求解无约束优化问题的新谱共轭梯度法及其收敛性

强Wolfe条件不能保证标准CD共轭梯度法全局收敛．本文通过建立新的共轭参数,提出无约束优化问题的一个新谱共轭梯度法,该方法在精确线搜索下与标准CD共轭梯度法等价,在标准wolfe线搜索下具有下降性和全局收敛性．初步的数值实验结果表明新方法是有效的,适合于求解非线性无约束优化问题．     

Applying the Powell's Symmetrical Technique to Conjugate Gradient Methods with the Generalized Conjugacy Condition

This article proposes new conjugate gradient method for unconstrained optimization by applying the Powell symmetrical technique in a defined sense. Using the Wolfe line search conditions, the global convergence property of the method is also obtained based on the spectral analysis of the conjugate gradient iteration matrix and the Zoutendijk condition for steepest descent methods. Preliminary numerical results for a set of 86 unconstrained optimization test problems verify the performance of the algorithm and show that the Generalized Descent Symmetrical Hestenes-Stiefel algorithm is competitive with the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP⁺) algorithms.     

A modified Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization

A modified Polak–Ribière–Polyak conjugate gradient algorithm which satisfies both the sufficient descent condition and the conjugacy condition is presented. These properties are independent of the line search. The algorithms use the standard Wolfe line search. Under standard assumptions, we show the global convergence of the algorithm. Numerical comparisons with conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the CUTE library, show that this computational scheme outperforms the known Polak–Ribière–Polyak algorithm, as well as some other unconstrained optimization algorithms.     

Nonmonotone spectral method for large-scale symmetric nonlinear equations

In this paper, by the use of the residual vector and an approximation to the steepest descent direction of the norm function, we develop a norm descent spectral method for solving symmetric nonlinear equations. The method based on the nomonotone line search techniques is showed to be globally convergent. A specific implementation of the method is given which exploits the recently developed cyclic Barzilai–Borwein (CBB) for unconstrained optimization. Preliminary numerical results indicate that the method is promising.     

A short note on the global convergence of the unmodified PRP method

To guarantee global convergence of the standard (unmodified) PRP nonlinear conjugate gradient method for unconstrained optimization, the exact line search or some Armijo type line searches which force the PRP method to generate descent directions have been adopted. In this short note, we propose a non-descent PRP method in another way. We prove that the unmodified PRP method converges globally even for nonconvex minimization by the use of an approximate descent inexact line search.     

两类下降的非线性共轭梯度法的全局收敛性

本文在文献[1]中提出了一类新共轭梯度法的基础上,给出求解无约束优化问题的两类新的非线性下降共轭梯度法,此两类方法在无任何线搜索下,能够保证在每次迭代中产生下降方向.对一般非凸函数,我们在Wolfe线搜索条件下证明了两类新方法的全局收敛性.     

一类带线搜索的非单调信赖域新算法

结合非单调信赖域方法,和非单调线搜索技术,提出了一种新的无约束优化算法.信赖域方法的每一步采用线搜索,使得迭代每一步都充分下降加快了迭代速度.在一定条件下,证明了算法具有全局收敛性和局部超线性.收敛速度.数值试验表明算法是十分有效的.     

Automatic Deformation of Riemann–Hilbert Problems with Applications to the Painlevé II Transcendents

The stability and convergence rate of Olver’s collocation method for the numerical solution of Riemann–Hilbert problems (RHPs) are known to depend very sensitively on the particular choice of contours used as data of the RHP. By manually performing contour deformations that proved to be successful in the asymptotic analysis of RHPs, such as the method of nonlinear steepest descent, the numerical method can basically be preconditioned, making it asymptotically stable. In this paper, however, we will show that most of these preconditioning deformations, including lensing, can be addressed in an automatic, completely algorithmic fashion that would turn the numerical method into a black-box solver. To this end, the preconditioning of RHPs is recast as a discrete, graph-based optimization problem: the deformed contours are obtained as a system of shortest paths within a planar graph weighted by the relative strength of the jump matrices. The algorithm is illustrated for the RHP representing the Painlevé II transcendents.     

New accelerated conjugate gradient algorithms as a modification of Dai-Yuan’s computational scheme for unconstrained optimization

New accelerated nonlinear conjugate gradient algorithms which are mainly modifications of Dai and Yuan’s for unconstrained optimization are proposed. Using the exact line search, the algorithm reduces to the Dai and Yuan conjugate gradient computational scheme. For inexact line search the algorithm satisfies the sufficient descent condition. Since the step lengths in conjugate gradient algorithms may differ from 1 by two orders of magnitude and tend to vary in a very unpredictable manner, the algorithms are equipped with an acceleration scheme able to improve the efficiency of the algorithms. Computational results for a set consisting of 750 unconstrained optimization test problems show that these new conjugate gradient algorithms substantially outperform the Dai-Yuan conjugate gradient algorithm and its hybrid variants, Hestenes-Stiefel, Polak-Ribière-Polyak, CONMIN conjugate gradient algorithms, limited quasi-Newton algorithm LBFGS and compare favorably with CG_DESCENT. In the frame of this numerical study the accelerated scaled memoryless BFGS preconditioned conjugate gradient ASCALCG algorithm proved to be more robust.     

Preconditioned Spectral Projected Gradient Method on Convex Sets

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.     

A nonmonotone trust-region line search method for large-scale unconstrained optimization

We consider an efficient trust-region framework which employs a new nonmonotone line search technique for unconstrained optimization problems. Unlike the traditional nonmonotone trust-region method, our proposed algorithm avoids resolving the subproblem whenever a trial step is rejected. Instead, it performs a nonmonotone Armijo-type line search in direction of the rejected trial step to construct a new point. Theoretical analysis indicates that the new approach preserves the global convergence to the first-order critical points under classical assumptions. Moreover, superlinear and quadratic convergence are established under suitable conditions. Numerical experiments show the efficiency and effectiveness of the proposed approach for solving unconstrained optimization problems.     

A conjugate gradient method with descent direction for unconstrained optimization

A modified conjugate gradient method is presented for solving unconstrained optimization problems, which possesses the following properties: (i) The sufficient descent property is satisfied without any line search; (ii) The search direction will be in a trust region automatically; (iii) The Zoutendijk condition holds for the Wolfe–Powell line search technique; (iv) This method inherits an important property of the well-known Polak–Ribière–Polyak (PRP) method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. The global convergence and the linearly convergent rate of the given method are established. Numerical results show that this method is interesting.     

On preconditioned eigensolvers and Invert-Lanczos processes

This paper deals with the convergence analysis of various preconditioned iterations to compute the smallest eigenvalue of a discretized self-adjoint and elliptic partial differential operator. For these eigenproblems several preconditioned iterative solvers are known, but unfortunately, the convergence theory for some of these solvers is not very well understood.The aim is to show that preconditioned eigensolvers (like the preconditioned steepest descent iteration (PSD) and the locally optimal preconditioned conjugate gradient method (LOPCG)) can be interpreted as truncated approximate Krylov subspace iterations. In the limit of preconditioning with the exact inverse of the system matrix (such preconditioning can be approximated by multiple steps of a preconditioned linear solver) the iterations behave like Invert-Lanczos processes for which convergence estimates are derived.     

A three-term conjugate gradient algorithm for large-scale unconstrained optimization problems

In this paper, a three-term conjugate gradient algorithm is developed for solving large-scale unconstrained optimization problems. The search direction at each iteration of the algorithm is determined by rectifying the steepest descent direction with the difference between the current iterative points and that between the gradients. It is proved that such a direction satisfies the approximate secant condition as well as the conjugacy condition. The strategies of acceleration and restart are incorporated into designing the algorithm to improve its numerical performance. Global convergence of the proposed algorithm is established under two mild assumptions. By implementing the algorithm to solve 75 benchmark test problems available in the literature, the obtained results indicate that the algorithm developed in this paper outperforms the existent similar state-of-the-art algorithms.     

CFD-based optimization of fuel injection strategies in a diesel engine using an adaptive gradient method

A computationally efficient computational fluid dynamics (CFD)-based optimization method with the capability of finding optimal engine operating conditions with respect to emissions and fuel consumption has been developed. The approach taken uses a steepest descent method for an adaptive cost function, where the line search is performed with a backtracking algorithm. The backtracking algorithm utilizes quadratic and cubic polynomials to accelerate the convergence, and the initial backtracking step employs an adaptive step size mechanism which depends on the steepness of the search direction. The adaptive cost function is based on the penalty method such that the penalty term is stiffened after every line search. The engine simulations are performed with a KIVA-3-based CFD code which is equipped with well-established spray, combustion and emission models. The application of this optimization tool is demonstrated for a non-road, medium-speed DI diesel engine which, for these simulations, utilizes a multi-orifice, asynchronous injection system. It has been demonstrated that this new injection method has a large potential for reducing emissions while maintaining a low fuel consumption. In addition, this optimization approach is computationally very efficient when good enough initial values are available.     

修正PRP共轭梯度方法求解无约束最优化问题

基于著名的PRP共轭梯度方法,利用CG_DESCENT共轭梯度方法的结构,本文提出了一种求解大规模无约束最优化问题的修正PRP共轭梯度方法。该方法在每一步迭代中均能够产生一个充分下降的搜索方向,且独立于任何线搜索条件。在标准Wolfe线搜索条件下,证明了修正PRP共轭梯度方法的全局收敛性和线性收敛速度。数值结果展示了修正PRP方法对给定的测试问题是非常有效的。     

一个充分下降的改进HS共轭梯度法

共轭梯度法是求解大规模无约束优化问题最有效的方法之一.对HS共轭梯度法参数公式进行改进,得到了一个新公式,并以新公式建立一个算法框架.在不依赖于任何线搜索条件下,证明了由算法框架产生的迭代方向均满足充分下降条件,且在标准Wolfe线搜索条件下证明了算法的全局收敛性.最后,对新算法进行数值测试,结果表明所改进的方法是有效的.     

The Global Convergence of Self-Scaling BFGS Algorithm with Nonmonotone Line Search for Unconstrained Nonconvex Optimization Problems

The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex （or even uniformly convex）. We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems.     

修正PRP共轭梯度方法求解无约束最优化问题

基于著名的PRP共轭梯度方法,利用CG_DESCENT共轭梯度方法的结构,本文提出了一种求解大规模无约束最优化问题的修正PRP共轭梯度方法。该方法在每一步迭代中均能够产生一个充分下降的搜索方向,且独立于任何线搜索条件。在标准Wolfe线搜索条件下,证明了修正PRP共轭梯度方法的全局收敛性和线性收敛速度。数值结果展示了修正PRP方法对给定的测试问题是非常有效的。     

Wolfe线搜索下一类混合共轭梯度法的全局收敛性

本文给出了一个新的共轭梯度公式,新公式在精确线搜索下与DY公式等价,并给出了新公式的相关性质.结合新公式和DY公式提出了一个新的混合共轭梯度法,新算法在Wolfe线搜索下产生一个下降方向,并证明了算法的全局收敛性,并给出了数值例子.