Solving Conic Systems via Projection and Rescaling

Abstract

We propose a simple projection and rescaling algorithm to solve the feasibility problem

$$\begin{aligned} \text { find } x \in L \cap \Omega , \end{aligned}$$

where L and $\Omega$ are respectively a linear subspace and the interior of a symmetric cone in a finite-dimensional vector space V. This projection and rescaling algorithm is inspired by previous work on rescaled versions of the perceptron algorithm and by Chubanov’s projection-based method for linear feasibility problems. As in these predecessors, each main iteration of our algorithm contains two steps: a basic procedure and a rescaling step. When $L \cap \Omega \ne \emptyset$, the projection and rescaling algorithm finds a point $x \in L \cap \Omega$ in at most $\mathcal {O}(\log (1/\delta (L \cap \Omega )))$ iterations, where $\delta (L \cap \Omega ) \in (0,1]$ is a measure of the most interior point in $L \cap \Omega$. The ideal value $\delta (L\cap \Omega ) = 1$ is attained when $L \cap \Omega$ contains the center of the symmetric cone $\Omega$. We describe several possible implementations for the basic procedure including a perceptron scheme and a smooth perceptron scheme. The perceptron scheme requires $\mathcal {O}(r^4)$ perceptron updates and the smooth perceptron scheme requires $\mathcal {O}(r^2)$ smooth perceptron updates, where r stands for the Jordan algebra rank of V.