Java实现算法导论中线性规划单纯形算法 转

贝克街的亡灵sf

` `

package sk.mlib;

import java.util.Random;

/*************************************************************************

* Compilation: javac Simplex.java

* Execution: java Simplex

*

* Given an M-by-N matrix A, an M-length vector b, and an

* N-length vector c, solve the LP { max cx : Ax <= b, x >= 0 }.

* Assumes that b >= 0 so that x = 0 is a basic feasible solution.

*

* Creates an (M+1)-by-(N+M+1) simplex tableaux with the

* RHS in column M+N, the objective function in row M, and

* slack variables in columns M through M+N-1.

*

*************************************************************************/

public class Simplex {

private static final double EPSILON = 1.0E-10;

private double[][] a; // tableaux

private int M; // number of constraints

private int N; // number of original variables

private int[] basis; // basis[i] = basic variable corresponding to row i

// only needed to print out solution, not book

// sets up the simplex tableaux

public Simplex(double[][] A, double[] b, double[] c) {

M = b.length;

N = c.length;

a = new double[M+1][N+M+1];

for (int i = 0; i < M; i++)

for (int j = 0; j < N; j++)

a[i][j] = A[i][j];

for (int i = 0; i < M; i++) a[i][N+i] = 1.0;

for (int j = 0; j < N; j++) a[M][j] = c[j];

for (int i = 0; i < M; i++) a[i][M+N] = b[i];

basis = new int[M];

for (int i = 0; i < M; i++) basis[i] = N + i;

solve();

// check optimality conditions

assert check(A, b, c);//初始化判断输入的线性规划是否存在初始基本可行解、对偶最优解判断、无限解

}

// run simplex algorithm starting from initial BFS

private void solve() {

while (true) {

// find entering column q

int q = bland();

if (q == -1) break; // optimal

// find leaving row p

int p = minRatioRule(q);

if (p == -1) throw new RuntimeException("Linear program is unbounded");

// pivot

pivot(p, q);

// update basis

basis[p] = q;

}

}

// lowest index of a non-basic column with a positive cost

private int bland() {

for (int j = 0; j < M + N; j++)

if (a[M][j] > 0) return j;

return -1; // optimal

}

// index of a non-basic column with most positive cost

private int dantzig() {

int q = 0;

for (int j = 1; j < M + N; j++)

if (a[M][j] > a[M][q]) q = j;

if (a[M][q] <= 0) return -1; // optimal

else return q;

}

// find row p using min ratio rule (-1 if no such row)

private int minRatioRule(int q) {

int p = -1;

for (int i = 0; i < M; i++) {

if (a[i][q] <= 0) continue;

else if (p == -1) p = i;

else if ((a[i][M+N] / a[i][q]) < (a[p][M+N] / a[p][q])) p = i;

}

return p;

}

// pivot on entry (p, q) using Gauss-Jordan elimination，主元操作，交换变量

private void pivot(int p, int q) {

// everything but row p and column q

for (int i = 0; i <= M; i++)

for (int j = 0; j <= M + N; j++)

if (i != p && j != q) a[i][j] -= a[p][j] * a[i][q] / a[p][q];

// zero out column q

for (int i = 0; i <= M; i++)

if (i != p) a[i][q] = 0.0;

// scale row p

for (int j = 0; j <= M + N; j++)

if (j != q) a[p][j] /= a[p][q];

a[p][q] = 1.0;

}

// return optimal objective value

public double value() {

return -a[M][M+N];

}

// return primal solution vector

public double[] primal() {

double[] x = new double[N];

for (int i = 0; i < M; i++)

if (basis[i] < N) x[basis[i]] = a[i][M+N];

return x;

}

// return dual solution vector

public double[] dual() {

double[] y = new double[M];

for (int i = 0; i < M; i++)

y[i] = -a[M][N+i];

return y;

}

// is the solution primal feasible?构造辅助线性规划

private boolean isPrimalFeasible(double[][] A, double[] b) {

double[] x = primal();

// check that x >= 0

for (int j = 0; j < x.length; j++) {

if (x[j] < 0.0) {

System.out.println("x[" + j + "] = " + x[j] + " is negative");

return false;

}

}

// check that Ax <= b

for (int i = 0; i < M; i++) {

double sum = 0.0;

for (int j = 0; j < N; j++) {

sum += A[i][j] * x[j];

}

if (sum > b[i] + EPSILON) {

System.out.println("not primal feasible");

System.out.println("b[" + i + "] = " + b[i] + ", sum = " + sum);

return false;

}

}

return true;

}

// is the solution dual feasible?构造对偶线性规划

private boolean isDualFeasible(double[][] A, double[] c) {

double[] y = dual();

// check that y >= 0

for (int i = 0; i < y.length; i++) {

if (y[i] < 0.0) {

System.out.println("y[" + i + "] = " + y[i] + " is negative");

return false;

}

}

// check that yA >= c

for (int j = 0; j < N; j++) {

double sum = 0.0;

for (int i = 0; i < M; i++) {

sum += A[i][j] * y[i];

}

if (sum < c[j] - EPSILON) {

System.out.println("not dual feasible");

System.out.println("c[" + j + "] = " + c[j] + ", sum = " + sum);

return false;

}

}

return true;

}

// check that optimal value = cx = yb

private boolean isOptimal(double[] b, double[] c) {

double[] x = primal();

double[] y = dual();

double value = value();

// check that value = cx = yb

double value1 = 0.0;

for (int j = 0; j < x.length; j++)

value1 += c[j] * x[j];

double value2 = 0.0;

for (int i = 0; i < y.length; i++)

value2 += y[i] * b[i];

if (Math.abs(value - value1) > EPSILON || Math.abs(value - value2) > EPSILON) {

System.out.println("value = " + value + ", cx = " + value1 + ", yb = " + value2);

return false;

}

return true;

}

private boolean check(double[][]A, double[] b, double[] c) {

return isPrimalFeasible(A, b) && isDualFeasible(A, c) && isOptimal(b, c);

}

// print tableaux

public void show() {

System.out.println("M = " + M);

System.out.println("N = " + N);

for (int i = 0; i <= M; i++) {

for (int j = 0; j <= M + N; j++) {

System.out.printf("%7.2f ", a[i][j]);

}

System.out.println();

}

System.out.println("value = " + value());

for (int i = 0; i < M; i++)

if (basis[i] < N) System.out.println("x_" + basis[i] + " = " + a[i][M+N]);

System.out.println();

}

public static void test(double[][] A, double[] b, double[] c) {

Simplex lp = new Simplex(A, b, c);

System.out.println("value = " + lp.value());

double[] x = lp.primal();

for (int i = 0; i < x.length; i++)

System.out.println("x[" + i + "] = " + x[i]);

double[] y = lp.dual();

for (int j = 0; j < y.length; j++)

System.out.println("y[" + j + "] = " + y[j]);

}

public static void test1() {

double[][] A = {

{ -1, 1, 0 },

{ 1, 4, 0 },

{ 2, 1, 0 },

{ 3, -4, 0 },

{ 0, 0, 1 },

};

double[] c = { 1, 1, 1 };

double[] b = { 5, 45, 27, 24, 4 };

test(A, b, c);

}

// x0 = 12, x1 = 28, opt = 800

public static void test2() {

double[] c = { 13.0, 23.0 };

double[] b = { 480.0, 160.0, 1190.0 };

double[][] A = {

{ 5.0, 15.0 },

{ 4.0, 4.0 },

{ 35.0, 20.0 },

};

test(A, b, c);

}

// unbounded

public static void test3() {

double[] c = { 2.0, 3.0, -1.0, -12.0 };

double[] b = { 3.0, 2.0 };

double[][] A = {

{ -2.0, -9.0, 1.0, 9.0 },

{ 1.0, 1.0, -1.0, -2.0 },

};

test(A, b, c);

}

// degenerate - cycles if you choose most positive objective function coefficient

public static void test4() {

double[] c = { 10.0, -57.0, -9.0, -24.0 };

double[] b = { 0.0, 0.0, 1.0 };

double[][] A = {

{ 0.5, -5.5, -2.5, 9.0 },

{ 0.5, -1.5, -0.5, 1.0 },

{ 1.0, 0.0, 0.0, 0.0 },

};

test(A, b, c);

}

// test client

public static void main(String[] args) {

try { test1(); }

catch (Exception e) { e.printStackTrace(); }

System.out.println("--------------------------------");

/*try { test2(); }

catch (Exception e) { e.printStackTrace(); }

System.out.println("--------------------------------");

try { test3(); }

catch (Exception e) { e.printStackTrace(); }

System.out.println("--------------------------------");

try { test4(); }

catch (Exception e) { e.printStackTrace(); }

System.out.println("--------------------------------");

int M = Integer.parseInt(args[0]);

int N = Integer.parseInt(args[1]);

double[] c = new double[N];

double[] b = new double[M];

double[][] A = new double[M][N];

for (int j = 0; j < N; j++)

c[j] = new Random().nextInt(1000);

for (int i = 0; i < M; i++)

b[i] = new Random().nextInt(1000);

for (int i = 0; i < M; i++)

for (int j = 0; j < N; j++)

A[i][j] = new Random().nextInt(100);

Simplex lp = new Simplex(A, b, c);

System.out.println(lp.value());*/

}

}

//***************************************

` `

value = 22.0

x[0] = 9.000000000000002

x[1] = 9.0

x[2] = 4.0

y[0] = -0.0

y[1] = 0.1428571428571428

y[2] = 0.4285714285714286

y[3] = -0.0

y[4] = 1.0

贝克街的亡灵sf

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