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Motivated by a recent work of Setiono, a path-following algorithm for linear pro- gramming using both logarithmic and quadratic penalty functions is proposed. In the algorithm, a logarithmic and a quadratic penalty is placed on, respectively, the nonnegativity constraints and an arbitrary subset of the equality constraints; Newton's method is applied to solve the penalized problem, and after each Newton step the penalty parameters are decreased. This algorithm main- tains neither primal nor dual feasibility and does not require a Phase I. It is shown that if the initial iterate is chosen appropriately and the penalty parameters are decreased to zero in a particular way, then the algorithm is linearly convergent. Numerical results are also presented suggesting that the algorithm may be competitive with interior point algorithms in practice, requiring typically between 30-45 iterations to accurately solve each Netlib problem tested.

A Path-Following Algorithm for Linear Programming Using Quadratic and Logarithmic Penalty Functions