Solving Conic Systems via Projection and Rescaling

We propose a simple projection and rescaling algorithm to solve the feasibility problem $$\begin{aligned} \text { find } x \in L \cap \Omega , \end{aligned}$$findx∈L∩Ω,where L and $$\Omega $$Ω 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 $$L \cap \Omega \ne \emptyset $$L∩Ω≠∅, the projection and rescaling algorithm finds a point $$x \in L \cap \Omega $$x∈L∩Ω in at most $$\mathcal {O}(\log (1/\delta (L \cap \Omega )))$$O(log(1/δ(L∩Ω))) iterations, where $$\delta (L \cap \Omega ) \in (0,1]$$δ(L∩Ω)∈(0,1] is a measure of the most interior point in $$L \cap \Omega $$L∩Ω. The ideal value $$\delta (L\cap \Omega ) = 1$$δ(L∩Ω)=1 is attained when $$L \cap \Omega $$L∩Ω contains the center of the symmetric cone $$\Omega $$Ω. We describe several possible implementations for the basic procedure including a perceptron scheme and a smooth perceptron scheme. The perceptron scheme requires $$\mathcal {O}(r^4)$$O(r4) perceptron updates and the smooth perceptron scheme requires $$\mathcal {O}(r^2)$$O(r2) smooth perceptron updates, where r stands for the Jordan algebra rank of V.