Let's consider the problem of finding a step length that satisfies the strong Wolfe conditions for the function $f(y) = \frac{1}{2} \| \max(a + Qy, 0) \|_2^2$. The function is a squared norm of the projection of an affine function onto the non-negative orthant, which can cause challenges due to the non-smooth behavior introduced by the $\max(\cdot, 0)$ operation.

The Strong Wolfe Conditions

The strong Wolfe conditions are typically used to ensure sufficient decrease and curvature in line search methods. These conditions are:

1. Sufficient decrease (Armijo condition): $f(y + \alpha p) \leq f(y) + c_1 \alpha \nabla f(y)^T p$ for some $0 < c_1 < 1$.

2. Curvature condition: $|\nabla f(y + \alpha p)^T p| \leq c_2 |\nabla f(y)^T p|$ for $0 < c_1 < c_2 < 1$, where $p$ is a descent direction, and $\alpha$ is the step length.

Step Length Existence

Non-Smooth Behavior

The challenge in this problem arises from the $\max(a + Qy, 0)$ term. This introduces non-smoothness due to the presence of the ReLU-type operation $\max(\cdot, 0)$, which changes the gradient abruptly when any component of $a + Qy$ crosses zero. This can make it harder to ensure the curvature condition holds, especially around points where some components of $a + Qy$ switch between being positive and zero.

However, the function itself is piecewise quadratic, and in regions where $a + Qy > 0$ (i.e., no components are zero), the function is smooth and behaves like a typical quadratic function, for which a step length satisfying the Wolfe conditions can always be found. In these smooth regions, typical line search algorithms can find a step length.

At Non-Smooth Points

At points where some components of $a + Qy = 0$, the gradient may not be well defined in the traditional sense. In such cases, we need to consider generalized gradients. Nonetheless, the Wolfe conditions can still be applied using subgradients in non-smooth optimization, though this complicates the analysis. The key point is that even if the function is non-smooth, there may still exist a step length that satisfies the Wolfe conditions, but verifying this requires careful attention to the nature of the subgradients.

Conclusion

In general, a step length that satisfies the strong Wolfe conditions may not always exist for non-smooth functions like the one given. However, in regions where the function is smooth (i.e., when all components of $a + Qy$ are strictly positive), such a step length can typically be found. For non-smooth regions, finding a step length satisfying both Wolfe conditions may become more challenging and is not guaranteed.

Would you like more details or have further questions on any part of this?

Here are 5 related questions that build on this concept:

1. What are the weak Wolfe conditions, and how do they differ from the strong Wolfe conditions?
2. How does one compute subgradients for non-smooth functions?
3. Can the Wolfe conditions be adapted to non-smooth optimization problems?
4. How does the Armijo condition ensure sufficient decrease in non-smooth settings?
5. What are typical methods for handling non-smoothness in optimization algorithms?

Tip: In optimization of non-smooth functions, it's often useful to look into specialized algorithms like subgradient methods or bundle methods that handle non-differentiability explicitly.