共轭梯度法-湖南大学.ppt

luluoxx@hnu.cn 第4章 无约束多维问题的最优化方法(第一讲) 湖南大学车辆工程 上机实践中存在的问题(10-8) 如何求目标函数f (x)的最大值问题? 处理方法:-f (x) 或1/f (x) 约束非正(=0) 数值计算方法的特点 初始值x(0)的敏感性?算法 结果的误差 初始值(initial guess)接近最优解一定收敛的越快吗? 何时收敛??数值最优算法迭代过程的终止条件 Optimization Algorithm Optimization Algorithms are iterative They begin with an initial guess of the optimal values of the variables and generate a sequence of improved estimates until they reach a solution The strategy used to move from one iterate to the next distinguishes one algorithm from another Most strategies make use of the values of the objective function f, the constraints c, and possibly the first and second derivatives of these functions Some algorithms accumulate information gathered at previous iterations, while others use only local information from the current point Regardless of these specifics, all good algorithms should possess the following properties Robustness: they should perform well on a wide variety of problems in their class, for all reasonable choices of the initial variables Efficiency: they should not require too much computer time and storage Accuracy: they should be able to identify a solution with precision, without being overly sensitive to errors in the data or to the arithmetic rounding errors that occur when the algorithm is implemented on a computer 数学上讨论(Discussion) Re.search = do search again, and search repeatly (discovery) 比较(一般?特殊) 最优解(optimum) 等值线(contour) 根(root) 迭代终止/=收敛? ? exitflag=1 最优化方法的收敛 即近似解X(k)逼近精确解X* 迭代的终止条件 设计变量的变化量趋于无穷小TolX=1.0e-6 (不能TolX=0) 目标或约束函数的变化量趋于无穷小TolFun,TolCon 目标函数的梯度的变化量趋于无穷小TolPCG 函数计算最大次数MaxFunEvals 最大迭代次数MaxIter=400 Various kinds of computer errors Truncation Error: Round-off Error: caused by representing /storing numeric data in finite bits Overflow/Underflow: Negligible addition: Loss of Significance: caused by a “bad subtraction”, which means a subtraction of a number from another one that is almost equal in value Error Magnification: caused and magnified /propagated by multiplying/dividi