A new polynomial-time algorithm for linear programming
Combinatorica
A polynomial-time algorithm, based on Newton's method, for linear programming
Mathematical Programming: Series A and B
Cutting planes and column generation techniques with the projective algorithm
Journal of Optimization Theory and Applications
A "build-down" scheme for linear programming
Mathematical Programming: Series A and B
Interior path following primal-dual algorithms. Part I: Linear programming
Mathematical Programming: Series A and B
An OL(n3) potential reduction algorithm for linear programming
Mathematical Programming: Series A and B
A polynomial method of approximate centers for linear programming
Mathematical Programming: Series A and B
Karmarkar's algorithm and combinatorial optimization problems
Karmarkar's algorithm and combinatorial optimization problems
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We propose a strategy for building up the linear program while using a logarithmic barrier method. The method starts with a (small) subset of the dual constraints, and follows the corresponding central path until the iterate is close to (or violates) one of the constraints, which is in turn added to the current system. This process is repeated until an optimal solution is reached. If a constraint is added to the current system, the central path will, of course, change. We analyze the effect on the barrier function value if a constraint is added. More importantly, we give an upper bound for the number of iterations needed to return to the new path. We prove that in the worst case the complexity is the same as that of the standard logarithmic barrier method. In practice this build-up scheme is likely to save a great deal of computation.