# General Unconstrained Minimization

Ceres 求解器除了能求解非线性最小二乘法问题外，还能使用目标函数值和梯度求解一般无约束问题。在本章中，我们将了解如何做到这一点。

## Rosenbrock’s Function

考虑尝试最小化 [Rosenbrock’s function](http://en.wikipedia.org/wiki/Rosenbrock_function)。最简单方法是定义一个模板函数来评估误差函数，然后使用 Ceres Solver 的自动微分来计算其导数。我们首先定义一个模板化的函数，然后使用 `AutoDiffFirstOrderFunction` 构造一个 `FirstOrderFunction` 接口的实例。这个对象负责计算目标函数值和梯度（如果需要）。这与 Ceres 中定义非线性最小二乘法问题时的 [`CostFunction`](http://ceres-solver.org/nnls_modeling.html#_CPPv4N5ceres12CostFunctionE)类似。

```cpp
// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
struct Rosenbrock {
  template <typename T>
  bool operator()(const T* parameters, T* cost) const {
    const T x = parameters[0];
    const T y = parameters[1];
    cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
    return true;
  }

  static ceres::FirstOrderFunction* Create() {
    constexpr int kNumParameters = 2;
    return new ceres::AutoDiffFirstOrderFunction<Rosenbrock, kNumParameters>();
  }
};
```

然后，只需构建一个 [`GradientProblem`](http://ceres-solver.org/gradient_solver.html#_CPPv4N5ceres15GradientProblemE) 对象并调用 [`Solve()`](http://ceres-solver.org/gradient_solver.html#_CPPv4N5ceres5SolveERKN21GradientProblemSolver7OptionsERK15GradientProblemPdPN21GradientProblemSolver7SummaryE) 即可将其最小化。

```cpp
double parameters[2] = {-1.2, 1.0};

ceres::GradientProblem problem(Rosenbrock::Create());

ceres::GradientProblemSolver::Options options;
options.minimizer_progress_to_stdout = true;
ceres::GradientProblemSolver::Summary summary;
ceres::Solve(options, problem, parameters, &summary);

std::cout << summary.FullReport() << "\n";
```

执行该代码，使用 limited memory [BFGS](http://en.wikipedia.org/wiki/Broyden-Fletcher-Goldfarb-Shanno_algorithm)算法解决问题。

```cpp
   0: f: 2.420000e+01 d: 0.00e+00 g: 2.16e+02 h: 0.00e+00 s: 0.00e+00 e:  0 it: 1.19e-05 tt: 1.19e-05
   1: f: 4.280493e+00 d: 1.99e+01 g: 1.52e+01 h: 2.01e-01 s: 8.62e-04 e:  2 it: 7.30e-05 tt: 1.72e-04
   2: f: 3.571154e+00 d: 7.09e-01 g: 1.35e+01 h: 3.78e-01 s: 1.34e-01 e:  3 it: 1.60e-05 tt: 1.93e-04
   3: f: 3.440869e+00 d: 1.30e-01 g: 1.73e+01 h: 1.36e-01 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 1.97e-04
   4: f: 3.213597e+00 d: 2.27e-01 g: 1.55e+01 h: 1.06e-01 s: 4.59e-01 e:  1 it: 1.19e-06 tt: 2.00e-04
   5: f: 2.839723e+00 d: 3.74e-01 g: 1.05e+01 h: 1.34e-01 s: 5.24e-01 e:  1 it: 9.54e-07 tt: 2.03e-04
   6: f: 2.448490e+00 d: 3.91e-01 g: 1.29e+01 h: 3.04e-01 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 2.05e-04
   7: f: 1.943019e+00 d: 5.05e-01 g: 4.00e+00 h: 8.81e-02 s: 7.43e-01 e:  1 it: 9.54e-07 tt: 2.08e-04
   8: f: 1.731469e+00 d: 2.12e-01 g: 7.36e+00 h: 1.71e-01 s: 4.60e-01 e:  2 it: 2.15e-06 tt: 2.11e-04
   9: f: 1.503267e+00 d: 2.28e-01 g: 6.47e+00 h: 8.66e-02 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 2.14e-04
  10: f: 1.228331e+00 d: 2.75e-01 g: 2.00e+00 h: 7.70e-02 s: 7.90e-01 e:  1 it: 0.00e+00 tt: 2.16e-04
  11: f: 1.016523e+00 d: 2.12e-01 g: 5.15e+00 h: 1.39e-01 s: 3.76e-01 e:  2 it: 1.91e-06 tt: 2.25e-04
  12: f: 9.145773e-01 d: 1.02e-01 g: 6.74e+00 h: 7.98e-02 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 2.28e-04
  13: f: 7.508302e-01 d: 1.64e-01 g: 3.88e+00 h: 5.76e-02 s: 4.93e-01 e:  1 it: 9.54e-07 tt: 2.30e-04
  14: f: 5.832378e-01 d: 1.68e-01 g: 5.56e+00 h: 1.42e-01 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 2.33e-04
  15: f: 3.969581e-01 d: 1.86e-01 g: 1.64e+00 h: 1.17e-01 s: 1.00e+00 e:  1 it: 1.19e-06 tt: 2.36e-04
  16: f: 3.171557e-01 d: 7.98e-02 g: 3.84e+00 h: 1.18e-01 s: 3.97e-01 e:  2 it: 1.91e-06 tt: 2.39e-04
  17: f: 2.641257e-01 d: 5.30e-02 g: 3.27e+00 h: 6.14e-02 s: 1.00e+00 e:  1 it: 1.19e-06 tt: 2.42e-04
  18: f: 1.909730e-01 d: 7.32e-02 g: 5.29e-01 h: 8.55e-02 s: 6.82e-01 e:  1 it: 9.54e-07 tt: 2.45e-04
  19: f: 1.472012e-01 d: 4.38e-02 g: 3.11e+00 h: 1.20e-01 s: 3.47e-01 e:  2 it: 1.91e-06 tt: 2.49e-04
  20: f: 1.093558e-01 d: 3.78e-02 g: 2.97e+00 h: 8.43e-02 s: 1.00e+00 e:  1 it: 2.15e-06 tt: 2.52e-04
  21: f: 6.710346e-02 d: 4.23e-02 g: 1.42e+00 h: 9.64e-02 s: 8.85e-01 e:  1 it: 8.82e-06 tt: 2.81e-04
  22: f: 3.993377e-02 d: 2.72e-02 g: 2.30e+00 h: 1.29e-01 s: 4.63e-01 e:  2 it: 7.87e-06 tt: 2.96e-04
  23: f: 2.911794e-02 d: 1.08e-02 g: 2.55e+00 h: 6.55e-02 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.00e-04
  24: f: 1.457683e-02 d: 1.45e-02 g: 2.77e-01 h: 6.37e-02 s: 6.14e-01 e:  1 it: 1.19e-06 tt: 3.03e-04
  25: f: 8.577515e-03 d: 6.00e-03 g: 2.86e+00 h: 1.40e-01 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.06e-04
  26: f: 3.486574e-03 d: 5.09e-03 g: 1.76e-01 h: 1.23e-02 s: 1.00e+00 e:  1 it: 1.19e-06 tt: 3.09e-04
  27: f: 1.257570e-03 d: 2.23e-03 g: 1.39e-01 h: 5.08e-02 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.12e-04
  28: f: 2.783568e-04 d: 9.79e-04 g: 6.20e-01 h: 6.47e-02 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.15e-04
  29: f: 2.533399e-05 d: 2.53e-04 g: 1.68e-02 h: 1.98e-03 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.17e-04
  30: f: 7.591572e-07 d: 2.46e-05 g: 5.40e-03 h: 9.27e-03 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.20e-04
  31: f: 1.902460e-09 d: 7.57e-07 g: 1.62e-03 h: 1.89e-03 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.23e-04
  32: f: 1.003030e-12 d: 1.90e-09 g: 3.50e-05 h: 3.52e-05 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.26e-04
  33: f: 4.835994e-17 d: 1.00e-12 g: 1.05e-07 h: 1.13e-06 s: 1.00e+00 e:  1 it: 1.19e-06 tt: 3.34e-04
  34: f: 1.885250e-22 d: 4.84e-17 g: 2.69e-10 h: 1.45e-08 s: 1.00e+00 e:  1 it: 9.54e-07 tt: 3.37e-04

Solver Summary (v 2.2.0-eigen-(3.4.0)-lapack-suitesparse-(7.1.0)-metis-(5.1.0)-acceleratesparse-eigensparse)

Parameters                                  2
Line search direction              LBFGS (20)
Line search type                  CUBIC WOLFE


Cost:
Initial                          2.420000e+01
Final                            1.955192e-27
Change                           2.420000e+01

Minimizer iterations                       36

Time (in seconds):

  Cost evaluation                    0.000000 (0)
  Gradient & cost evaluation         0.000000 (44)
  Polynomial minimization            0.000061
Total                                0.000438

Termination:                      CONVERGENCE (Parameter tolerance reached. Relative step_norm: 1.890726e-11 <= 1.000000e-08.)

Initial x: -1.2 y: 1
Final   x: 1 y: 1
```

如果由于某种原因无法使用自动微分（例如因为需要调用外部库），那么可以使用数值微分。在这种情况下，函数的定义如下

```cpp
// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
struct Rosenbrock {
  bool operator()(const double* parameters, double* cost) const {
    const double x = parameters[0];
    const double y = parameters[1];
    cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
    return true;
  }

  static ceres::FirstOrderFunction* Create() {
    constexpr int kNumParameters = 2;
    return new ceres::NumericDiffFirstOrderFunction<Rosenbrock,
                                                    ceres::CENTRAL,
                                                    kNumParameters>();
  }
};
```

最后，如果你更愿意手工计算导数（比如因为参数向量太大而无法自动微分）。那么就应该定义 FirstOrderFunction 的实例，它与 [`CostFunction`](http://ceres-solver.org/nnls_modeling.html#_CPPv4N5ceres12CostFunctionE) 类似，适用于非线性最小二乘法问题

```cpp
// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
class Rosenbrock final  : public ceres::FirstOrderFunction {
  public:
    bool Evaluate(const double* parameters,
                           double* cost,
                           double* gradient) const override {
       const double x = parameters[0];
       const double y = parameters[1];

       cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
       if (gradient) {
         gradient[0] = -2.0 * (1.0 - x) - 200.0 * (y - x * x) * 2.0 * x;
         gradient[1] = 200.0 * (y - x * x);
       }
      return true;
   }

   int NumParameters() const override { return 2; }
};
```

## Footnotes

* [examples/rosenbrock.cc](https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock.cc)
* [examples/rosenbrock\_numeric\_diff.cc](https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock_numeric_diff.cc)
* [examples/rosenbrock\_analytic\_diff.cc](https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock_analytic_diff.cc)


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://ceres-solver-tutorial-cn.gitbook.io/ceres/optimization-tutorial/general-unconstrained-minimization/general-unconstrained-minimization.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.