Algorithms for projecting a point onto the convex hull spanned by a set of vectors

Given a set of vectors $$V = \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \subset \mathbb{R}^d$$, I want to project a point $$\mathbf{x}_0 \in \mathbb{R}^d$$ onto the convex hull $$\text{conv}(V)$$ of the vectors in $$V$$.

I know this is a quadratic program, to find $$\mathbf{z}^*$$ that minimizes $$\frac{1}{2}\|\mathbf{x}_0 - \mathbf{z}\|^2$$ subject to $$\mathbf{z} \in \text{conv}(V)$$.

I also know that $$\text{conv}(V)$$ is a polytope and expressible as a set $$S = \{ \mathbf{x} : A\mathbf{x} \le \mathbf{b} \}$$.

However, I don't know how to derive the constraint matrix and constraint vector $$(A,\mathbf{b})$$ from the vectors in $$V$$.

Secondarily, I'm wondering if there are simple and fast algorithms to solve this problem. The number of vectors in $$V$$ will be less than $$250$$ and the dimensionality will be less than $$50$$.

And finally, I am hoping to express the solution $$\mathbf{z}^* = \text{Proj}(\mathbf{x}_0)$$ in something like barycentric coordinates with respect to the vectors in $$V$$. In other words, I'd like to express $$\mathbf{z}^*$$ as the vector $$(\alpha_1, \ldots, \alpha_n)$$ such that $$\mathbf{z}^* = \sum_i \alpha_i \mathbf{v}_i$$ with $$\alpha_i \ge 0$$ and $$\sum_i \alpha_i = 1$$. Given that the set $$V$$ won't be linearly independent (because $$n > d$$), I know that such barycentric coordinates are not well defined. I'm hoping to use something akin to a least norm solution $$\|\alpha\|^2$$ here.

You do not need to express the convex hull explicitly as a system of inequalities $$\mathbf{A x} \leq \mathbf{b}$$. The convex hull, by definition, is $$\operatorname{conv}(\mathbf{v}_1, \dots, \mathbf{v}_m) = \left \{ \sum_{i=1}^m w_i \mathbf{v}_i : \sum_{i=1}^m w_i=1, w_i \geq 0 \right\}$$
Therefore, the projection onto the convex hull of a vector $$\mathbf{y}$$ can be computed from the solution of $$\min_{\mathbf{w}} \quad \left \| \sum_{i=1}^m w_i \mathbf{v}_i - \mathbf{y} \right \|_2^2 \quad \text{s.t.} \quad \sum_{i=1}^m w_i=1, w_i \geq 0$$ The vector $$\mathbf{w}$$ contains the barycentric coordinates of the projection. I am not aware of a specialized algorithm of computing the optimal $$\mathbf{w}$$, but any QP solver can do so.
A simple solution method which you can implement yourself can be obtained from a fast projected gradient algorithm ,such as FISTA, and remembering that the constraints are exactly the unit simplex, and there is a simple $$O(m \log m)$$ algorithm for projecting a point onto the unit simplex. Moreover, you also have the well-known Mirror Descent algorithm for optimizing over the unit simplex, which can be quite fast in practice.
• $+1$ for an elegant answer .. I think the optimal way would be a brute-force of order $O(N^m)$ where $N$ would be the size of the grid, or more generally $O(\prod_{k=1}^m N_k)$ if each $w_k$ is searched on a different grid of size $N_k$. – Ahmad Bazzi Feb 12 at 12:10
• @AhmadBazzi, what grid are you referring to? A grid defined on $[0,1]^m$? I believe that this becomes impractical even for a small value of $m$, say, $m \geq 5$. – Alex Shtof Feb 12 at 12:24