We propose a simple projection and rescaling algorithm that finds {\em most interior} solutions to the pair of feasibility problems \[ \text{find} x\in L\cap \R^n_{+} \text{ and } \text{find} \; \hat x\in L^\perp\cap\R^n_{+}, \] where $L$ is a linear subspace of $\R^n$ and $L^\perp$ is its orthogonal complement. The algorithm complements a basic procedure that … Read more
This paper documents a computational implementation of a {\em projection and rescaling algorithm} for finding most interior solutions to the pair of feasibility problems find $x\in L\cap\mathbb{R}^n_{+} $ and find $x\in L^\perp\cap\mathbb{R}^n_{+},$ where $L$ denotes a linear subspace in $\mathbb{R}^n$ and $L^\perp$ denotes its orthogonal complement. The projection and rescaling algorithm is a recently developed … Read more
We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix $A\in \R^{m\times n}$, the {\em kernel problem} requires a positive vector in the kernel of $A$, and the {\em image problem} requires a positive vector in the image of $A^\T$. Both algorithms iterate between simple first order steps and rescaling steps. … Read more
We propose a simple {\em projection and algorithm} to solve the feasibility problem \[ \text{ find } x \in L \cap \Omega, \] 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 … Read more
Recently Chubanov proposed a method which solves homogeneous linear equality systems with positive variables in polynomial time. Chubanov’s method can be considered as a column-wise rescaling procedure. We adapt Chubanov’s method to the von Neumann problem, and so we design a polynomial time column-wise rescaling von Neumann algorithm. This algorithm is the first variant of … Read more
The perceptron algorithm is a simple iterative procedure for finding a point in a convex cone $F$. At each iteration, the algorithm only involves a query of a separation oracle for $F$ and a simple update on a trial solution. The perceptron algorithm is guaranteed to find a feasible point in $F$ after $\Oh(1/\tau_F^2)$ iterations, … Read more