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A PrimalDual Potential Reduction Method for Problems Involving Matrix Inequalities
 in Protocol Testing and Its Complexity", Information Processing Letters Vol.40
, 1995
"... We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worstcase analysis shows that the number of iterations ..."
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Cited by 87 (21 self)
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We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worstcase analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interiorpoint methods the overall computational effort is therefore dominated by the leastsquares system that must be solved in each iteration. A type of conjugategradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugategradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration.
Solving RealWorld Linear Ordering Problems . . .
, 1995
"... Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear prog ..."
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Cited by 21 (8 self)
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Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear programming relaxations. A point which is a good warm start for a simplexbased cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some realworld problems; the algorithm appears to be competitive with a simplexbased cutting plane algorithm.
INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION
, 1995
"... Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivale ..."
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Cited by 14 (9 self)
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Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.
Interior point algorithms for integer programming
 Advances in Linear and Integer Programming, chapter 6
, 1996
"... Research on using interior point algorithms to solve integer programming problems is surveyed. This paper concentrates on branch and bound and cutting plane methods; a potential function method is also briefly mentioned. The principal difficulty with using an interior point algorithm in a branch a ..."
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Cited by 7 (4 self)
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Research on using interior point algorithms to solve integer programming problems is surveyed. This paper concentrates on branch and bound and cutting plane methods; a potential function method is also briefly mentioned. The principal difficulty with using an interior point algorithm in a branch and cut method to solve integer programming problems is in warm starting the algorithm efficiently. Methods for overcoming this difficulty are described and other features of the algorithms are given. This paper focuses on the techniques necessary to obtain an efficient computational implementation; there is a short discussion of theoretical issues.
The Synchronization Problem
 in Protocol Testing and its Complexityā€¯, Inf. Proc. Letters, Vol.40
, 1991
"... primaldual potential reduction method for ..."
A potential reduction algorithm with userspecified Phase I  Phase II balance, for solving a linear program from an infeasible warm start
, 1991
"... This paper develops a potential reduction algorithm for solving a linearprogramming problem directly from a "warm start " initial point that is neither feasible nor optimal. The algorithm is of an "interior point " variety that seeks to reduce a single potential function which s ..."
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Cited by 1 (1 self)
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This paper develops a potential reduction algorithm for solving a linearprogramming problem directly from a "warm start " initial point that is neither feasible nor optimal. The algorithm is of an "interior point " variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement (Phase I) and objective value improvement (Phase II). The key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense. Given a prespecified balancing parameter /3> 0, the algorithm maintains the following Phase I Phase II "/3balancing constraint " throughout: (cTx Z*) < /3TX, where cTx is the objective function, z * is the (unknown) optimal objective value of the linear program, and Tx measures the infeasibility of the current iterate x. This balancing constraint can be used to either emphasize rapid attainment of feasibility (set large) at the possible expense of good objective function values or to emphasize rapid attainment of good objective values (set /3 small) at the possible expense of a lower infeasibility gap. The algorithm exhibits the following advantageous features: (i) the iterate solutions monotonically decrease the infeasibility measure, (ii) the iterate solutions satisy the /3balancing constraint, (iii) the iterate solutions achieve constant improvement in both Phase I and Phase II in O(n) iterations, (iv) there is always a possibility of finite termination of the Phase I problem, and (v) the algorithm is amenable to acceleration via linesearch of the potential function.
A Note on an Infeasible Start Interior Point Method for Linear Programming
, 1999
"... The paper is a simplified exposition of an early combined phase Iphase II method for linear programming. The method works from an infeasible start. Besides, there is no need for regularity conditions if the method is applied to a primaldual formulation. Keywords Interior point methods. Infeasible ..."
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The paper is a simplified exposition of an early combined phase Iphase II method for linear programming. The method works from an infeasible start. Besides, there is no need for regularity conditions if the method is applied to a primaldual formulation. Keywords Interior point methods. Infeasible start. Karmarkar's method. 1 Introduction In this note we revisit an early contribution in the field of interior point methods [3], which presented a combined phase Iphase II interior point method for linear program. The method extends the original contribution of Karmarkar [8]. Like [13], it relaxes the assumption of a known optimal value, but it is more general, since it does not require knowledge of an initial interior feasible point. It also operates on the problem without any prior data manipulation. The field of infeasible start methods became popular later, in particular in the framework of the primaldual method, when it appeared that this method turned out to be more efficient th...