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.