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On the Complexity of a Practical Interior-Point Method
SIAM Journal on Optimization, 1998Summary: The theory of self-concordance in convex optimization has been used to analyze the complexity of interior-point methods based on Newton's method. For large problems, it may be impractical to use Newton's method; here we analyze a truncated-Newton method, in which an approximation to the Newton search direction is used.
Stephen G. Nash, Ariela Sofer
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Journal of Optimization Theory and Applications, 1998
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1996
Abstract The interest in interior point methods for linear programming emerged from Karmarkar’s contribution in 1984. This field has soon become one of the most active in the area of mathematical programming. It introduced new ideas and techniques that now have received their own place among the basic tools in optimization.
Cornelis Roos, Jean-Philippe Vial
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Abstract The interest in interior point methods for linear programming emerged from Karmarkar’s contribution in 1984. This field has soon become one of the most active in the area of mathematical programming. It introduced new ideas and techniques that now have received their own place among the basic tools in optimization.
Cornelis Roos, Jean-Philippe Vial
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Barrier Functions in Interior Point Methods
Mathematics of Operations Research, 1996We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone.
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The Kantorovich Theorem and interior point methods
Mathematical Programming, 2004zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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2008
Linear programs can be viewed in two somewhat complementary ways. They are, in one view, a class of continuous optimization problems each with continuous variables defined on a convex feasible region and with a continuous objective function. They are, therefore, a special case of the general form of problem considered in this text.
David G. Luenberger, Yinyu Ye
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Linear programs can be viewed in two somewhat complementary ways. They are, in one view, a class of continuous optimization problems each with continuous variables defined on a convex feasible region and with a continuous objective function. They are, therefore, a special case of the general form of problem considered in this text.
David G. Luenberger, Yinyu Ye
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2013
The ellipsoid method has an undeniable historical relevance (due to its role in establishing polynomial time for linear programming with integer data). In addition, its underlying idea is simple and elegant. Unfortunately, it is not efficient in practice compared with both the simplex method and the more recent interior-point methods.
Peter Bürgisser, Felipe Cucker
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The ellipsoid method has an undeniable historical relevance (due to its role in establishing polynomial time for linear programming with integer data). In addition, its underlying idea is simple and elegant. Unfortunately, it is not efficient in practice compared with both the simplex method and the more recent interior-point methods.
Peter Bürgisser, Felipe Cucker
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Ragnar Frisch and interior-point methods
Optimization Letters, 2014zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Olav Bjerkholt, Sjur Didrik Flåm
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The interior-point method for linear programming
IEEE Software, 1992A robust, reliable, and efficient implementation of the primal-dual interior-point method for linear programs, which is based on three well-established optimization algorithms, is presented. The authors discuss the theoretical foundation for interior-point methods which consists of three crucial building blocks: Newton's method for solving nonlinear ...
Greg Astfalk +3 more
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2013
As was known, the simplex method moves on the underlying polyhedron, from vertex to adjacent vertex along descent edges, until an optimal vertex is reached, or unboundedness of the problem is detected. Nevertheless, it would go through an exponential number of vertices of the polyhedron (Sect. 3.8), and even stall at a vertex forever because of cycling
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As was known, the simplex method moves on the underlying polyhedron, from vertex to adjacent vertex along descent edges, until an optimal vertex is reached, or unboundedness of the problem is detected. Nevertheless, it would go through an exponential number of vertices of the polyhedron (Sect. 3.8), and even stall at a vertex forever because of cycling
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