外点惩处函数法·约束优化问题

外点法惩处函数（r添加，SUMT.java）用于求解约束优化问题。解题过程例如以下：

Step1 输入目标函数与约束方程，构建外点惩处函数法求解方程，求解初始化。

Step2 对求解方程进行一次无约束优化方法求解（鲍威尔BWE），得到新解。

Step3 新解与原解求误差。如误差满足精度要求，则输出解，否则添加因子r，运行Step 2。

鲍威尔法（BWE.java）是N维无约束求解方法。须要调用一维求解方法。一维求解方法採用黄金切割法（GSM.java）。

在实现算法的代码中，我去掉了输入处理，人为地将输入确定下来。可降低代码篇幅。

我会将文件打包放入我的下载，欢迎大家一起交流。

（1）外点法惩处函数 SUMT.java：

package ODM.Method;

import java.util.Arrays;
/*
 * 无约束优化方法：惩处函数法·外点法
 */
public class SUMT { private int n = 6; // 维数，变量个数 private final double eps = 1e-5; // 精度 private final double c = 5; // 递增系数 private double r = 0.1; // 惩处因子，趋向无穷 public SUMT(){ Finit(); AlgorithmProcess(); AnswerOutput(); } // 结果 private double[] xs; private double ans; private void Finit(){ xs = new double[n]; Arrays.fill(xs, 0); ans = -1; //xs[0] = xs[1] = xs[2] = xs[4] = 1; xs[3] = 3; xs[5] = 5; } // 算法主要流程 private void AlgorithmProcess(){ int icnt = 0; // 迭代次数 double[] x = new double[n]; // 转化为无约束优化问题的解 while(true){ icnt++; BWE temp = new BWE(n, r, xs); // 採用鲍威尔方法求函数最优解 x = temp.retAns(); if(retOK(x) <= eps){ // 满足精度要求 for(int i = 0; i < n; i++) xs[i] = x[i]; ans = temp.mAns(); break; } r = c * r; for(int i = 0; i < n; i++) xs[i] = x[i]; } System.out.println("迭代次数：" + icnt); } // 收敛条件（仅仅有一个，不完好） private double retOK(double[] x){ double sum = 0; for(int i = 0; i < n; i++){ sum += Math.pow(x[i] - xs[i], 2); } return Math.sqrt(sum); } // 结果输出 private void AnswerOutput(){ for(int i = 0; i < n; i++) System.out.printf("%.6f\t", xs[i]); System.out.printf("%.6f\n", ans); } public static void main(String[] args) { // TODO Auto-generated method stub new SUMT(); } }

（2）鲍威尔法 BWE.java：

package ODM.Method;

import java.util.Arrays;

public class BWE { private double r; // 初始化变量 private double[] x0; // 初始解集 private double[][] e; // 初始方向 private int N; final private double eps = 1e-5; private Func F; // 初始化：初始点, 初始矢量(n 个，n*n 矩阵), 维数 private void Init(int n){ this.x0 = new double[n]; if(r == -1) Arrays.fill(this.x0, 0); else{ } this.e = new double[n][n]; for(int i = 0; i < n; i++){ for(int j = 0; j < n; j++){ if(i != j)e[i][j] = 0; else e[i][j] = 1; } } this.N = n; if(r != -1) F = new Func(r); else F = new Func(); } // 搜索点, 方向矢量 private double[][] x; private double[][] d; // 方向重排, 队列操作 private void queueDir(double[] X){ // 删去首方向 for(int i = 0; i < N-1; i++){ for(int j = 0; j < N; j++){ d[i][j] = d[i+1][j]; } } // 新方向插入队尾 for(int i = 0; i < N; i++) d[N-1][i] = X[i]; } private void Process(){ x = new double[N+1][N]; d = new double[N][N]; for(int j = 0; j < N; j++) x[0][j] = x0[j]; for(int i = 0; i < N; i++){ for(int j = 0; j < N; j++){ d[i][j] = e[i][j]; } } int k = 0; // 迭代次数 while(k < N){ for(int i = 1; i <= N; i++){ GSM t = new GSM(F, x[i-1], d[i-1]); x[i] = t.getOs(); } double[] X = new double[N]; for(int i = 0; i < N; i++) X[i] = x[N][i] - x[0][i]; queueDir(X); GSM t = new GSM(F, x[N], X); x[0] = t.getOs(); k++; } } // 答案打印 private void AnswerOutput(){ for(int i = 0; i < N; i++){ System.out.printf("x[%d] = %.6f\n", i+1, x[0][i]); // System.out.print(x[0][i] + " "); } System.out.printf("最小值：%.6f\n", F.fGetVal(x[0])); // System.out.println(": " + F.fGetVal(x[0])); } public BWE(int n){ this.r = -1; Init(n); Process(); AnswerOutput(); } public BWE(int n, double r, double[] x){ this.r = r; Init(n); for(int i = 0; i < n; i++) x0[i] = x[i]; Process(); } // 返回结果，解向量和最优值 public double[] retAns(){ return x[0]; } public double mAns(){ return F.fGetVal(x[0], 0); } /* public static void main(String[] args) { // TODO Auto-generated method stub new BWE(2); }*/ }

（3）黄金切割 GSM.java：

package ODM.Method;
/*
 * 黄金切割法
 */
public class GSM { private int N; // 维度 private final double landa = (Math.sqrt(5)-1)/2; // 0.618 private double[] x1; private double[] x2; private double[] os; private final double eps = 1e-5; // 解精度 private ExtM EM; // 用于获取外推法结果 // 最优值输出 public double[] getOs() { return os; } // 函数, 初始点, 方向矢量 public GSM(Func Sample, double[] x, double[] e) { //for(int i = 0; i < e.length; i++)System.out.print(e[i] + " ");System.out.println(); initial(Sample, x, e); process(Sample); AnswerPrint(Sample); } // 结果打印 private void AnswerPrint(Func Sample) { os = new double[N]; for(int i = 0; i < N; i++) os[i] = 0.5*(x1[i] + x2[i]); // System.out.println("os = " + os[0] + " " + os[1]); // System.out.println("ans = " + Sample.fGetVal(os)); } // 向量范值 private double FanZhi(double[] b, double[] a){ double sum = 0; for(int i = 0; i < N; i++){ if(b[i] - a[i] != 0 && b[i] == 0)return eps*(1e10); if(b[i] == 0)continue; sum += Math.pow((b[i] - a[i]) / b[i], 2); } return Math.pow(sum, 0.5); } // 算法主流程 private void process(Func Sample) { double[] xx1 = new double[N]; SubArraysCopy(xx1); double yy1 = Sample.fGetVal(xx1); double[] xx2 = new double[N]; AddArraysCopy(xx2); double yy2 = Sample.fGetVal(xx2); // 迭代过程 while(true){ if(yy1 >= yy2){ ArraysCopy(xx1, x1); ArraysCopy(xx2, xx1); yy1 = yy2; AddArraysCopy(xx2); yy2 = Sample.fGetVal(xx2); }else{ ArraysCopy(xx2, x2); ArraysCopy(xx1, xx2); yy2 = yy1; SubArraysCopy(xx1); yy1 = Sample.fGetVal(xx1); } //System.out.println(FanZhi(x2, x1) + " / " + Math.abs((yy2 - yy1)/yy2)); if(FanZhi(x2, x1) < eps && Math.abs(yy2 - yy1) < eps)break; } } // 获得外推法结果：左右边界 private void initial(Func Sample, double[] x, double[] e) { N = x.length; EM = new ExtM(Sample, x, e); x1 = EM.getX1(); x2 = EM.getX3(); } // 向量赋值 private void ArraysCopy(double[] s, double[] e){ for(int i = 0; i < N; i++) e[i] = s[i]; } // + landa private void AddArraysCopy(double[] arr){ for(int i = 0; i < N; i++) arr[i] = x1[i] + landa*(x2[i] - x1[i]); } // - landa private void SubArraysCopy(double[] arr){ for(int i = 0; i < N; i++) arr[i] = x2[i] - landa*(x2[i] - x1[i]); } /* public static void main(String[] args) { // TODO Auto-generated method stub double[] C = {0, 0}; double[] d = {1, 0}; new GSM(new Func(), C, d); } */ }

以上算法文件包括函数方程，黄金切割时有一维搜索的外推法确定“高低高”区间。

函数方程 Func.java。外推法 ExtM.java。

Func.java：

package ODM.Method;

public class Func { private int N; // N 维 private double[] left; // 函数左边界 private double[] right; // 函数右边界 private double r; public Func() { r = -1; } public Func(double r) { this.r = r; } // 定义函数与函数值返回 public double fGetVal(double[] x){ if(r != -1)return fGetVal(x, r); // 10*(x1+x2-5)^2 + (x1-x2)^2 return 10*Math.pow(x[0]+x[1]-5, 2) + Math.pow(x[0]-x[1], 2); // } private double max(double a, double b){return a > b ? a : b;} public double fGetVal(double[] x, double r){ double ret = 0; // function f1 // ret = Math.pow(x[0]-5, 2) + 4*Math.pow(x[1]-6, 2) // + r * ( // + Math.pow(max(64-x[0]*x[0]-x[1]*x[1], 0), 2) // + Math.pow(max(x[1]-x[0]-10, 0), 0) // + Math.pow(max(x[0]-10, 0), 2) // ); // function f2 // ret = x[0]*x[0] + x[1]*x[1] + r*(1-x[0]>0 ? 1-x[0] : 0)*(1-x[0]>0 ?

1-x[0] : 0); // function f3 ret = Math.pow(x[0]-x[3], 2) + Math.pow(x[1]-x[4], 2) + Math.pow(x[2]-x[5], 2) + r * ( + Math.pow(max(x[0]*x[0]+x[1]*x[1]+x[2]*x[2]-5, 0), 2) + Math.pow(max(Math.pow(x[3]-3, 2)+x[4]*x[4]-1, 0), 2) + Math.pow(max(x[5]-8, 0), 2) + Math.pow(max(4-x[5], 0), 2) ); return ret; } }

ExtM.java：

package ODM.Method;

/*
 * 外推法确定“高-低-高”区间
 */
public class ExtM { private int N; // 函数维数 private double[] x1; private double[] x2; private double[] x3; private double y1; private double y2; private double y3; private double h; // 步长 private double[] d; // 方向矢量 public double[] getX1() { return x1; } public double[] getX2() { return x2; } public double[] getX3() { return x3; } public double getH() { return h; } // 函数, 初始点，方向 public ExtM(Func Sample, double[] x, double[] e) { initial(Sample, x, e); process(Sample); AnswerPrint(); } // 初始化阶段 private void initial(Func Sample, double[] x, double[] e) { N = x.length; x1 = new double[N]; x2 = new double[N]; x3 = new double[N]; h = 0.01; d = new double[N]; ArraysCopy(e, 0, d); //for(int i = 0; i < d.length; i++)System.out.print(d[i]);System.out.println(); ArraysCopy(x, 0, x1); y1 = Sample.fGetVal(x1); ArraysCopy(x, h, x2); y2 = Sample.fGetVal(x2); } // 循环部分 private void process(Func Sample) { if(y2 > y1){ h = -h; ArraysCopy(x1, 0, x3); y3 = y1; }else{ ArraysCopy(x2, h, x3); y3 = Sample.fGetVal(x3); } while(y3 < y2){ h = 2*h; // System.out.println("h = " + h); ArraysCopy(x2, 0, x1); y1 = y2; ArraysCopy(x3, 0, x2); y2 = y3; ArraysCopy(x2, h, x3); y3 = Sample.fGetVal(x3); // System.out.println("x1 = " + x1[0] + " " + x1[1] + " y1 = " + y1); // System.out.println("x2 = " + x2[0] + " " + x2[1] + " y2 = " + y2); // System.out.println("x3 = " + x3[0] + " " + x3[1] + " y3 = " + y3); } } // 打印算法结果 private void AnswerPrint() { // System.out.println("x1 = " + x1[0] + " " + x1[1] + " y1 = " + y1); // System.out.println("x2 = " + x2[0] + " " + x2[1] + " y2 = " + y2); // System.out.println("x3 = " + x3[0] + " " + x3[1] + " y3 = " + y3); } // 向量转移 private void ArraysCopy(double[] s, double c, double[] e){ for(int i = 0; i < s.length; i++) e[i] = d[i]*c + s[i]; } /* public static void main(String[] args) { // TODO Auto-generated method stub // new ExtM(); }*/ }