Given a convex compact set $K\subset\mathbb{R}^D$ and convex function $f:K\rightarrow \mathbb{R}$, the sublevel set $$ X_\alpha = \{ x \in K : f(x) \leq \alpha \} $$ defines a convex closed set (assuming it is non-empty). I would like to compute the projection operator $\Pi_{X_\alpha}$ to $X_\alpha$ for any point in $K$. Are there known methods and iteration complexity results for computing $\Pi_{X_\alpha}$ when f is convex or strongly convex?

I couldn't find references on this, even though such projections could appear frequently in constrained convex optimisation. I'm especially curious if strong convexity could lead to linear convergence when computing such projections.


1 Answer 1


The projection can be written as a convex optimization problem \begin{align*} \Pi_{X_{\alpha}}(\bar{x}) \in \arg\min_{x} \quad & \| x - \bar{x} \|\\ \text{s.t.} \quad & f(x) \leq \alpha \end{align*}

where $\bar{x} \in K$ is the point you want to project, and $\|\cdot\|$ is the norm you use for the projection. Note that depending on the nature of $f$ and your norm (especially if they are polyhedral), the projection may not be unique.

You can solve this problem using any algorithm for convex optimization, e.g., an interior-point method which has polynomial complexity. This is the most general case.

There are special cases where you can have closed-form solutions, or more efficient algorithms. For instance, if $f$ is a quadratic function, and you use Euclidean distances, then you can compute the projection by solving a linear system. If $f$ is a polyhedral function, the above problem is a quadratic optimization problem, which you can also solve using simplex.


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