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We present a new polynomial-time algorithm for linear programming. The running-time of this algorithm is <italic>O</italic>(<italic>n</italic><supscrpt>3-5</supscrpt><italic>L</italic><supscrpt>2</supscrpt>), as compared to <italic>O</italic>(<italic>n</italic><supscrpt>6</supscrpt><italic>L</italic><supscrpt>2</supscrpt>) for the ellipsoid algorithm. We prove that given a polytope <italic>P</italic> and a strictly interior point <italic>a</italic> ε <italic>P</italic>, there is a projective transformation of the space that maps <italic>P</italic>, <italic>a</italic> to <italic>P'</italic>, <italic>a'</italic> having the following property. The ratio of the radius of the smallest sphere with center <italic>a'</italic>, containing <italic>P'</italic> to the radius of the largest sphere with center <italic>a'</italic> contained in <italic>P'</italic> is <italic>O</italic> (<italic>n</italic>). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial-time.

A new polynomial-time algorithm for linear programming