ortools中的迭代算法pdlp

• ortools 中提供了一个文档介绍快速具体算法 https://developers.google.cn/optimization/lp/lp_advanced?hl=en
  其中实现了一个迭代法 PDLP, 对于低精度，以及给定初值的情况应该比较有利。

文档没有提供例子， 但代码路中有例子：

• Python版 https://github.com/google/or-tools/blob/main/ortools/pdlp/samples/simple_pdlp_program.py

• c 版本 https://github.com/google/or-tools/blob/main/ortools/pdlp/samples/simple_pdlp_program.cc

• java 版 或许可以参考实现. java 直接调用 MPSolver solver = MPSolver.createSolver("PDLP");

  • 例子 https://github.com/or-tools/java_or-tools

• 通用用法（Java可用） https://groups.google.com/g/or-tools-discuss/c/EulhYdMudBw

  • 可以控制迭代精度和步数

      MPSolver solver = MPSolver.createSolver("PDLP");
    String pdlp_params_str = "termination_criteria: { iteration_limit: 500 "+
            "simple_optimality_criteria {eps_optimal_absolute: 1e-4 eps_optimal_relative: 1e-4 }}";
    solver.setSolverSpecificParametersAsString(pdlp_params_str);
  

• python 专用的调用方法

  • https://github.com/google/or-tools/blob/stable/ortools/pdlp/samples/simple_pdlp_program.py

  • https://github.com/google/or-tools/blob/stable/ortools/pdlp/python/pywrap_pdlp_test.py

  • 这个方法，有几个好处：

    1. 可以控制迭代精度和步数，
    2. 可以传入初始值 (注意需要提供对偶变量、原变量)
    3. 以矩阵形式更高效的构造数据

    def tiny_lp():
      """Returns a small test LP.
    
      The LP:
        min 5 x_1 + 2 x_2 + x_3 +   x_4 - 14 s.t.
        2 x_1 +   x_2 + x_3 + 2 x_4  = 12
        x_1 +         x_3         >=  7
                      x_3 -   x_4 >=  1
      0 <= x_1 <= 2
      0 <= x_2 <= 4
      0 <= x_3 <= 6
      0 <= x_4 <= 3
    
    Optimum solutions:
      Primal: x_1 = 1, x_2 = 0, x_3 = 6, x_4 = 2. Value: 5 + 0 + 6 + 2 - 14 = -1.
      Dual: [0.5, 4.0, 0.0]  Value: 6 + 28 - 3.5*6 - 14 = -1
      Reduced costs: [0.0, 1.5, -3.5, 0.0]
      """
      qp = pywrap_pdlp.QuadraticProgram()
      qp.objective_offset = -14
      qp.objective_vector = [5, 2, 1, 1]
      qp.constraint_lower_bounds = [12, 7, 1]
      qp.constraint_upper_bounds = [12, np.inf, np.inf]
      qp.variable_lower_bounds = np.zeros(4)
      qp.variable_upper_bounds = [2, 4, 6, 3]
      constraint_matrix = np.array([[2, 1, 1, 2], [1, 0, 1, 0], [0, 0, 1, -1]])
      qp.constraint_matrix = scipy.sparse.csr_matrix(constraint_matrix)
      return qp
    
    def test_starting_point(self):
        params = solvers_pb2.PrimalDualHybridGradientParams()
        opt_criteria = params.termination_criteria.simple_optimality_criteria
        opt_criteria.eps_optimal_relative = 0.0
        opt_criteria.eps_optimal_absolute = 1.0e-10
        params.l_inf_ruiz_iterations = 0
        params.l2_norm_rescaling = False
    
        start = pywrap_pdlp.PrimalAndDualSolution()
        start.primal_solution = [1.0, 0.0, 6.0, 2.0]
        start.dual_solution = [0.5, 4.0, 0.0]
        result = pywrap_pdlp.primal_dual_hybrid_gradient(
            tiny_lp(), params.SerializeToString(), initial_solution=start)
        solve_log = solve_log_pb2.SolveLog.FromString(result.solve_log_str)
        self.assertEqual(solve_log.termination_reason,
                         solve_log_pb2.TERMINATION_REASON_OPTIMAL)
        self.assertEqual(solve_log.iteration_count, 0)
    

• 对偶问题理解 https://en.wikipedia.org/wiki/Dual_linear_program