  Major progress has been made in the previous decade to characterize the
asymptotic behavior of regularized M-estimators in high-dimensional regression
problems in the proportional asymptotic regime where the sample size $n$ and
the number of features $p$ are increasing simultaneously such that $n/p\to
\delta \in(0,\infty)$, using powerful tools such as Approximate Message Passing
or the Convex Gaussian Min-Max Theorem (CGMT). The asymptotic error and
behavior of the regularized M-estimator is then typically described by a system
of nonlinear equations with a few scalar unknowns, and the solution to this
system precisely characterize the asymptotic error. Application of the CGMT and
related machinery requires the existence of a solution to this low-dimensional
system of equations.
  This paper resolves the question of existence of solution to this
low-dimensional system for the case of linear models with independent additive
noise, when both the data-fitting loss function and regularization penalty are
separable and convex. Such existence result for solution to the nonlinear
system were previously known under strong convexity for specific estimators
such as the Lasso. The main idea behind this existence result is inspired by an
argument developed \cite{montanari2019generalization,celentano2020lasso} in
different contexts: By constructing an ad-hoc convex minimization problem in an
infinite dimensional Hilbert space, the existence of the Lagrange multiplier
for this optimization problem makes it possible to construct explicitly
solutions to the low-dimensional system of interest.
  The conditions under which we derive this existence result exactly correspond
to the side of the phase transition where perfect recovery $\hat x= x_0$ fails,
so that these conditions are optimal.
