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An \(O(n^ 3L)\) potential reduction algorithm for linear programming. (English) Zbl 0734.90057
Summary: We describe a primal-dual potential function for linear programming: \[ \phi (x,s)=\rho \ln (x^ Ts)-\sum^{n}_{j=1}\ln (x_ js_ j), \] where \(\rho\geq n\), x is the primal variable, and s is the dual-slack variable. As a result, we develop an interior point algorithm seeking reductions in the potential function with \(\rho =n+\sqrt{n}\). Neither tracing the central path nor using the projective transformation, the algorithm converges to the optimal solution set in O(\(\sqrt{n}L)\) iterations and uses \(O(n^ 3L)\) total arithmetic operations. We also suggest a practical approach to implementing the algorithm.

MSC:
90C05 Linear programming
90C60 Abstract computational complexity for mathematical programming problems
90-08 Computational methods for problems pertaining to operations research and mathematical programming
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