the application of the spectral projected gradient method in. Introduction a pioneering paper bybarzilai and borwein1988 proposed a gradient method for the unconstrained minimization of a di erentiable function f. Jul 31, 2006 2012 a nonmonotone spectral projected gradient method for largescale topology optimization problems. Oct 01, 2005 in a recent paper, a nonmonotone spectral projected gradient spg method was introduced by birgin et al. Birgin jose mario martnez marcos raydan june 7, 1999 abstract nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets.
A conjugate gradient method to solve convex constrained monotone. A nonmonotone spectral projected gradient method for tensor. A nonmonotone spectral projected gradient method for. Partial spectral projected gradient method with activeset. Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method jinzuo ye, ayang du, yu an, b chongwei chi, a, and jie tian a, a chinese academy of sciences, key laboratory of molecular imaging of chinese academy of sciences, institute of automation, no. Convex constrained optimization, projected gradient, nonmonotone line search, spectral gradient, dykstras algorithm. This paper proposes an adaptive nonmonotone line search. On the application of the spectral projected gradient method. The spectral projected gradient spg method birgin, mart nez, and raydan2000,2001, 2003b, for solving convex constrained problems, was born from the marriage of the global barzilaborwein spectral nonmonotone scheme raydan1997 with the classical projected.
Spectral projected gradient method with inexact restoration. Convergence properties of nonmonotone spectral projected. The spectral projected gradient spg method birgin, mart. In this paper, we propose a twostep iterative algorithm based on projection technique for solving system of monotone nonlinear equations with convex. Reconstruction of fluorescence molecular tomography via a. Automatic differentiation and nonmonotone spectral projected gradient techniques are used for solving optimal control problems. Pdf a nonmonotone line search technique and its application. In a very recent work, zhang and hager 22 have proposed to replace the sup in 1. Abstract we investigate the application of the non monotone spectral projected gradient spg method to a regionbased variational model for image segmenta. A lagrangian dual spectral projected gradient method for. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the grippolampariellolucidi nonmonotone line search.
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