  The composite $L_q~(0<q<1)$ minimization problem over a general polyhedron
has received various applications in machine learning, wireless communications,
image restoration, signal reconstruction, etc. This paper aims to provide a
theoretical study on this problem. Firstly, we show that for any fixed $0<q<1$,
finding the global minimizer of the problem, even its unconstrained
counterpart, is strongly NP-hard. Secondly, we derive Karush-Kuhn-Tucker (KKT)
optimality conditions for local minimizers of the problem. Thirdly, we propose
a smoothing sequential quadratic programming framework for solving this
problem. The framework requires a (approximate) solution of a convex quadratic
program at each iteration. Finally, we analyze the worst-case iteration
complexity of the framework for returning an $\epsilon$-KKT point; i.e., a
feasible point that satisfies a perturbed version of the derived KKT optimality
conditions. To the best of our knowledge, the proposed framework is the first
one with a worst-case iteration complexity guarantee for solving composite
$L_q$ minimization over a general polyhedron.
