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/*
* This file is part of choco-solver, http://choco-solver.org/
*
* Copyright (c) 2022, IMT Atlantique. All rights reserved.
*
* Licensed under the BSD 4-clause license.
*
* See LICENSE file in the project root for full license information.
*/
package org.chocosolver.lp;
import java.util.Arrays;
import java.util.HashMap;
import java.util.Optional;
/**
* A linear program, equipped with a Simplex method.
* This is based on "Introduction to Algorithms, Third Edition",
* By Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein,
* 29.3 The simplex algorithm.
* There are various ways to declare a LP.
* Either, by giving it as a standard form providing the A nxm-matrix, the m-vector b and the n-vector c.
*
{@code
* double[] c = {5, 7};
* double[][] A = {{4, 3}, {2, 3}};
* double[] b = {36, 48};
* LinearProgram lp = new LinearProgram(A, b, c);
* }
*
* Or, by declaring the variables first and then adding some constraints and the objective function.
* In that case, all the variables must be declared first and have to be nonnegative (≥ 0).
*
{@code
* LinearProgram lp = new LinearProgram(false);
* lp.makeVariables(2);
* lp.addLeq(new double[]{4, 3}, 36);
* lp.addLeq(new double[]{2, 3}, 48);
* lp.setObjective(true, new double[]{5, 7});
* }
*
* Next, a call to {@link #simplex()} runs the Simplex algorithm and returns the status of the resolution.
* If the status is {@link Status#FEASIBLE}
* then the value assigned to each variable is accessible with {@code lp.lp.value(i);}
* and the value of the objective function with {@code lp.objective();}.
*
*
*
* @author Charles Prud'homme
* @since 04/07/2022
*/
public class LinearProgram {
private static class Slack {
// number of non-basic variables
private final int n;
// number of basic variables
private final int m;
// a mxn matrix
private final double[][] A;
// an m-vector
private final double[] b;
// an n-vector
private final double[] c;
// an m-vector, the basic variables
private final int[] B;
// an n-vector, the non-basic variables
private final int[] N;
// an integer
private double v;
/**
* Slack form of a linear program
*
* @param a an m.n matrix
* @param b an m-vector
* @param c an n-vector
*/
public Slack(double[][] a, double[] b, double[] c) {
this.m = b.length;
this.n = c.length;
this.A = a.clone();
this.b = b.clone();
this.c = c.clone();
this.N = new int[n];
this.B = new int[m];
this.v = 0;
initialBasicSolution();
}
/**
* Slack form of a linear program with N and B already defined
*
* @param a an m.n matrix
* @param b an m-vector
* @param c an n-vector
* @param N an n-vector
* @param B an m-vector
* @param v a double
*/
public Slack(double[][] a, double[] b, double[] c, int[] N, int[] B, double v) {
this.m = b.length;
this.n = c.length;
this.A = a;
this.b = b;
this.c = c;
this.N = N;
this.B = B;
this.v = v;
}
/**
* Initialize N and B wrt basic solution
*/
private void initialBasicSolution() {
this.v = 0;
for (int i = 0; i < n; i++) {
this.N[i] = i;
}
for (int i = 0; i < m; i++) {
this.B[i] = i + n;
}
}
/**
* The method takes as input a slack form and it modifies in-place A, b, c, v, N and B.
*
* @param l index of the leaving variable
* @param e index of the entering variable
*/
void pivot(int l, int e, boolean trace) {
if (trace) System.out.printf("[pivot] e: x%d, l: x%d\n", B[l], N[e]);
// Compute the coefficients of the equation for new basic variable x_e
b[l] /= A[l][e];
for (int j = 0; j < n; j++) {
if (j != e) {
A[l][j] /= A[l][e];
}
}
A[l][e] = 1 / A[l][e];
// Compute the coefficients of the remaining constraints
for (int i = 0; i < m; i++) {
if (i != l) {
b[i] -= A[i][e] * b[l];
for (int j = 0; j < n; j++) {
if (j != e) {
A[i][j] = A[i][j] - A[i][e] * A[l][j];
}
}
A[i][e] *= -A[l][e];
}
}
// Compute the objective function
v += c[e] * b[l];
for (int j = 0; j < n; j++) {
if (j != e) {
c[j] -= c[e] * A[l][j];
}
}
c[e] *= -A[l][e];
// Update basic and non-basic variables set
int x = N[e];
N[e] = B[l];
B[l] = x;
}
private void setValues(double[] x) {
for (int i = 0; i < m; ++i) {
if (B[i] < n) x[B[i]] = b[i];
}
}
@Override
public String toString() {
StringBuilder st = new StringBuilder();
st.append("z = ").append(v);
for (int i = 0; i < n; i++) {
st.append(c[i] >= 0. ? " + " : " - ")
.append(String.format("%.4f", Math.abs(c[i])))
.append(" * x").append(N[i]);
}
st.append("\n");
for (int j = 0; j < m; j++) {
st.append("x")
.append(B[j])
.append(" = ")
.append(b[j]);
for (int i = 0; i < n; i++) {
st.append(A[j][i] <= 0. ? " + " : " - ")
.append(String.format("%.4f", Math.abs(A[j][i])))
.append(" * x")
.append(N[i]);
}
st.append("\n");
}
return st.toString();
}
}
public enum Status {
UNKNOWN,
FEASIBLE,
INFEASIBLE,
UNBOUNDED
}
// number of variables
private int n;
// number of constraints
private int m;
// a mxn matrix
private double[][] A;
// an m-vector
private double[] b;
// an n-vector
private double[] c;
// an n-vector
private double[] x;
private double z;
// feasibility of the LP
private Status status = Status.UNKNOWN;
// trace the resolution
private final boolean trace;
/**
* Create a LinearProgram instance that take a linear program in standard form as input.
*
* @param matA is an mxn matrix
* @param vecB is an m-vector
* @param vecC is n-vector
* @param trace set to true to trace the resolution
*/
public LinearProgram(double[][] matA, double[] vecB, double[] vecC, boolean trace) {
super();
this.m = vecB.length;
this.n = vecC.length;
this.A = matA.clone();
this.b = vecB.clone();
this.c = vecC.clone();
this.x = new double[n];
this.trace = trace;
}
/**
* Create a LinearProgram instance that take a linear program in standard form as input.
*
* @param matA is an mxn matrix
* @param vecB is an m-vector
* @param vecC is n-vector
*/
public LinearProgram(double[][] matA, double[] vecB, double[] vecC) {
this(matA, vecB, vecC, false);
}
/**
* Initialize a LinearProgram instance.
*
* @param trace set to true to trace the resolution
*/
public LinearProgram(boolean trace) {
this(new double[0][0], new double[0], new double[0], trace);
}
/**
* Initialize a LinearProgram instance.
*/
public LinearProgram() {
this(false);
}
/**
* Declare a new variable.
* A variable is supposed to be nonnegative (≥ 0).
*
* @return the index of the variable
*/
public int makeVariable() {
if (m > 0) {
throw new UnsupportedOperationException("Some constraints are alreadey declared");
}
return n++;
}
/**
* Declare n new variables
*/
public void makeVariables(int n) {
if (m > 0) {
throw new UnsupportedOperationException("Some constraints are alreadey declared");
}
this.n += n;
}
private void checkLength(double[] a) {
if (a.length != n) {
throw new UnsupportedOperationException("" +
"The number of coefficients in the objective function differs from " +
"the number of variables declared.");
}
}
/**
* Set the objective function to optimize
*
* @param maximize set to true for maximization, false otherwise
* @param ci coefficients of the objective function
*/
public void setObjective(boolean maximize, double[] ci) {
checkLength(ci);
this.c = Arrays.copyOf(ci, n);
if (!maximize) {
for (int i = 0; i < n; i++) {
c[i] *= -1;
}
}
}
/**
* Add a linear inequality (≤) to the system.
*
* @param ci coefficients of the linear inequality, respecting variable indices
* @param b the right-hand side value
*/
public void addLeq(double[] ci, double b) {
checkLength(ci);
// increase capacity of A
double[][] At = this.A;
A = new double[this.m + 1][this.n];
System.arraycopy(At, 0, this.A, 0, m);
// then add the new constraint
System.arraycopy(ci, 0, this.A[m], 0, n);
// increase capacity of b
double[] bt = this.b;
this.b = new double[this.m + 1];
System.arraycopy(bt, 0, this.b, 0, m);
// and add the new rhs
this.b[m] = b;
m++;
}
/**
* Add a linear inequality (≤) to the system.
*
* @param map coefficients of the linear inequality, as a map
* @param b the right-hand side value
*/
public void addLeq(HashMap map, double b) {
double[] ci = new double[n];
map.forEach((v, c) -> ci[v] = c);
addLeq(ci, b);
}
/**
* Add a linear inequality (≥) to the system.
*
* @param ci coefficients of the linear inequality, respecting variable indices
* @param b the right-hand side value
* @implNote the (≥)-inequality is turned into a (≤)-inequality constraint
*/
public void addGeq(double[] ci, double b) {
addLeq(ci, b);
for (int i = 0; i < n; i++) {
this.A[m - 1][i] *= -1;
}
this.b[m - 1] *= -1;
}
/**
* Add a linear inequality (≥) to the system.
*
* @param map coefficients of the linear inequality, as a map
* @param b the right-hand side value
*/
public void addGeq(HashMap map, double b) {
double[] ci = new double[n];
map.forEach((v, c) -> ci[v] = c);
addGeq(ci, b);
}
/**
* Add a linear equality (=) to the system.
*
* @param ci coefficients of the linear equality, respecting variable indices
* @param b the right-hand side value
* @implNote the equality constraint is turned into two inequality constraints.
*/
public void addEq(double[] ci, double b) {
checkLength(ci);
addLeq(ci, b);
addGeq(ci, b);
}
/**
* Add a linear equality (=) to the system.
*
* @param map coefficients of the linear equality, as a map
* @param b the right-hand side value
*/
public void addEq(HashMap map, double b) {
double[] ci = new double[n];
map.forEach((v, c) -> ci[v] = c);
addEq(ci, b);
}
/**
* Apply the Simplex algorithm on this linear program.
* If the problem is infeasible, this method terminates.
* Otherwise, the optimal solution of this linear program is computed and values of the variables
* can be read calling {@link #value(int)}.
*
*
* @return true is this LP is feasible, false otherwise.
*/
public Status simplex() {
// slack form of the LP
Optional opt = initialize();
if (opt.isPresent()) {
Slack slack = opt.get();
if (Status.FEASIBLE.equals(status = solve(slack))) {
if (this.x.length != n) {
this.x = new double[n];
}
slack.setValues(x);
this.z = slack.v;
}
}
return status;
}
/**
* Main loop of the simplex.
*
* @param lipr a linear program, under slack form
* @return the status of the resolution
*/
private Status solve(Slack lipr) {
if (trace) System.out.printf("%s", lipr);
int e = enteringVariable(lipr);
while (e < Integer.MAX_VALUE) {
int l = leavingVariable(lipr, e);
if (l < 0) {
return Status.UNBOUNDED;
}
lipr.pivot(l, e, trace);
e = enteringVariable(lipr);
if (trace) System.out.printf("%s", lipr);
}
return Status.FEASIBLE;
}
/**
* Determines that the linear program is infeasible
* or returns a slack form for which the basic solution is feasible
*
* @return optional slack form of the linear program defined is this
*/
private Optional initialize() {
this.status = Status.UNKNOWN;
Arrays.fill(this.x, 0.);
this.z = 0.;
int k = argmin(b);
if (b[k] >= 0) {
if (trace) System.out.println("Initial basic solution feasible");
return Optional.of(new Slack(A, b, c));
}
if (trace) System.out.println("Formulate an auxiliary linear program, adding x0");
int x0 = 0;
Slack laux = auxiliaryLinearProgram(A, b, c);
if (trace) System.out.print(laux);
laux.pivot(k, x0, trace);
// The basic solution is now feasible for laux
this.status = Status.INFEASIBLE;
if (Status.FEASIBLE.equals(solve(laux))) {
double[] x = new double[laux.n];
laux.setValues(x);
if (x[x0] == 0.) { // if the optimal solution to laux sets x0 to 0
// Since this solution has x0 = 0, we know that our initial problem was feasible
this.status = Status.FEASIBLE;
for (int i = 0; i < laux.m; i++) {
if (laux.B[i] == x0) { // if x0 is basic
// perform one (degenerate) pivot to make it nonbasic
// using any e in N such that a0e != 0
int e = -1;
for (int j = 0; j < laux.n; j++) {
if (laux.A[i][j] != 0) {
e = j;
break;
}
}
if (e == -1) throw new UnsupportedOperationException();
if (trace) System.out.println("Perform one (degenerate) pivot to make it nonbasic");
laux.pivot(i, e, trace);
break;
}
}
// from the final slack form of laux
if (isFeasible()) {
if (trace) System.out.println("Modified final slack form");
return Optional.of(finalSlack(laux, this.c));
}
}
}
return Optional.empty();
}
/**
* Determine the index of the constraint with the smallest b.
*
* @param b right-hand side coefficients
* @return the index of the smallest b
*/
private int argmin(double[] b) {
int k = 0;
double bk = b[k];
for (int i = 1; i < m; i++) {
if (bk > b[i]) {
bk = b[i];
k = i;
}
}
return k;
}
/**
* Form the auxiliary linear program by adding -x0 to the LHS of each constraint
* and setting the objective function to -x0.
*
* @return the resulting slack form of L_aux
*/
private static Slack auxiliaryLinearProgram(double[][] A, double[] b, double[] c) {
int n = c.length;
int m = b.length;
double[][] A2 = new double[m][n + 1];
for (int i = 0; i < m; i++) {
System.arraycopy(A[i], 0, A2[i], 1, n);
A2[i][0] = -1;
}
double[] c2 = new double[n + 1];
c2[0] = -1;
return new Slack(A2, b, c2);
}
/**
* remove x0 from the constraints and restore the original objective function of L,
* but replace each basic variable in this objective function by the right-hand side of its associated constraint.
*
* @param laux the auxiliary linear program to transform
* @return final slack form of laux
*/
private static Slack finalSlack(Slack laux, double[] oriC) {
int n = laux.n - 1;
int m = laux.m;
// since x0 = 0, we can just remove it from the set of constraints
double[][] A2 = new double[m][n];
for (int i = 0; i < m; i++) {
for (int j = 0, k = 0; j <= n; j++) {
if (laux.N[j] > 0) {
A2[i][k++] = laux.A[i][j];
}
}
}
double[] b2 = new double[m];
System.arraycopy(laux.b, 0, b2, 0, m);
// We then restore the original objective function,
double[] ct = new double[n + m + 1];
System.arraycopy(oriC, 0, ct, 1, n);
// with appropriate substitutions made to include only nonbasic variables
double z = 0.;
for (int j = 0; j < laux.m; j++) {
// for every nonbasic variables
int nx = laux.B[j];
if (nx <= n) {
double cx = ct[nx];
ct[nx] = 0.;
// do the substitutions
z += cx * laux.b[j];
for (int i = 0; i < laux.n; i++) {
ct[laux.N[i]] -= cx * laux.A[j][i];
}
}
}
double[] c2 = new double[n];
int[] N2 = new int[n];
for (int i = 0, j = 0; i < laux.n; i++) {
if (laux.N[i] > 0) {
c2[j] = ct[laux.N[i]];
N2[j++] = laux.N[i] - 1;
}
}
int[] B2 = new int[m];
for (int i = 0; i < laux.m; i++) {
B2[i] = laux.B[i] - 1;
}
return new Slack(A2, b2, c2, N2, B2, z);
}
/**
* @return true if the application of the Simplex algorithm computed the (optimal) solution.
*/
public boolean isFeasible() {
return status == Status.FEASIBLE;
}
/**
* Return the value of the ith variable in the linear program.
*
* If this is infeasible, return {@code -1.}, otherwise the value is returned.
*
*
* @param i index of the variable.
* @return the value assigned the ith variable in this linear program.
*/
public double value(int i) {
if (isFeasible()) {
return x[i];
} else return -1.;
}
/**
* Return the value of the objective function defined in this linear program.
*
* If this is not feasible, returns {@link Double#NEGATIVE_INFINITY},
* otherwise, the optimal value is returnd.
*
*
* @return the value of the objective function.
*/
public double objective() {
if (isFeasible()) {
return z;
} else return Double.NEGATIVE_INFINITY;
}
/**
* Choose the entering variable with c > 0
* and break ties by always choosing the variable
* with the smallest index (Bland's rule).
*
* @return a value between [0, n) if an entering variable is found, {@link Integer#MAX_VALUE} otherwise.
*/
private static int enteringVariable(Slack s) {
int j = Integer.MAX_VALUE;
for (int i = 0; i < s.n; i++) {
if (s.c[i] > 0. && s.N[i] < j) {
j = i;
}
}
return j;
}
/**
* Choose the leaving variable, the one with the smallest ratio b(i) / A(i,e).
* Ties are broken with index value (Bland's rule)
*
* @param s the slack
* @param e index of the entering variable
* @return index of the leaving variable
*/
private static int leavingVariable(Slack s, int e) {
double min_v = Double.POSITIVE_INFINITY;
int l = -1;
for (int i = 0; i < s.m; i++) {
if (s.A[i][e] > 0) {
double d = s.b[i] / s.A[i][e];
if (min_v > d) {
l = i;
min_v = d;
}
}
}
return l;
}
@Override
public String toString() {
return toLP(A, b, c, 0.);
}
private static String toLP(double[][] A, double[] b, double[] c, double v) {
StringBuilder st = new StringBuilder();
st.append("Maximize").append('\n');
st.append("\\ v = ").append(v).append("\n");
st.append(" obj:");
for (int i = 0; i < c.length; i++) {
st.append(c[i] >= 0 ? " +" : " ")
.append(c[i])
.append(" x")
.append(i + 1);
}
st.append("\nSubject to\n");
for (int j = 0; j < b.length; j++) {
st.append(" c").append(j + 1).append(": ");
st.append(A[j][0]).append(" x1");
for (int i = 1; i < c.length; i++) {
st.append(A[j][i] >= 0 ? " +" : " ")
.append(A[j][i])
.append(" x")
.append(i + 1);
}
st.append(" <= ").append(b[j]).append('\n');
}
st.append("End");
return st.toString();
}
}