Comparison of Wolfe conditions to other "weaker" conditions or facts about optimization techniques for conceptual understanding So in the book, it states that the first wolfe condition is the following,
$\begin{equation}p^Tg_k\leq-\eta_0|||p|||g_k||\end{equation}$, where $g_k=\nabla F(x_k)$.
Here it states that this is a stronger form of the descent property, $p^T\nabla F(x_k)<0$ and that this ensures that angle $\theta\in(-\pi/2,\pi/2)$ for $\eta_0\approx0.01$ for most cases. This makes sense, but I'm having trouble grasping the other two conditions, that is, the second Wolfe condition
$\begin{equation}F(x_k+sp)-F(x_k)\leq\eta_1sp^Tg_k\end{equation}$ for $\eta_1\in(0,0.5)$
Update: I believe that the second Wolfe condition is a stronger form of the weak line search definition, that is, $F(x_k+sp_k)-F(x_k)<0$
and the third Wolfe condition
$\begin{equation}|F(x_k+_sp)-F(x_k)-sp^Tg_k|\geq\eta_2|sp^Tg_k|\end{equation}$ for $\eta_2\in(0,0.5)$
Update: This comes from the Taylor series approximation, so it says that it descends below the region covered by the line at $x_k$, $F(x_k)+sp^T\nabla F(x_k)$ which are the first two terms in the Taylor series
Are these conditions comparable to any other conditions placed upon, or properties inherent within, the problem of the minimization of a quadratic function? whether it be constrained or unconstrained?
 A: The Wolf conditions can be said to be of two types:

*

*A sufficient decrease condition:
\begin{equation}F(x_k+s_k p_k)-F(x_k)\leq\eta_1 s_k p_k^Tg_k\end{equation} for $\eta_1\in(0,0.5).$

*“Big steps” condition:
$$s_k\geq \eta_2 \dfrac{ |p_k^{T} g_k |}{\|p_k\|^2}.$$
As it is not hard to see, these two conditions always ensures the Zoutendijk condition:
\begin{equation}
\text{the sum }\sum_{k=0} \cos^{2} \theta_k \| g_k \|^2 \text{ converges,}
\end{equation} where $cos \theta_k = \dfrac{ p_k^{T} g_k }{\|p_k\| \| g_k\| }.$
The Zoutendijk condtion says that, if the step directions are not that bad, i.e., the $\cos^{2} \theta_k \geq \delta > 0$, for big enough $k$, the algorithm is actually finding stationary points of the objective function asymptotically, i.e.,  $$\lim \| g_k \| = 0.$$  This "angle condition" is locally guaranteed by Quasi-Newton methods. In your case, the bad directions are avoided by your first Wolfe conditons:
$\begin{equation}p^Tg_k \leq -\eta_0\| p_k \| \|g_k\|\end{equation}$. It prevents the methods from "walking" in circles, since, if $\lim \cos^{2} \theta_k = 0$, the method is nearly walking over some level-set of the objective function $F.$
The “big step” condition is usually ensured by the Wolfe conditions and strong Wolfe conditions, and its main purpose is to ensure that the algorithm is not preventing itself from finding better minimizers due to some too small choices of step size $s_k$. This condition can be linked to an algorithm that is nearly finding stationary points. Incredibly, in your case, you satisfy this by asking your third Wolf condition. To see it, suppose the sequence $\{x^{k}\}_{k\in\mathbb{N}}$ is contained in a bounded set. Hence, for two times continuously differentiable map, there exist an $L>0$ so that
\begin{align}
L s_k^2 \|p_k \|^2 & \geq  |F(x_k+ s_k p_k)-F(x_k)-s_k p_k ^Tg_k| \\ 
& \geq \eta_2 |s_k p^Tg_k|.
\end{align} Hence, \begin{equation}
s_k \geq \dfrac{\eta_2}{L} \dfrac{| p^T g_k|}{ \|p_k\|^2},
\end{equation} guaranteeing the “big steps” condition.
