# "Solution path" for quadratic program as regularizer changes

I am solving a quadratic program with regularization parameter $\alpha\geq0$ to get the solution to a problem of the form $$p(\alpha):= \arg\min_{p\in\mathbb{R}^n}\ [\alpha(v^\top p)+f(p)],$$ where $f(\cdot)$ is a strictly convex, continuous, and piecewise quadratic function of $p\in\mathbb{R}^n$.

I'm hoping to say something about the behavior of $p$ and/or the optimal objective value as $\alpha$ changes, in the style of solution path papers like this one. Sadly, the paper I linked does not seem to apply since the function next to $\alpha$ is not bounded below.

Does anyone have pointers for theorems that characterize the "solution path" ($p$ as a function of the regularizer $\alpha$) of problems in this form?

• If it would be a quadratic program (as the title suggests), the optimal its optimality condition would be linear and everything would be very easy. But the body of the question states "piecewise quadratic" which makes the problem harder. You may want to change your title (and the respective part in the body).
– Dirk
Oct 16 '16 at 0:53
• If it's piecewise quadratic, consider first the individual quadratic problems $f_i$ with constraints on $p$ being in the appropriate subdomain. Characterize the effect of $\alpha$ there and then look at the behavior of $\alpha$ as $p$ crosses boundaries, i.e., $p_i(\alpha)$. Nov 1 '16 at 15:50