Results 1 
9 of
9
A PrimalDual Potential Reduction Method for Problems Involving Matrix Inequalities
 in Protocol Testing and Its Complexity&quot;, 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 iterat ..."
Abstract

Cited by 102 (20 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 30 (8 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 16 (9 self)
 Add to MetaCart
(Show Context)
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.
Potential reduction algorithms
 Interior Point Methods in Mathematical Programming
, 1996
"... Potential reduction algorithms have a distinguished role in the area of interior point methods for mathematical programming. Karmarkar’s [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
Potential reduction algorithms have a distinguished role in the area of interior point methods for mathematical programming. Karmarkar’s [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a
Interior Point Algorithms for Integer Programming
, 1994
"... 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 ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
This paper develops a potential reduction algorithm for solving a linearprogramming problem directly from a &quot;warm start &quot; initial point that is neither feasible nor optimal. The algorithm is of an &quot;interior point &quot; 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 &quot;/3balancing constraint &quot; 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.