  In this paper, we study the problem of finding the Euclidean distance to a
convex cone generated by a set of discrete points in $\mathbb{R}^n_+$. In
particular, we are interested in problems where the discrete points are the set
of feasible solutions of some binary linear programming constraints. This
problem has applications in manufacturing, machine learning, clustering,
pattern recognition, and statistics. Our problem is a high-dimensional
constrained optimization problem. We propose a Frank-Wolfe based algorithm to
solve this non-convex optimization problem with a convex-noncompact feasible
set. Our approach consists of two major steps: presenting an equivalent convex
optimization problem with a non-compact domain, and finding a compact-convex
set that includes the iterates of the algorithm. We discuss the convergence
property of the proposed approach. Our numerical work shows the effectiveness
of this approach.
