Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix $A\in \mathbb{R}^{m\times n}$, the kernel problem requires a positive vector in the kernel of $A$, and the image problem requires a positive vector in the image of $A^\top$. Both algorithms iterate beten simple first order steps and rescaling steps. These rescalings steps improve natural geometric potentials in the domain and image spaces, respectively. If Goffin's condition measure $\hat \rho_A$ is negative, then the kernel problem is feasible and the worst-case complexity of the kernel algorithm is $O\left((m^3n+mn^2)\log{|\hat \rho_A|^{-1}}\right)$; if $\hat\rho_A>0$, then the image problem is feasible and the image algorithm runs in time $O\left(m^2n^2\log{\hat \rho_A^{-1}}\right)$. #R##N# also address the degenerate case $\hat\rho_A=0$:  extend our algorithms for finding maximum support nonnegative vectors in the kernel of $A$ and in the image of $A^\top$.  obtain the same running time bounds, with $\hat\rho_A$ replaced by appropriate condition numbers. In case the input matrix $A$ has integer entries and total encoding length $L$, all algorithms are polynomial. Both full support and maximum support kernel algorithms run in time $O\left((m^3n+mn^2)L\right)$, whereas both image algorithms run in time $O\left(m^2n^2L\right)$. The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for Linear Programming.
['Daniel Dadush', 'László A. Végh', 'Giacomo Zambelli']
I am writing a research paper on a idea Rescaling Algorithms for Linear Programming - Part I: Conic feasibility
[{"from": "human", "value": "I am writing a research paper on a idea Rescaling Algorithms for Linear Programming - Part I: Conic feasibility"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix $A\\in \\mathbb{R}^{m\\times n}$, the kernel problem requires a positive vector in the kernel of $A$, and the image problem requires a positive vector in the image of $A^\\top$. Both algorithms iterate beten simple first order steps and rescaling steps. These rescalings steps improve natural geometric potentials in the domain and image spaces, respectively. If Goffin's condition measure $\\hat \\rho_A$ is negative, then the kernel problem is feasible and the worst-case complexity of the kernel algorithm is $O\\left((m^3n+mn^2)\\log{|\\hat \\rho_A|^{-1}}\\right)$; if $\\hat\\rho_A>0$, then the image problem is feasible and the image algorithm runs in time $O\\left(m^2n^2\\log{\\hat \\rho_A^{-1}}\\right)$. #R##N# also address the degenerate case $\\hat\\rho_A=0$:  extend our algorithms for finding maximum support nonnegative vectors in the kernel of $A$ and in the image of $A^\\top$.  obtain the same running time bounds, with $\\hat\\rho_A$ replaced by appropriate condition numbers. In case the input matrix $A$ has integer entries and total encoding length $L$, all algorithms are polynomial. Both full support and maximum support kernel algorithms run in time $O\\left((m^3n+mn^2)L\\right)$, whereas both image algorithms run in time $O\\left(m^2n^2L\\right)$. The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for Linear Programming."}]