// Paste into an online Java IDE as Main.java and run.
// Verifies infeasibility of the LP:
//   min z = -3 x1 + x2
//   s.t. x1 - 2 x2 >= 2
//        -x1 + x2 >= 3
//        x1, x2 >= 0
public class Main {
  static boolean feasible(double x1, double x2) {
    return x1 >= 0 && x2 >= 0
        && (x1 - 2*x2 >= 2 - 1e-12)
        && (-x1 + x2 >= 3 - 1e-12);
  }
 
  public static void main(String[] args) {
    System.out.println("LP to check:");
    System.out.println("  minimize  z = -3 x1 + x2");
    System.out.println("  subject to x1 - 2 x2 >= 2");
    System.out.println("             -x1 + x2 >= 3");
    System.out.println("             x1, x2 >= 0");
    System.out.println();
 
    // 1) Algebraic proof (one-line contradiction)
    System.out.println("Algebraic check:");
    System.out.println("  Add the two >= constraints:");
    System.out.println("    (x1 - 2x2) + (-x1 + x2) >= 2 + 3  =>  -x2 >= 5  =>  x2 <= -5");
    System.out.println("  But bounds require x2 >= 0. Contradiction.");
    System.out.println("  ==> The LP is INFEASIBLE.");
    System.out.println();
 
    // 2) Small brute-force sanity check over a grid (non-proof, just confirmation)
    boolean anyFeasible = false;
    double bestZ = Double.POSITIVE_INFINITY, bestX1 = Double.NaN, bestX2 = Double.NaN;
 
    // scan a modest grid to illustrate there are no feasible points
    for (int i1 = 0; i1 <= 50; i1++) {
      for (int i2 = 0; i2 <= 50; i2++) {
        double x1 = i1 * 0.5; // step 0.5
        double x2 = i2 * 0.5;
        if (feasible(x1, x2)) {
          anyFeasible = true;
          double z = -3*x1 + x2;
          if (z < bestZ) {
            bestZ = z; bestX1 = x1; bestX2 = x2;
          }
        }
      }
    }
 
    if (!anyFeasible) {
      System.out.println("Brute-force grid confirmation: no feasible points found on a 0.5 grid in [0,25]^2.");
      System.out.println("Conclusion: INFEASIBLE.");
    } else {
      System.out.printf("Found feasible point (unexpected): x1=%.4f, x2=%.4f, z=%.4f%n", bestX1, bestX2, bestZ);
    }
  }
}
 
 

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