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Page 579 of Mathematical Reviews Vol. , Issue 88a [page]

1988 Mathematical Reviews  
Bhatia (New Delhi) 88a:90124 90C05 68Q25 Blum, Lenore (1-CA) Towards an asymptotic analysis of Karmarkar’s algorithm. Inform. Process. Lett. 23 (1986), no. 4, 189-194.  ...  The author gives an analysis of Karmarkar’s algorithm for linear programming. She gives a new and simplified proof of the number of iterative steps required by the algorithm in the worst case. K. G.  ... 

Page 1772 of Mathematical Reviews Vol. , Issue 94c [page]

1994 Mathematical Reviews  
Furthermore, the paper provides an algorithm for moving towards consistency and shows mathematically why the algorithm works.”  ...  projected gradients produced by a variant of Karmarkar’s interior- point algorithm known as the affine-scaling primal algorithm.  ... 

Degeneracy in interior point methods for linear programming: a survey

O. Güler, D. den Hertog, C. Roos, T. Terlaky, T. Tsuchiya
1993 Annals of Operations Research  
The publication of Karmarkar's paper has resulted in intense research activity into Interior Point Methods (IPMs) for linear programming.  ...  Roughly speaking, we shall deal with the effect of degeneracy on the following: the convergence of IPMs, the trajectories followed by the algorithms, numerical performance, and finding basic solutions.  ...  Thus, we can apply Karmarkar's analysis to study the behavior of the algorithm near degenerate faces.  ... 
doi:10.1007/bf02096259 fatcat:wfohc5pjibbotho4z6bo6kzukq

Linear Programming (1986)

N Megiddo
1987 Annual Review of Computer Science  
Karmarkar used a "potential function" in the design and analysis of his algorithm.  ...  as Karmarkar's algorithm.  ... 
doi:10.1146/annurev.cs.02.060187.001003 fatcat:2bv2mgpyvnhwlf5xfo6yu5jary

Page 6123 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
The basic steps of the algorithm are Newton steps toward a point on the central path of the problem, but typically the iterates are out of the feasible region.  ...  An essential component is the addition of a single constraint, motivated by Shaw and Goldfarb’s analysis, which makes the standard form algorithm strictly monotone in the true objective.” 94j:90021 90C05  ... 

Page 502 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews  
The corresponding geometrical interpretation to the iterative procedure is shown and the con- nection between our algorithm and Karmarkar’s algorithm in the theory of linear programming is noted.” 97a:  ...  This paper gives a detailed description of the use of pivotal infer- ence in the analysis of Darwin’s data on the heights of matched pairs of maize plants.  ... 

Isospectral flows and linear programming

U. Helmke
1993 The Journal of the Australian Mathematical Society Series B Applied Mathematics  
An interior point algorithm for the standard simplex is analysed in detail and a comparison is made with a continuous time version of Karmarkar algorithm.  ...  We show that the flow converges exponentially fast to the optimal solution of the programming problem and we give explicit estimates for the time needed by the flow to approach an e-neighbourhood of the  ...  Acknowledgement This work was completed during two months in 1991 which the author spent at the Department of Systems Engineering, A. N. U. Canberra.  ... 
doi:10.1017/s0334270000009048 fatcat:f6cjqunrwbeobeutnnzx7z46bq

Large scale inequality constrained optimization and control

1998 IEEE Control Systems  
ince Karmarkar's landmark 1984 paper 13 I], interior point S methods in linear programming have triggered a tremendous amount of activity.  ...  We conclude with a review of current status and a discussion of future directions.  ...  Acknowledgments The authors would like to thank George Staus, Purt Tanartkit, and Joao Albuquerque for their input, which helped shape up some of the ideas in this paper.  ... 
doi:10.1109/37.736012 fatcat:jvbmi566mrcdrk5qkkwi7trjla

The interior-point revolution in optimization: History, recent developments, and lasting consequences

Margaret H. Wright
2004 Bulletin of the American Mathematical Society  
programming was unthinkable because of the total dominance of the simplex method.  ...  Although interior-point techniques, primarily in the form of barrier methods, were widely used during the 1960s for problems with nonlinear constraints, their use for the fundamental problem of linear  ...  A detailed analysis was given in [37] of the structure of the primal barrier Hessian (16) in an entire neighborhood of the solution.  ... 
doi:10.1090/s0273-0979-04-01040-7 fatcat:rl4mg5zfm5gzrde75sya6pmg7y

Page 3030 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews  
Faibusovich], On the phase portrait of the Karmarkar’s flow (203-210); Max D. Gunzburger and Janet S.  ...  Backx, Engineering aspects of industrial applications of model-based control techniques and system theory (79-109); M. Gevers, Towards a joint design of identification and control? (111-151); A.  ... 

Asymptotic analysis of the exponential penalty trajectory in linear programming

R. Cominetti, J. San Martín
1994 Mathematical programming  
These results are completed by an asymptotic analysis when r tends to ~: the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution.  ...  For r close to 0, the unconstrained minimizer x(r) offr admits an asymptotic expansion of the form x (r) = x* + rd* + *l(r) where x* is a particular optimal solution of the linear program and the error  ...  The previous results are completed by an asymptotic analysis when r tends to ~.  ... 
doi:10.1007/bf01582220 fatcat:rgbnwwjv7jeeph363yyejnplla

Superlinear convergence of the affine scaling algorithm

T. Tsuchiya, R. D. C. Monteiro
1996 Mathematical programming  
In this paper we show that a variant of the long-step affine scaling algorithm (with variable stepsizes) is two-step superlinearly convergent when applied to general linear programming (LP) problems.  ...  Superlinear convergence of the sequence of dual estimates is also established.  ...  , Houston, USA from March of 1992 to February of 1993.  ... 
doi:10.1007/bf02592206 fatcat:3laoszvbzfe2lj3ks4mi7udaem

Boundary Behavior of Interior Point Algorithms in Linear Programming

Nimrod Megiddo, Michael Shub
1989 Mathematics of Operations Research  
The algorithms considered are Karmarkar's projective rescaling algorithm, the linear rescaling algorithm which w as proposed as a variation on Karmarkar's algorithm, and the logarithmic barrier technique  ...  It is shown that all the trajectories have a unique asymptotic direction of convergence to the optimum. * This work was done in part while the authors were members at the Mathematical Proposition 2.3.  ...  Michael Shub thanks Earl Barnes for introducing him to the linear rescaling algorithm and also acknowledges conversations with Lenore Blum.  ... 
doi:10.1287/moor.14.1.97 fatcat:xhutb4e47rhbbmf2gwcm4wj4wm

Computation of the collapse state in limit analysis using the LP primal affine scaling algorithm

E. Christiansen, K.O. Kortanek
1991 Journal of Computational and Applied Mathematics  
Kortanek, Computation of the collapse state in limit analysis using the LP primal affine scaling algorithm, Journal of Computational and Applied Mathematics 34 (1991) 47-63.  ...  The first goal is to demonstrate that for the duality problem of limit analysis with linearized yield condition the LP primal affine scaling algorithm shows properties, which are significantly different  ...  Acknowledgement The CRAY X-MP/48 computer runs were performed at the NCSA at the University of Illinois, Urbana-Champaign.  ... 
doi:10.1016/0377-0427(91)90147-c fatcat:ea4wl66ihzfrffiibaj25taoeu

The Nonlinear Geometry of Linear Programming. III Projective Legendre Transform Coordinates and Hilbert Geometry

J. C. Lagarias
1990 Transactions of the American Mathematical Society  
This paper studies projective scaling trajectories, which are the trajectories obtained by following the infinitesimal version of Karmarkar's linear programming algorithm.  ...  A nonlinear change of variables, projective Legendre transform coordinates, is introduced to study these trajectories.  ...  One of these algorithms is the projective scaling algorithm (Karmarkar's algorithm) , and the associated curves are called projective scaling trajectories or P-trajectories.  ... 
doi:10.2307/2001758 fatcat:6v6edxrd6rdofi7x5mf53eqyam
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