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The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of $m$ linear inequalities in $n$ variables $(P): A^{top}x le u$ when its set of solutions has positive volume. However, when $(P)$ is infeasible, the ellipsoid algorithm has no mechanism for proving that $(P)$ is infeasible. This is in contrast to the other two fundamental algorithms for tackling $(P)$, namely the simplex method and interior-point methods, each of which can be easily implemented in a way that either produces a solution of $(P)$ or proves that $(P)$ is infeasible by producing a solution to the alternative system $mathrm{({it Alt})}: Alambda= 0$, $u^{top}lambda < 0$, $lambda ge 0$. This paper develops an Oblivious Ellipsoid Algorithm (OEA) that either produces a solution of $(P)$ or produces a solution of $mathrm{({it Alt})}$. Depending on the dimensions and on other natural condition measures, the computational complexity of the basic OEA may be worse than, the same as, or better than that of the standard ellipsoid algorithm. We also present two modifi
We propose an iterative improvement method for the Harrow-Hassidim-Lloyd (HHL) algorithm to solve a linear system of equations. This is a quantum-classical hybrid algorithm. The accuracy is essential to solve the linear system of equations. However,
In this paper, we develop a new algorithm combining the idea of ``boosting with the first-order algorithm to approximately solve a class of (Integer) Linear programs(LPs) arisen in general resource allocation problems. Not only can this algorithm sol
Based on the geometric {it Triangle Algorithm} for testing membership of a point in a convex set, we present a novel iterative algorithm for testing the solvability of a real linear system $Ax=b$, where $A$ is an $m times n$ matrix of arbitrary rank.
Column generation is often used to solve multi-commodity flow problems. A program for column generation always includes a module that solves a linear equation. In this paper, we address three major issues in solving linear problem during column gener
With the development of robotics, there are growing needs for real time motion planning. However, due to obstacles in the environment, the planning problem is highly non-convex, which makes it difficult to achieve real time computation using existing