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## 线搜索

In numerical analysis, Brent’s method is a root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable methods. The algorithm tries to use the potentially fast-converging secant method or inverse quadratic interpolation if possible, but it falls back to the more robust bisection method if necessary. Brent’s method is due to Richard Brent[1] and builds on an earlier algorithm by Theodorus Dekker.[2] Consequently, the method is also known as the Brent–Dekker method.

### 精确线搜索

$$\nabla f(x_k+\alpha_kd_k)d_k=0$$

## 有约束规划

$$\forall x,y\in\Omega, \alpha\in(0,1)$$

$$f(\alpha x+(1-\alpha)y)\leq \alpha f(x)+(1-\alpha)f(y)$$