求解非线性方程组的一种并行算法

提出了一种在分布式环境下求解非线性方程组的并行算法,该算法将Newton迭代法中的Jacobi矩阵进行适当的分裂,使得Newton迭代法具有很好的并行性。并在理论上进行了收敛性分析。在HP rx2600集群上进行的数值实验结果表明并行效率达70%以上。     

GRAPES动力框架中大规模稀疏线性系统并行求解及优化

赫姆霍兹方程求解是GRAPES数值天气预报系统动力框架中的核心部分,可转换为大规模稀疏线性系统的求解问题,但受限于硬件资源和数据规模,其求解效率成为限制系统计算性能提升的瓶颈。分别通过MPI、MPI+OpenMP、CUDA三种并行方式实现求解大规模稀疏线性方程组的广义共轭余差法,并利用不完全分解LU预处理子（ILU）优化系数矩阵的条件数,加快迭代法收敛。在CPU并行方案中,MPI负责进程间粗粒度并行和通信,OpenMP结合共享内存实现进程内部的细粒度并行,而在GPU并行方案中,CUDA模型采用数据传输、访存合并及共享存储器方面的优化措施。实验结果表明,通过预处理优化减少迭代次数对计算性能提升明显,MPI+OpenMP混合并行优化较MPI并行优化性能提高约35%,CUDA并行优化较MPI+OpenMP混合并行优化性能提高约50%,优化性能最佳。     

一种高效的函数可微的全局优化模拟退火算法

针对函数可微的全局优化问题,将最速下降法,Newton法和罚函数法引入模拟退火算法中,提出了一种高效的模拟退火算法.该算法可以求得可微函数优化问题的全局最优解,且具有计算量小,效率高的特点.利用罚函数将约束优化问题转化为无约束优化问题后,可以利用提出的算法进行求解.数值算例表明,提出的算法能够高效地求解无约束及带约束的函数可微的全局优化问题.     

基于SIP 算法的暂态稳定约束最优潮流计算

提出一种基于约束转换方法的建立暂态稳定约束最优潮流(OTS)的算法.运用函数转换技术等价变换OTS为半无限优化(SIP)问题,转换后的SIP问题与常规的最优潮流(OPF)具有相同的维数;基于有效集策略设计了转换后SIP问题的迭代法,该方法减少了子问题中不等式约束的个数.电力系统的2个数值仿真例子验证了所提出算法的有效性.     

一种求解非线性方程组的混合优化算法及程序实现

非线性方程组求解是工程实践与理论研究中的一个典型问题。传统的方法主要有梯度法、Newton迭代法等。该文综合修正Newton法与梯度法的各自优势,对非线性方程组的求解问题提出了一种混合方法并用C语言编码实现该算法。将两种方法相结合,使其相互取长补短,在迭代初始值不太好的情况下也能保证收敛性,同时加快收敛速度,数值结果表明该算法是有效的。     

基于变量转换的并行优化算法

针对大规模边界约束优化问题,现有并行变量转换(PVT)算法不适于直接求解。基于此,采用内点法和逐步下降的思想,提出一个并行求解边界约束最优化问题的可行算法。在下降方向满足梯度相关、步长满足Goldstein规则的条件下,证明该算法的收敛性。当约束失效时,该算法退化为求解无约束的PVT算法,从而成为原有算法向约束优化问题的一个推广。     

MISO系统中上下行链路对偶性研究

TDD系统中,利用上下行链路的对偶性可以将非凸的下行优化问题转换到上行链路,从而极大简化问题的分析和数值求解。上下行链路在波束成形和容量域两方面都存在对偶性。本文在统一的系统模型下,针对总功率约束和每天线功率约束两种情况,利用拉格朗日对偶法分别对波束成形对偶、容量域对偶做了推导、分析。此外,用凸优化软件包CVX简化了问题的建模。分析表明,上下行对偶等价于拉格朗日对偶,对偶问题可更一般地表示为极大极小问题。针对每天线功率约束下的波束成形问题,用迭代法和内点法分别对其进行了MATLAB仿真。仿真结果表明,非凸的下行优化问题可以利用对偶性转化为上行链路中的凸优化问题,从而得以解决。     

基于并行约束规划的最大团识别研究

为提高大数据平台下大规模图例的最大团问题求解效率,提出一种基于并行约束规划的最大团识别算法.通过BMT图划分策略将一个复杂图例分割为若干个可独立计算的子图,并将其分配给Spark集群中的计算节点,每个计算节点采用约束规划方法对分割产生的子问题分别进行建模和求解,实现最大团问题的并行化处理.引入时间预测模型,设计基于任务运行时间预测模型的并行图划分方法,从而有效解决计算节点的负载均衡问题.实验结果表明,与基于BMC图划分策略的最大团并行识别算法相比,该算法具有更高的求解效率,可取得近似线性的加速比.     

三维热传导方程的Krylov子空间方法并行分析

热传导方程在地下水流动数值模拟、油藏数值模拟等工程计算中有着广泛应用,其并行实现是加速问题求解速度、提高问题求解规模的重要手段,因此热传导方程的并行求解具有重要意义。对Krylov子空间方法中的CG和GMRES算法进行并行分析,并对不同的预处理CG算法作了比较。在Linux集群系统上,以三维热传导模型为例进行了数值实验。实验结果表明,CG算法比GMRES算法更适合建立三维热传导模型的并行求解。此外,CG算法与BJACOBI预条件子的整合在求解该热传导模型时,其并行程序具有良好的加速比和效率。因此,采用BJACOBI预处理技术的CG算法是一种较好的求解三维热传导模型的并行方案。     

一种求解约束优化问题的混合算法

提出一种基于修改增广Lagrange函数和PSO的混合算法用于求解约束优化问题。将约束优化问题转化为界约束优化问题,混合算法由两层迭代结构组成,在内层迭代中,利用改进PSO算法求解界约束优化问题得到下一个迭代点。外层迭代主要修正Lagrange乘子和罚参数,检查收敛准则是否满足,重构下次迭代的界约束优化子问题,检查收敛准则是否满足。数值实验结果表明该混合算法的有效性。     

大规模过程系统优化方法

基于非线性约束的序列界无约束极小化方法,对大规模过程系统稳态优化的序列界约束极小化方法(SBCMM)进行了研究.对工程模型引进松弛变量处理后,SBCMM的罚函数仅包含等式约束的惩罚项,不包含界约束及不等式约束的惩罚项.原问题的解由求解一系列界约束极小化子问题而非无约束极小化子问题来获得.最后,用一类规模可变的非线性规划...     

基于极大熵差分进化混合算法求解非线性方程组*

针对非线性方程组,给出了一种新的算法——极大熵差分进化混合算法。首先把非线性方程组转换为一个不可微优化问题;然后用一个称之为凝聚函数的光滑函数直接代替不可微的极大值函数,从而可把非线性方程组的求解转换为无约束优化问题,利用差分进化算法对其进行求解。计算结果表明,该算法在求解的准确性和有效性均优于其他算法。     

Greedy algorithms for a class of knapsack problems with binary weights

In this article we identify a class of two-dimensional knapsack problems with binary weights and related three-criteria unconstrained combinatorial optimization problems that can be solved in polynomial time by greedy algorithms. Starting from the knapsack problem with two equality constraints we show that this problem can be solved efficiently by using an appropriate partitioning of the items with respect to their binary weights. Based on the results for this problem we derive an algorithm for the three-criteria unconstrained combinatorial optimization problem with two binary objectives that explores the connectedness of the set of efficient knapsacks with respect to a combinatorial definition of adjacency. Furthermore, we prove that our approach is asymptotically optimal and provide extensive computational experiments that shows that we can solve the three-criteria problem with up to one million items in less than half an hour. Finally, we derive an efficient algorithm for the two-dimensional knapsack problems with binary constraints that only takes into account the results we obtained for the unconstrained three-criteria problem with binary weights.     

一种新的遗传算法求解有等式约束的优化问题

针对有等式约束的优化问题,提出了一种新的遗传算法.该算法是在种群初始化、交叉、变异操作过程中使用求解参数方程的方法处理等式约束,违反不等式约束的个体用死亡罚函数进行惩罚设计出的实数编码遗传算法.数值实验结果表明,新算法性能优于现有其它算法;它不仅可以处理线性等式约束,而且还可以处理非线性等式约束,同时提高了收敛速度和解的精度,是一种通用强、高效稳健的智能算法.     

A hybrid cooperative search algorithm for constrained optimization

Many engineering design problems can be formulated as constrained optimization problems which often consist of many mixed equality and inequality constraints. In this article, a hybrid coevolutionary method is developed to solve constrained optimization problems formulated as min–max problems. The new method is fast and capable of global search because of combining particle swarm optimization and gradient search to balance exploration and exploitation. It starts by transforming the problem into unconstrained one using an augmented Lagrangian function, then using two groups to optimize different components of the solution vector in a cooperative procedure. In each group, the final stage of the search procedure is accelerated by via a simple local search method on the best point reached by the preceding exploration based search. We validated the effectiveness and robustness of the proposed algorithm using several engineering problems taken from the specialised literature.     

A particle filter via constrained sampling for nonlinear dynamic systems

This article is concentrated on the particle filtering problem for nonlinear systems with nonlinear equality constraints. Considering the constraint information incorporated into filters can improve the state estimation accuracy, we propose an adaptive constrained particle filter via constrained sampling. First, in order to obtain particles drawn from the constrained important density function, we construct and solve a general optimization function theoretically fusing equality constraints and the importance density function. Furthermore, to reduce the computation time caused by the number of particles, the constrained Kullback‐Leiler distance sampling method is given to online adapt the number of particles needed for state estimation. A simulation study in the context of road‐confined vehicle tracking demonstrates that the proposed filter outperforms the typical constrained ones for equality constrained dynamic systems.     

Numerical experience with a derivative-free trust-funnel method for nonlinear optimization problems with general nonlinear constraints

A trust-funnel method is proposed for solving nonlinear optimization problems with general nonlinear constraints. It extends the one presented by Gould and Toint [Nonlinear programming without a penalty function or a filter. Math. Prog. 122(1):155–196, 2010], originally proposed for equality-constrained optimization problems only, to problems with both equality and inequality constraints and where simple bounds are also considered. As the original one, our method makes use of neither filter nor penalty functions and considers the objective function and the constraints as independently as possible. To handle the bounds, an active-set approach is employed. We then exploit techniques developed for derivative-free optimization (DFO) to obtain a method that can also be used to solve problems where the derivatives are unavailable or are available at a prohibitive cost. The resulting approach extends the DEFT-FUNNEL algorithm presented by Sampaio and Toint [A derivative-free trust-funnel method for equality-constrained nonlinear optimization. Comput. Optim. Appl. 61(1):25–49, 2015], which implements a derivative-free trust-funnel method for equality-constrained problems. Numerical experiments with the extended algorithm show that our approach compares favourably to other well-known model-based algorithms for DFO.     

基于差分算法的反渗透海水淡化优化调度

以系统运行费用为目标的反渗透海水淡化优化调度是一类带有约束的非线性优化问题。针对这一问题,提出一种改进的差分进化算法。该算法对基本差分进化算法中的变异因子和交叉因子进行改进;定义约束违反度函数,将约束优化问题转化为无约束的优化问题。以24小时为一个周期,通过改进的差分进化算法对系统模型进行优化调度。仿真结果表明,改进的算法可以对机组进行优化操作,有效的降低了系统的生产成本。     

A Dual‐Type Method Based Algorithm for Nonlinear Large Network Optimization Problems

In previous research, we have proposed a Dual Projected Pseudo Quasi Newton (DPPQN) method which differs from the conventional Lagrange relaxation method by treating the inequality constraints as the domain of the primal variables in the dual function and using Projection Theory to handle the inequality constraints. We have combined this dual‐type method with a Projected Jacobi (PJ) method to solve nonlinear large network optimization problems with decomposable inequality constraints, and have achieved several attractive features. To retain the attractive features and to remedy the flaw of the previous method, in the current paper, we propose an active set strategy based DPPQN method to solve the projection problem formed by coupling functional inequality constraints. This method associated with the DPPQN method and the PJ method can be used to solve general nonlinear large network optimization problems. We present this algorithm, demonstrate its computational efficiency through numerical simulations and compare it with the previous method.     

一类非线性极大极小问题的极大熵社会认知算法

针对一类非线性极大极小问题目标函数非光滑的特点给求解带来的困难,利用社会认知算法并结合极大熵函数法给出了此类问题的一种新的有效算法。首先利用极大熵函数将原问题转化为一个光滑无约束优化问题,然后利用社会认知算法对其进行求解。该算法是基于社会认知理论,通过一系列的学习代理来模拟人类的社会性以及智能性从而完成对目标的优化。数值结果表明,该算法收敛快,数值稳定性好,是求解非线性极大极小问题的一种有效算法。