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# Introduction

Ceres 可以鲁棒的解决以下形式的带有边界约束的非线性最小二乘问题

$$
\min\_{\mathbf{x}}\frac{1}{2}\sum\_{i}^{}\tau *{i}\left ( \parallel f\_i\left(x*{i1}, ..., x\_{ik}\right)\parallel ^2 \right )\ s.t.\ \ l\_j \le x\_j \le \mu\_{j}
$$

这种形式的问题广泛出现在科学研究和工程领域中，从统计学中的 [fitting curves](http://en.wikipedia.org/wiki/Nonlinear_regression) 问题到计算机视觉中的 [3D models from photographs](http://en.wikipedia.org/wiki/Bundle_adjustment) 问题。

在本章中，我们将学习如何使用 Ceres Solver 求解上述问题，本章中描述的所有示例以及更完整工作代码可以在 [examples](https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/) 目录中找到。

表达式 $$\tau *{i}\left(\left|f*{i}\left(x\_{i\_{1}}, \ldots, x\_{i\_{k}}\right)\right|^{2}\right)$$ 在 Ceres 中是一个 `ResidualBlock` ，其中 $$f\_{i}(\cdot)$$ 在 Ceres 中是一个 `CostFunction` ，它的计算又需要 $$x\_{ij}$$ 这些参数的参与， $$\left\[x\_{i\_{1}}, \ldots, x\_{i\_{k}}\right]$$ 统称一个参数块。在大多数优化问题中，小组标量一起出现。例如，平移向量的三个分量和定义相机姿态的四元数的四个分量。我们将这样一组小标量称为 `ParameterBlock`。当然 `ParameterBlock` 也可以只有一个分量。$$l\_{j}$$ and $$u\_{j}$$ 是参数块 $$x\_{j}$$ 的上下界。 $$\tau \_{i}$$ 是一个 `Lossfunction`。它是一个标量函数，用于减小非线性优化问题中的异常值对优化的影响。

在特殊情况下，当 $$\tau *{i}(x)=x$$，是一个恒等映射，并且 $$l*{j}=-\infty$$ and $$u\_{j}=\infty$$ 我们就能得到常见的非线性优化问题形式如下：

$$
\frac{1}{2} \sum\_{i}\left|f\_{i}\left(x\_{i\_{1}}, \ldots, x\_{i\_{k}}\right)\right|^{2}
$$


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