A new polynomial-time algorithm for linear programming

Karmarkar, Narendra

doi:10.1007/bf02579150

Cited by 3,582 publications

(711 citation statements)

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“…Indeed, it was well known that during the period of serial processors, the speed-up in computation due to improved hardware-the exponential graph predicted by "Moore's law" [152]-was matched by a similar graph of speed-up due to the development of novel computational algorithms. A few examples are in order: the QR algorithm for computing eigensystems [80] and the fast Fourier transform FFT [52], which gave the impetus for the development of spectral methods during the 1960s; the development of multigrid and MATLAB in the 1970s [24,148,36]; wavelets, linear programming interior point methods, and the fast multipole method (FMM) [61,149,117,94] in the 1980s; high-resolution methods for discontinuous solutions in the 1990s [100,49]; and curvelets, greedy algorithms, compressive sensing and other "optimal algorithms" of finite-dimensional approximations, which have matured during recent years [32,69,33,208,66].…”

Section: Mathematical Models and Modern Mathematical Toolsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

143

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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Section: Mathematical Models and Modern Mathematical Toolsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

143

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“…Karmarkar's method. In 1984, Narendra Karmarkar [21] announced a polynomial-time LP method for which he reported solution times that were consistently 50 times faster than the simplex method. This event, which received publicity around the world throughout the popular press and media, marks the beginning of the interior-point revolution.…”

Section: Margaret H Wrightmentioning

confidence: 99%

“…Now that the dust has settled, derivations of interior methods typically involve barrier functions or their properties, such as perturbed complementarity (19). Readers interested in Karmarkar's method should consult his original paper [21] or any of the many comprehensive treatments published since 1984 (e.g., [19,30,41,35,44]). …”

Section: Margaret H Wrightmentioning

confidence: 99%

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

Wright

2004

Bull. Amer. Math. Soc.

277

131

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Abstract. Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. 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 programming was unthinkable because of the total dominance of the simplex method. During the 1970s, barrier methods were superseded, nearly to the point of oblivion, by newly emerging and seemingly more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost universally regarded as a closed chapter in the history of optimization.This picture changed dramatically in 1984, when Narendra Karmarkar announced a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have continued to transform both the theory and practice of constrained optimization. We present a condensed, unavoidably incomplete look at classical material and recent research about interior methods.

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“…In 1984, Karmarkar presented a new solution algorithm for linear programming problems that did not solve for the optimal solution by following a series of points that were on the "constraint boundary" but rather followed a path through the interior of the constraints directly toward the optimal solution on the constraint boundary [25]. This solution was achieved much more rapidly than with conventional LP algorithms.…”

Section: Interior Point Methods (Ip)mentioning

confidence: 99%

“…The interior point methods appeared in the literature in the early 1950's and they have been formally studied in detail by Fiacco and McCormick [73]. Since then, the big breakthrough of interior point methods was accomplished in 1984 by Karmarker's famous paper [25]. The excitement brought by this paper was due partly to a theoretical property of interior point methods, the primal-dual interior point method [74][75][76][77] proves to be the most elegant theoretically and the most successful computationally for LP.…”

Section: Interior Point Methods In Optimal Power Flow (Opf)mentioning

confidence: 99%

Interior point based optimal voltage stability, oscillatory stability and ATC margin boundary tracing

Zhou

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A new polynomial-time algorithm for linear programming

Cited by 3,582 publications

References 1 publication

A review of numerical methods for nonlinear partial differential equations

A review of numerical methods for nonlinear partial differential equations

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

Interior point based optimal voltage stability, oscillatory stability and ATC margin boundary tracing

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