#Open Hemisphere


    # Create the linear programming problem

    # Define the decision variables

    # Define the objective function

    # Define the constraints

    # Add the additional constraint for at least K solutions

    # Solve the problem

        # Problem is feasible, return the optimal solution
        # Problem is infeasible

# Example usage


if __name__ == '__main__':
    from pulp import LpProblem, LpVariable, LpMaximize
    def solve_open_hemisphere(X, K):
        prob = LpProblem("Open Hemisphere Problem", LpMaximize)
        y = [LpVariable(f"y{i}", lowBound=0, cat="Continuous") for i in range(len(X[0]))]
        prob += sum(y)
        for x in X:
            prob += sum(xi * yi for xi, yi in zip(x, y)) >= 1
        prob += sum(1 for x in X for xi in x for yi in y if xi * yi > 0) >= K
        prob.solve()
        if prob.status == 1:
            solution = [v.varValue for v in y]
            return solution
        else:
            return None
    X = [[1, 2, 3], [-1, -2, 1], [0, 1, -1]]
    K = 2
    solution = solve_open_hemisphere(X, K)
    if solution:
        print("Feasible solution:", solution)
    else:
        print("No feasible solution found.")
