Solving Conic Systems via Projection and Rescaling

We propose a simple projection and rescaling algorithm to solve the feasibility problem $$\text{ find }x\in L\cap \mathrm{\Omega },$$ where $L$ and $\mathrm{\Omega }$ 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 {\em basic procedure} and a {\em rescaling} step. When $L \cap \Omega \ne \emptyset$,theprojectionandrescalingalgorithmfindsapoint$x \in L \cap \Omega$inatmost$O(\log(1/\delta(L \cap \Omega)))$iterations,where$\delta(L \cap \Omega) \in (0,1]$isameasureofthemostinteriorpointin$L \cap \Omega$\mathrm{.}Theidealvalue$\delta(L\cap \Omega) = 1$isattainedwhen$L \cap \Omega$containsthecenterofthesymmetriccone$\Omega$. We describe several possible implementations for the basic procedure including a perceptron scheme and a smooth perceptron scheme. The perceptron scheme requires $O(r^4)$ perceptron updates and the smooth perceptron scheme requires $O(r^2)$ smooth perceptron updates, where $r$ stands for the Jordan algebra rank of $V$.