  The $p$-regularized subproblem (p-RS) is a regularisation technique in
computing a Newton-like step for unconstrained optimization, which globally
minimizes a local quadratic approximation of the objective function while
incorporating with a weighted regularisation term $\frac{\sigma}{p} \|x\|^p$.
The global solution of the $p$-regularized subproblem for $p=3$, also known as
the cubic regularization, has been characterized in literature. In this paper,
we resolve both the global and the local non-global minimizers of (p-RS) for
$p>2$ with necessary and sufficient optimality conditions. Moreover, we prove a
parallel result of Mart\'{\i}nez \cite{Mar} that the (p-RS) for $p>2$,
analogous to the trust region subproblem, can have at most one local non-global
minimizer. When the (p-RS) is subject to a fixed number $m$ additional linear
inequality constraints, we show that the uniqueness of the local solution of
the (p-RS) (if exists at all), especially for $p=4$, can be applied to solve
such an extension in polynomial time.

Optimization and Control