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Interiorpoint Methods
, 2000
"... The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadrati ..."
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Cited by 463 (16 self)
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The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by selfconcordant barrier functions.
Smoothed Analysis of Termination of Linear Programming Algorithms
"... We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng ..."
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Cited by 23 (4 self)
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We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng
Smoothed analysis of Renegar’s condition number for linear programming
, 2003
"... We perform a smoothed analysis of Renegar’s condition number for linear programming. In particular, we show that for every nbyd matrix Ā, nvector ¯ b and dvector ¯c satisfying ∥ Ā, ¯ b, ¯c ∥ ∥ F ≤ 1 and every σ ≤ 1 / √ dn, the expectation of the logarithm of C(A,b,c) is O(log(nd/σ)), where A, ..."
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Cited by 22 (6 self)
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We perform a smoothed analysis of Renegar’s condition number for linear programming. In particular, we show that for every nbyd matrix Ā, nvector ¯ b and dvector ¯c satisfying ∥ Ā, ¯ b, ¯c ∥ ∥ F ≤ 1 and every σ ≤ 1 / √ dn, the expectation of the logarithm of C(A,b,c) is O(log(nd/σ)), where A, b and c are Gaussian perturbations of Ā, ¯ b and ¯c of variance σ 2. From this bound, we obtain a smoothed analysis of Renegar’s interior point algorithm. By combining this with the smoothed analysis of finite termination Spielman and Teng (Math. Prog. Ser. B, 2003), we show that the smoothed complexity of linear programming is O(n 3 log(nd/σ)).
Probabilistic Analysis of an InfeasibleInteriorPoint Algorithm for Linear Programming
, 1998
"... We consider an infeasibleinteriorpoint algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal ..."
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Cited by 11 (3 self)
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We consider an infeasibleinteriorpoint algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal solution in the initialization of the algorithm. Our main result is that the expected number of iterations before termination with an exact optimal solution is O(n ln(n)). Keywords: Linear Programming, AverageCase Behavior, InfeasibleInteriorPoint Algorithm. Running Title: Probabilistic Analysis of an LP Algorithm 1 Dept. of Management Sciences, University of Iowa. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 2 Dept. of Mathematics, Valdosta State University. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 3 Dept. of Mathematics, University of Iowa. Supported by ...
Smoothed Analysis of Condition Numbers and Complexity Implications for Linear Programming
, 2009
"... We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to illposedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every nbyd matrix Ā, nvector ¯ b, and dvector ¯c satis ..."
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Cited by 7 (0 self)
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We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to illposedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every nbyd matrix Ā, nvector ¯ b, and dvector ¯c satisfying ∥ ∥ Ā, ¯ b, ¯c ∥ ∥ F ≤ 1 and every σ ≤ 1, E [log C(A, b, c)] = O(log(nd/σ)), A,b,c where A, b and c are Gaussian perturbations of Ā, ¯ b and ¯c of variance σ 2 and C(A, b, c) is the condition number of the linear program defined by (A, b, c). From this bound, we obtain a smoothed analysis of interior point algorithms. By combining this with the smoothed analysis of finite termination of Spielman and Teng (Math. Prog. Ser. B, 2003), we show that the smoothed complexity of interior point algorithms for linear programming is O(n 3 log(nd/σ)).
Smoothed Analysis of InteriorPoint Algorithms: Condition Number
, 2003
"... A linear program is typically specified by a matrix A together with two vectors b and c, where A is an nbyd matrix, b is an nvector and c is a dvector. There are several canonical forms for defining a linear program using (A,b,c). One commonly used canonical form is: max c T x s.t. Ax ≤ b and it ..."
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A linear program is typically specified by a matrix A together with two vectors b and c, where A is an nbyd matrix, b is an nvector and c is a dvector. There are several canonical forms for defining a linear program using (A,b,c). One commonly used canonical form is: max c T x s.t. Ax ≤ b and its dual min b T y s.t A T y = c, y ≥ 0. In [Ren95b, Ren95a, Ren94], Renegar defined the condition number C(A,b,c) of a linear program and proved that an interior point algorithm whose complexity was O(n 3 log(C(A,b,c)/ǫ)) could solve a linear program in this canonical form to relative accuracy ǫ, or determine that the program was infeasible or unbounded. In this paper, we prove that for any ( Ā, ¯ b, ¯c) such that ∥ ∥ Ā, ¯ b, ¯c ∥ ∥ F ≤ 1, where ∥ ∥ Ā, ¯ b, ¯c ∥ ∥