A nonmonotone spectral projected gradient method for largescale topology optimization problems article pdf available in numerical algebra 22. The original problem is reduced to a nonlinear programming one using general rungekutta integration formulas. Spectral projected gradient method with inexact restoration. A nonmonotone spectral projected gradient method for largescale topology optimization problems authors. Energy minimization of point charges on a sphere with a. By combining barzilaiborwein nonmonotone ideas with traditional projected gradient algorithm, birgin et al. A variant of the spectral projected gradient spg method proposed by birgin, martinez and raydan is proposed to solve semidefinite programs with \log. Convex constrained optimization, projected gradient, nonmonotone line search, spectral gradient, dykstras algorithm. A nonmonotone gradient algorithm for total variation image. 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. Reconstruction of fluorescence molecular tomography via a. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the grippolampariellolucidi nonmonotone line search. To ensure the global convergence, an adaptive nonmonotone line search is performed along the direction that.
Pdf a nonmonotone spectral projected gradient method for. Pdf nonmonotone spectral projected gradient methods on. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a. Raydan 2000 articletitle nonmonotone spectral projected gradient methods on convex sets siam journal of optimization 10. A cyclic projected gradient method mathematical image analysis.
Spectral projected gradient method with inexact restoration for minimization with nonconvex constraints m. Associate professor jianfeng cai assistant professor weiyu xu. Monotone spectral projected gradient method for the dual problem and its convergence the variant of the non. Experiments on synthetic and real data sets are presented in section 6. The almost sure convergence results are achieved without imposing explicit sample growth condition. Ruhollah tavakoli, hongchao zhang submitted on 3 jun 2010. Spectral projected gradient method for stochastic optimization. A nonmonotone spectral projected gradient method that has been intensively used in many applications is presented. These methods combine the classical projected gradient with two recently developed ingredients in optimization. It is assumed that the feasible set is convex, closed and easy to project on. Nonmonotone spectral projected gradient methods on convex sets.
Pdf nonmonotone projected gradient techniques are considered for the minimization of dieren tiable functions on closed convex sets. Section 4 investigates speci c but essential modi cations for applying the proposed projected gradients methods to nmf. Nonmonotone trustregion methods for boundconstrained. The classical projected gradient schemes are extended to include a. Jun 30, 2018 we consider the spectral projected gradient method for solving constrained optimization problems with the objective function in the form of mathematical expectation. Convergence properties and extensive numerical results are presented. 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. Projected wirtinger gradient descent for spectral compressed sensing by suhui liu a thesis submitted in partial ful llment of the requirements for the doctor of philosophy degree in mathematics in the graduate college of the university of iowa august 2017 thesis supervisors. The lemma 2 ii shows that if d k 0 at kthiteration, then z kis a stationary point of problem 3. In 2000, by combining the projected gradient method and the nonmonotone line search technique, birgin, mart nez and raydan 6 extended the spectral gradient method to convex constrained optimization.
A modified spectral conjugate gradient methods with the new. Spectralprojected gradient methods january17, 2007 keywords. Keywords constrained optimization projected gradient methods fast explicit. On the application of the spectral projected gradient. Birgin jos e mario mart nez y marcos raydan z july 2004 updated abstract nonmonotone projected gradient techniques are considered for the minimization of di erentiable functions on closed convex sets. Optimization over sparse symmetric sets via a nonmonotone projected gradient method zhaosong lu november 21, 2015 abstract ew consider the problem of minimizing a lipschitz di erentiable function over a class of sparse symmetric sets that has wide applications in engineering and. Numerical results show the e ciency of the proposed method. A nonmonotone spectral projected gradient method for largescale topology optimization problems. In this work we extend the spectral approach to solve nonlinear systems of equations.
Nonmonotone projected gradient techniques are considered for the minimization of differentiable. In this paper, combining the nonmonotone and monotone line search,a spectral conjugate gradient methods are used in this paper. Spectral projected gradient methods, nonmonotone line search, large scale prob lems, convex constrained problems. At each iteration, a spectral projected gradient method approximately minimizes a leastsquares problem with an explicit onenorm constraint. 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 classical projected gra.
Subsampled nonmonotone spectral gradient methods in. The adaptive cyclic reuse of the barzilaiborwein step is applied as the initial stepsize. Thomson problem, which is one of the problems concerning of optimal configuration on the sphere, is defined. A lagrangian dual spectral projected gradient method for. Inexact spectral projected gradient methods on convex sets. Nonmonotone spectral projected gradient methods on convex sets ernesto g. The spectral gradient method has proved to be effective for solving largescale optimization problems. Abstract this paper deals with subsampled spectral gradient methods for. Research article a nonmonotone gradient algorithm for. A lagrangian dual spectral projected gradient method 277 ii the vector g tz. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process.
Our new nonmonotone spectral projected gradient method different from algorithm 2. Optimization over sparse symmetric sets via a nonmonotone projected gradient method zhaosong lu november 21, 2015 abstract ew consider the problem of minimizing a lipschitz di erentiable function over a class of sparse symmetric sets that has wide applications in engineering and science. Spectral projected gradient methods projected gradients nonmonotone line search large scale problems convex constrained problems. On the application of the spectral projected gradient method. Research article a nonmonotone gradient algorithm for total. In this paper, we proposed a spectral gradientnewton two phase method for constrained semismooth equations.
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 tiana, achinese academy of sciences, key laboratory of molecular imaging of chinese academy of sciences, institute of automation, no. Nonmonotone spectral projected gradient methods on convex. 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. 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 tiana, achinese academy of sciences, key laboratory of molecular imaging of chinese academy of sciences. In a recent paper, a nonmonotone spectral projected gradient spg method was introduced by birgin et al. A nonmonotone spectral projected gradient method for largescale topology optimization problems rouhollahtavakoli1 department of material science and engineering sharif university of technology tehran, p. Birgin jos e mario mart nez y marcos raydan z july 2004 updated abstract nonmonotone projected gradient techniques are considered for the minimization of di erentiable functions on.
On the application of the spectral projected gradient method in. Nonmonotone spectral projected gradient method for. The spg is modified in the orthogonal projection of the iterates onto the convex feasible set in order to obtain a cheap computation. By combining barzilaiborwein nonmonotone ideas with. Each iterate of this algorithm is obtained by the projection of a barzilaiborwein step onto the.
Convergence properties of nonmonotone spectral projected. Santos september 17, 2007 abstract this work takes advantage of the spectral projected gradient direction within the in. A nonmonotone spectral projected gradient problems arxiv. Another objective is to test the behavior of the spectral projected gradient methods introduced in 5. The objective function is approximated by a sequence of different sample average approximation functions with different sample sizes. Furthermore, traditional tv is handled by vectorizing, which makes. In particular, the nonmonotone strategy is combined with the spectral gradient choice of. 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, institute of automation, no.
Nonmonotone spectral methods for largescale nonlinear systems. An efficient gradientbased method to solve the volume constrained topology optimization problems is presented. A specific algorithm, the inexact spectral projected gradient method ispg, is implemented using inexact projections computed by dykstras alternating projection method and generates interior iterates. Jul 10, 2015 we investigate the application of the nonmonotone spectral projected gradient spg method to a regionbased variational model for image segmentation. Optimization over sparse symmetric sets via a nonmonotone. Solving bound constrained optimization via a new nonmonotone. Spectral projected gradient method for the procrustes problem. 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. The stopping conditions in an nmf code are discussed in section 5. A modified spectral conjugate gradient methods with the. In this paper, we give a more comprehensive theoretical analysis of the spg method. Nonmonotone projected gradient techniques are considered for the minimization of dieren tiable functions on closed convex sets. A spectral projected gradientnewton two phase method for. We consider a discretizethenoptimize approach and solve the resulting nonlinear optimization problem by an alternating minimization procedure that exploits the spg2 algorithm by birgin et al.
Spectral gradient methods for linearly constrained optimization. A nonmonotone spectral projected gradient method for large. 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 classical projected gradient strategies bertsekas1976. The ispg method is a generalization of the spectral projected gradient method spg, but can be used when projections are difficult to compute. Introduction to spectral projected gradient method. We consider a strategy based on nonmonotone line search techniques to guarantee global convergence. 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. Numerical results that indicate the effectiveness of this method to tackle the thomson problem are reported.
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