Lipschitz continuity conjecture for the convex QCQP reformulation

Let $\mathcal{X}$ be the parameter domain, let $K(x)\subseteq \mathcal{U}\subseteq \mathbb{R}^m$ be the feasible set defined by the paper's assumptions, and let $a_i(x)$, $b_i(x)$, and $\pi_{\rm f}(x)$ be as in the quadratic constraints. For fixed $k>0$, consider

\operatorname*{argmin}_{u\in\mathbb{R}^m}\ \left\\|u-\pi_{\rm des}(x)\right\\|^2

subject to

\left\\|u-c_i(x)\right\\|^2\leq d_i(x),

where

$$c_i(x)=\pi_{\rm f}(x)-ka_i(x),\qquad d_i(x)=k^2+2k\left(b_i(x)-a_i(x)^\top\pi_{\rm f}(x)\right).$$

Lipschitz continuity conjecture for the convex QCQP reformulation. If the paper's assumptions $Cref{assum:Ab}$ and $ref{assum:main}$ hold and $k>0$, then for every $x\in\mathcal{X}$ the problem has a unique minimizer, denoted by $\pi_{\rm qcqp}(x)$; moreover,

$$\pi_{\rm qcqp}(x)\in K(x)\quad\text{for all }x\in\mathcal{X},$$

and $\pi_{\rm qcqp}$ is Lipschitz on $\mathcal{X}$. The conjecture proposes a Lipschitz-continuous, constraint-satisfying solution map for this QCQP, in contrast with the generally only Hölder-continuous solution map of the preceding quadratic program. Its validity and the precise role of the paper's assumptions remain to be established.