用最速下降法求最优解
//最速下降法//请根据具体题目，修改本程序“//@”所在行的下一行代码。#include<math.h>#include<stdio.h>#include<stdlib.h>//@ 题目中方程是几元的，此处将LEN设为几#define LEN 4#define TYPE floatTYPE fAnswer(TYPE *x) { //求f(x)的值，并返回TYPE f;//@ 题目中的方程写与此f = (x[0] - 1) * (x[0] - 1) + 5 * (x[1] - 5) * (x[1] - 5) + (x[2] - 1) * (x[2] - 1) + 5 * (x[3] - 5) * (x[3] - 5);return f;}void vectorMultiply(TYPE *x, TYPE e) { //常数e乘x向量，赋值给x向量 【若求负梯度，可用梯度乘-1】int i;for(i = 0; i < LEN; i++) {x[i] = e * x[i];}}void vectorAdd(TYPE *x, TYPE *y, TYPE *z) { //向量加法操作int i;for(i = 0; i < LEN; i++) {z[i] = x[i] + y[i];}}void vectorSub(TYPE *x, TYPE *y, TYPE *z) { //向量减法操作int i;for(i = 0; i < LEN; i++) {z[i] = x[i] - y[i];}}void vectoreEqual(TYPE *x, TYPE *y) { //向量赋值操作int i;for(i = 0; i < LEN; i++) {y[i] = x[i];}}TYPE norm2_graded(TYPE *x) { //负梯度模的平方int i;TYPE norm2 = 0.0;for(i = 0; i < LEN; i++) {norm2 += x[i] * x[i];}return norm2;}void DSC(TYPE *x0, TYPE *stepLength) { //用D.S.C.法求下一个x落点TYPE x1[LEN];TYPE x2[LEN];TYPE x3[LEN];TYPE x4[LEN];TYPE x5[LEN];TYPE xa[LEN];TYPE xb[LEN];TYPE xc[LEN];vectorAdd(x0, stepLength, x1);if(fAnswer(x1) > fAnswer(x0))vectorMultiply(stepLength, -1);vectorAdd(x0, stepLength, x1);while(fAnswer(x1) <= fAnswer(x0)) {vectoreEqual(x1, x0);vectorAdd(stepLength, stepLength, stepLength);vectorAdd(x1, stepLength, x1);}vectorMultiply(stepLength, 0.5);vectorSub(x0, stepLength, x2);vectoreEqual(x1, x4);vectoreEqual(x0, x3);vectorSub(x1, stepLength, x5);if(fAnswer(x5) > fAnswer(x3))vectoreEqual(x3, xb);elsevectoreEqual(x5, xb);vectorSub(xb, stepLength, xa);vectorAdd(xb, stepLength, xc);vectorMultiply(stepLength, (fAnswer(xa) - fAnswer(xc)) / (2 * (fAnswer(xa) - 2 * fAnswer(xb) + fAnswer(xc))));vectorAdd(xb, stepLength, x0);}//@ 对题目的xi分别求偏倒，赋值给stepLength[i]void setStepLength(TYPE *stepLength, TYPE *x0) {stepLength[0] = 2 * (1 - x0[0]); stepLength[1] = 10 * (5 - x0[1]);stepLength[2] = 2 * (1 - x0[2]); stepLength[3] = 10 * (5 - x0[3]);}void main() { //方法主体TYPE x0[LEN]; //初始点TYPE stepLength[LEN]; //步长TYPE e = 0.001; //误差int i; //用于循环计数printf("请输入x的初始值：\n");for(i = 0; i < LEN; i++) { //初始化x0数组printf("x%d = ", i+1);scanf("%f", &x0[i]);}setStepLength(stepLength, x0); i = 1;while(norm2_graded(stepLength) > e * e) { //最速下降法主体DSC(x0, stepLength);setStepLength(stepLength, x0);i = i + 1;}printf("最速下降法循环结束，共循环%d次\n", i - 1);printf("使用最速下降法获得的最优点为：\n");for(i = 0; i < LEN; i++) {printf("x%d = %f\n", i+1, x0[i]);}printf("minf(x) = %f\n", fAnswer(x0));printf("最速下降法程序结束!!!\n");}
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总结：
? ? ? 本解法改进了一维搜索的D.S.C.法，将步长的一维初始值改为负梯度向量，D.S.C.算法中的其它一维参量也采用向量形式替换，使此D.S.C.法可适用于多维搜索。
? ? ? 经多次修改，此程序具有较好的复用性，主要体现在两个方面，一是只需根据具体题目，对本程序的三处代码稍加修改，即可复用；二是只需少量修改，便可将本最速下降法改写成共轭梯度法或牛顿法，具体操作，可参考我的用共轭梯度法球最优解和用牛顿法求最优解的两篇博文。
? ? ? 经编程验证，此方法无误。
