N. Karmarkar presents a new polynomial-time algorithm for linear programming, with a running time of \(O(n^3 L^2)\), improving over the ellipsoid algorithm's \(O(n^4 L^2)\). The algorithm involves repeated application of projective transformations followed by optimization over inscribed spheres to converge to the optimal solution in polynomial time. Karmarkar proves that for any polytope \(P\) and a strictly interior point \(a \in P\), there exists a projective transformation that maps \(P\) to a point \(a'\) such that the ratio of the radius of the smallest sphere containing \(a\) to the largest sphere contained in \(P\) is \(O(n)\). The algorithm's complexity is analyzed in detail, and its performance is compared to the simplex algorithm and other methods. The paper also discusses the global analysis of optimization algorithms, the significance of the factor \(L\) in the running time, and practical considerations such as numerical stability and parallel computation. The algorithm's effectiveness is demonstrated through theoretical bounds and experimental results, showing its potential for solving large-scale linear programming problems efficiently.N. Karmarkar presents a new polynomial-time algorithm for linear programming, with a running time of \(O(n^3 L^2)\), improving over the ellipsoid algorithm's \(O(n^4 L^2)\). The algorithm involves repeated application of projective transformations followed by optimization over inscribed spheres to converge to the optimal solution in polynomial time. Karmarkar proves that for any polytope \(P\) and a strictly interior point \(a \in P\), there exists a projective transformation that maps \(P\) to a point \(a'\) such that the ratio of the radius of the smallest sphere containing \(a\) to the largest sphere contained in \(P\) is \(O(n)\). The algorithm's complexity is analyzed in detail, and its performance is compared to the simplex algorithm and other methods. The paper also discusses the global analysis of optimization algorithms, the significance of the factor \(L\) in the running time, and practical considerations such as numerical stability and parallel computation. The algorithm's effectiveness is demonstrated through theoretical bounds and experimental results, showing its potential for solving large-scale linear programming problems efficiently.