基于信赖域技术的非单调非线性共轭梯度算法

共轭梯度算法由于其迭代简单和较小的存储在求解大规模无约束优化问题中起着特殊的作用.本文基于信赖域技术和修正拟牛顿方程,结合Zhang非单调策略,设计了一种新的求解无约束最优化问题的基于信赖域技术的非单调非线性共轭梯度算法.该算法每次迭代自动产生信赖域半径,并通过求解一个简单的子问题得到下一个迭代点,信赖域技术的应用保证...     

带有线搜索的新的非单调自适应信赖域算法

本文给出了一种新的信赖域算法。该算法以变化的速率来调整信赖域半径的大小。在由信赖域子问题产生的试探步不被接受的情况下,新算法采用线搜索的方法得到下一个迭代点。同时算法采用非单调的技术来加速算法的收敛效果。文中给出了新算法的全局收敛性分析和数值试验的结果。     

复合不可微最优化问题的非单调信赖域方法

对复合不可微最优化问题提出了一种新的非单调信赖域方法。算法在每个迭代点处构造带信赖域约束的二次规划子问题,新的迭代点采用非单调策略产生,在一般的假设条件下证明了算法的全局收敛性。数值试验表明：该算法能在一定程度上克服由非光滑性引起的Ｍａｒａｔｏｓ效应     

复合不可微最优化问题的非单调依赖域方法

对复合不可微最优化问题提出了一种新的非单调信赖域方法,算法在每个迭代点处构造带信域约束的二次规划子问题,新的迭代点采用了非单调策略产生,在一般的假设下证明了算法的全局收敛性,数值试验表明;该算法能在一定程度上克服由非光滑性引起的Ｍａｒａｔｏｓ效应。     

带回溯线搜索步的双子问题信赖域算法

无约束非线性优化问题广泛存在于工程、科学计算等实际应用领域。本文在信赖域算法的框架下提出无约束子问题,将它与信赖子问题相结合,构造了求解无约束优化问题的双子问题信赖域算法。同时利用信赖域子问题得到的试探步一定是目标函数充分下降方向的性质使得每次求解信赖域子问题之后均能得到使目标函数下降的步。在标准假设下证明了该算法具有全局收敛性和局部二次收敛速度。数值结果表明该算法比传统的信赖域算法速度更快更有效。     

求解非线性系统的信赖域方法

本文给出了一个求解非线性系统的信赖域方法。通过引入松驰变量将非线性系统问题转化为带非负约束的非线性最优化问题,新算法借助于KKT条件和F-BNCP函数,在每次迭代时,不必求解二次信赖域子问题,只需求解一个线性方程组。在一定的假设条件下,该算法还是全局收敛和局部超线性收敛的。数值试验结果表明该算法是有效的。     

线性约束LC1凸优化问题的内点信赖域算法

本文提出一种解线性约束凸规划的数值方法。通过将问题的KKT系统转化成一个约束方程,算法在每步迭代只需解一个线性方程组即可得到搜索方向。算法运用了信赖域方法利内点技术。在较弱的条件下,我们证明了算法的全局收敛性。     

凸组合投影算法中的组合因子对算法效率的影响

在求解生产生活中各类实际问题的优化模型的算法研究中,投影梯度算法在解决凸约束最优化问题上一直被学者所重视．本文考虑凸组合投影算法求解凸约束最优化问题,在此凸组合投影算法中,由投影梯度法得到的点与上一步迭代点的凸组合得到新的迭代点．此算法不仅利用投影算法得到的点的信息而且也利用了前一步点的信息．进一步,通过数值实验分析凸组合算法的效率及凸组合因子对算法的影响．数值试验结果表明,这种凸组合算法总体比原来投影梯度法更稳定,而且这种凸组合算法在适当的凸组合因子下较投影梯度法收敛更快．     

稳固非扩张映射不动点集处均衡问题的一种不精确次梯度算法（英文）

本文提出了稳固非扩张映射不动点集处均衡问题的一种新算法.该算法要求双函数是连续的,但不一定是单调的.首先,通过事先引入的参数确定一个闭凸集;其次,根据双函数的不精确次梯度在闭凸集上的投影构造中间迭代点;最后,下一个迭代点由当前迭代点和中间迭代点的凸组合在稳固非扩张算子的映射得到.在适当条件下,本文给出了该算法的全局收敛性证明.     

一类新的基于信赖域技术的非单调共轭梯度算法

为有效求解大规模无约束优化问题,本文基于信赖域技术和修正拟牛顿方程,同时结合Zhang H. C.策略和Gu N. Z.策略,设计了一种新的非单调共轭梯度算法,应用信赖域技术保证了算法的稳健性和收敛性,并给出了算法的全局收敛性分析．在适当条件下,证明了该算法具有线性收敛性．数值实验表明新算法能够有效求解病态和大规模问题．与单独结合其中一种非单调策略的算法相比,新算法需要较少的迭代次数和运行时间,利用其得到的函数值与最优值更接近．     

An Interior Method for Nonconvex Semidefinite Programs

In several applications, semidefinite programs arise in which the matrix depends nonlinearly on the unknown variables. We propose a new solution method for such semidefinite programs that also applies to other smooth nonconvex programs. The method is an extension of a primal predictor corrector interior method to nonconvex programs. The predictor steps are based on Dikin ellipsoids of a “convexified” domain. The corrector steps are based on quadratic subprograms that combine aspects of line search and trust region methods. Convergence results are given, and some preliminary numerical experiments suggest a high robustness of the proposed method.     

Adaptive accuracy trust region: using cross-validation in the optimization process

This article presents an adaptive accuracy trust region (AATR) optimization strategy where cross-validation is used by the trust region to reduce the number of sample points needed to construct metamodels for each step of the optimization process. Lower accuracy metamodels are initially used for the larger trust regions, and higher accuracy metamodels are used for the smaller trust regions towards the end of optimization. Various metamodelling strategies are used in the AATR algorithm: optimal and inherited Latin hypercube sampling to generate experimental designs; quasi-Newton, kriging and polynomial regression metamodels to approximate the objective function; and the leave-k-out method for validation. The algorithm is tested with two-dimensional single-discipline problems. Results show that the AATR algorithm is a promising method when compared to a traditional trust region method. Polynomial regression in conjunction with a new hybrid inherited-optimal Latin hypercube sampling performed the best.     

A continuation method for rigid‐cohesive fracture in a discontinuous Galerkin finite element setting

An energy minimization formulation of initially rigid cohesive fracture is introduced within a discontinuous Galerkin finite element setting with Nitsche flux. The finite element discretization is directly applied to an energy functional, whose term representing the energy stored in the interfaces is nondifferentiable at the origin. Unlike finite element implementations of extrinsic cohesive models that do not operate directly on the energy potential, activation of interfaces happens automatically when a certain level of stress encoded in the interface potential is reached. Thus, numerical issues associated with an external activation criterion observed in the previous literature are effectively avoided. Use of the Nitsche flux avoids the introduction of Lagrange multipliers as additional unknowns. Implicit time stepping is performed using the Newmark scheme, for which a dynamic potential is developed to properly incorporate momentum. A continuation strategy is employed for the treatment of nondifferentiability and the resulting sequence of smooth nonconvex problems is solved using the trust region minimization algorithm. Robustness of the proposed method and its capabilities in modeling quasistatic and dynamic problems are shown through several numerical examples.     

A new trust region–sequential quadratic programming approach for nonlinear systems based on nonlinear model predictive control

In this article, a superlinearly convergent trust region–sequential quadratic programming approach is first proposed, developed and investigated for nonlinear systems based on nonlinear model predictive control. The method incorporates a combination algorithm that allows both the trust region technique and the sequential quadratic programming method to be used. If the attempted search of the trust region method is not accepted, the line search rule will be adopted for the next iteration. Also, having to resolve the quadratic programming subproblem for nonlinear constrained optimization problems is avoided. This gives the potential for fast convergence in the neighbourhood of an optimal solution. Moreover, additional characteristics of the algorithm are that each quadratic programming subproblem is regularized and the quadratic programming subproblem always has a consistent point. The main result is illustrated on a nonlinear system with a variable parameter and a bipedal walking robot system through simulations and is utilized to achieve rapidly stability. Numerical results show that the trust region–sequential quadratic programming algorithm is feasible and effective for a nonlinear system with a variable parameter and a bipedal walking robot system. Therefore, the simulation results demonstrate the usefulness of the trust region–sequential quadratic programming approach with nonlinear model predictive control for real-time control systems.     

带非凸二次约束的二次规划问题的全局优化方法

利用二次函数的线形下界函数对带有非凸二次约束的二次规划(QP)提出一种新的求其全局最优解的分支定界算法.为改进算法的收敛性,根据问题的最优性和可行性提出一新的区域剪枝准则以排除(QP)的可行域中不存在全局解的部分.数值算例表明该准则能有效地加速算法的收敛性.     

一类极大极小优化问题的信赖域算法

借助于K-T条件和NCP函数,提出了处理一类极大极小优化问题的信赖域算法。数值实验结果表明该方法足行之有效的。