Linear programming.
Linear programming solvers.
- Simplex.java is a bare-bones versionof the simplex algorithm.
- QSoptis a Java linear program solver created by David Applegate, William Cook,Sanjeeb Dash, and Monika Mevenkamp.It can be used at no cost for research or education purposes.QSoptSolver.java solves a linear programin LP format, such as beer.lp.
- Matlabcontains a linear programming solver in the optimization toolbox.
[wayne:tombstone] ~> matlab
< M A T L A B (R) >
Copyright 1984-2009 The MathWorks, Inc.
Version 7.9.0.529 (R2009b) 64-bit (glnxa64)
August 12, 2009
>> A = [5 15; 4 4; 35 20];
>> b = [480; 160; 1190];
>> c = [13; 23];
>> lb = [0; 0];
>> ub = [inf; inf];
>> x = linprog(-c, A, b, [], [], lb, ub)
x =
12.0000
28.0000
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- CPLEXis a high-performance mathematical programming solverfor linear programming, mixed integer programming, and quadratic programming.It supports interfaces to C, C++, Java, Python, Matlab, and Microsoft Excel.It is also accessible via the modeling systems includingAIMMS, AMPL, GAMS, and MPL.
- AMPL is a modeling language for mathematical programming.The files beer.mod and beer.datspecify the model and data for the brewery problem.
[wayne:tombstone] ~> ampl
ILOG AMPL 9.100
AMPL Version 20021038 (SunOS 5.8)
ampl: model beer.mod;
ampl: data beer.dat;
ampl: solve;
ILOG CPLEX 9.100
CPLEX 9.1.0: optimal solution; objective 800
2 dual simplex iterations (1 in phase I)
ampl: display x;
x [*] := ale 12 beer 28 ;
ampl: display constraints.dual;
x [*] := corn 1 hops 2 malt 0 ;
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- Microsoft Excel has a primitive solver add-in for Windows, but it isno longer available for Mac.
Below is the syntax highlighted version of
Simplex.javafrom §6.5 Reductions.
/*************************************************************************
* 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);
}
// 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
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?
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) {
StdOut.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) {
StdOut.println("not primal feasible");
StdOut.println("b[" + i + "] = " + b[i] + ", sum = " + sum);
return false;
}
}
return true;
}
// is the solution dual feasible?
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) {
StdOut.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) {
StdOut.println("not dual feasible");
StdOut.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) {
StdOut.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() {
StdOut.println("M = " + M);
StdOut.println("N = " + N);
for (int i = 0; i <= M; i++) {
for (int j = 0; j <= M + N; j++) {
StdOut.printf("%7.2f ", a[i][j]);
}
StdOut.println();
}
StdOut.println("value = " + value());
for (int i = 0; i < M; i++)
if (basis[i] < N) StdOut.println("x_" + basis[i] + " = " + a[i][M+N]);
StdOut.println();
}
public static void test(double[][] A, double[] b, double[] c) {
Simplex lp = new Simplex(A, b, c);
StdOut.println("value = " + lp.value());
double[] x = lp.primal();
for (int i = 0; i < x.length; i++)
StdOut.println("x[" + i + "] = " + x[i]);
double[] y = lp.dual();
for (int j = 0; j < y.length; j++)
StdOut.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(); }
StdOut.println("--------------------------------");
try { test2(); }
catch (Exception e) { e.printStackTrace(); }
StdOut.println("--------------------------------");
try { test3(); }
catch (Exception e) { e.printStackTrace(); }
StdOut.println("--------------------------------");
try { test4(); }
catch (Exception e) { e.printStackTrace(); }
StdOut.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] = StdRandom.uniform(1000);
for (int i = 0; i < M; i++)
b[i] = StdRandom.uniform(1000);
for (int i = 0; i < M; i++)
for (int j = 0; j < N; j++)
A[i][j] = StdRandom.uniform(100);
Simplex lp = new Simplex(A, b, c);
StdOut.println(lp.value());
}
}
