基于修正拟牛顿方程的两阶段非单调稀疏对角变尺度梯度投影算法
|
摘要
基于修正拟牛顿方程,利用Goldstein-Levitin-Polyak(GLP)投影技术,建立了求解带凸集约束的优化问题的两阶段步长非单调变尺度梯度投影算法,证明了算法的全局收敛性和一定条件下的Q超线性收敛速率.数值结果表明新算法是有效的,适合求解大规模问题.
Based on modified quasi-Newton equation,by combining with Goldstein-LevitinPolyak(GLP) projection technique,a new non monotone two stages stepsize variable metric gradient projection method for convex set constrained optimization problem is presented.The global convergence property of the algorithm is proved.Under some reasonable conditions,it is proved that the algorithm has Q-superlinear convergence rate.The numerical results show that the new method is effective and is fit to solve large-scale problems.
Based on modified quasi-Newton equation,by combining with Goldstein-LevitinPolyak(GLP) projection technique,a new non monotone two stages stepsize variable metric gradient projection method for convex set constrained optimization problem is presented.The global convergence property of the algorithm is proved.Under some reasonable conditions,it is proved that the algorithm has Q-superlinear convergence rate.The numerical results show that the new method is effective and is fit to solve large-scale problems.
引文
[1]Goldstein A A.Convex programming in Hilbert space[J].Bulletin of the American Mathematical Society,1964,70(5):709-710.
[2]Levitin E S,Polyak B T.Constrained minimization methods[J].USSR Computational Mathematics and Mathematical Physics,1966,6(5):1-50.
[3]Calamai P H,More J J.Projected gradient methods for linearly constrained problems[J].Mathematical Programming,1987,39(1):93-116.
[4]Rustem B.Projection methods in constrained optimization and applications to optimal policy decisions[M].Springer-Vertag,Berlin,1981,31.
[5]Allwright J C.A feasible direction algorithm for convex optimization:global convergence rates[J].Journal of Optimization Theory and Applications,1980,30(1):1-18.
[6]Psenichny B N,Danilin Y M.Numerical methods in extremal problems[M].Mir Publishers,1978.
[7]Rustem B.A class of superlinearly convergent projection algorithms with relaxed stepsizes[J].Applied Mathematics and Optimization,1984,12(1):29-43.
[8]时贞军,孙国.无约束优化问题的对角稀疏拟牛顿算法[J].系统科学与数学,2006,26(1):101-112.
[9]Wei Z,Li G,Qi L.New quasi-Newton methods for unconstrained optimization problems[J].AppUed Mathematics and Computation,2006,175(2):1156-1188.
[10]Grippo L,Lampariello F,Lucidi S.A nonmonotone line search technique for Newton's method[J].SIAM Journal on Numerical Analysis,1986,23(4):707-716.
[11]Conn A R,Gould N I M,Toint P L.Testing a class of methods for solving minimization problems with simple bounds on the variables[J].Mathematics of Computation,1988,50(50):399-430.
[2]Levitin E S,Polyak B T.Constrained minimization methods[J].USSR Computational Mathematics and Mathematical Physics,1966,6(5):1-50.
[3]Calamai P H,More J J.Projected gradient methods for linearly constrained problems[J].Mathematical Programming,1987,39(1):93-116.
[4]Rustem B.Projection methods in constrained optimization and applications to optimal policy decisions[M].Springer-Vertag,Berlin,1981,31.
[5]Allwright J C.A feasible direction algorithm for convex optimization:global convergence rates[J].Journal of Optimization Theory and Applications,1980,30(1):1-18.
[6]Psenichny B N,Danilin Y M.Numerical methods in extremal problems[M].Mir Publishers,1978.
[7]Rustem B.A class of superlinearly convergent projection algorithms with relaxed stepsizes[J].Applied Mathematics and Optimization,1984,12(1):29-43.
[8]时贞军,孙国.无约束优化问题的对角稀疏拟牛顿算法[J].系统科学与数学,2006,26(1):101-112.
[9]Wei Z,Li G,Qi L.New quasi-Newton methods for unconstrained optimization problems[J].AppUed Mathematics and Computation,2006,175(2):1156-1188.
[10]Grippo L,Lampariello F,Lucidi S.A nonmonotone line search technique for Newton's method[J].SIAM Journal on Numerical Analysis,1986,23(4):707-716.
[11]Conn A R,Gould N I M,Toint P L.Testing a class of methods for solving minimization problems with simple bounds on the variables[J].Mathematics of Computation,1988,50(50):399-430.