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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 613 (15 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 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 26 (5 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)).
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 (3 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
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 12 (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 11 (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: Termination
, 2003
"... We perform a smoothed analysis of the termination phase of an interiorpoint method. By combining this analysis with the smoothed analysis of Renegar’s interiorpoint algorithm in [DST02], we show that the smoothed complexity of an interiorpoint algorithm for linear programming is O(m 3 log(m/σ)). ..."
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Cited by 3 (1 self)
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We perform a smoothed analysis of the termination phase of an interiorpoint method. By combining this analysis with the smoothed analysis of Renegar’s interiorpoint algorithm in [DST02], we show that the smoothed complexity of an interiorpoint algorithm for linear programming is O(m 3 log(m/σ)). In contrast, the best known bound on the worstcase complexity of linear programming is O(m 3 L), where L could be as large as m. We include an introduction to smoothed analysis and a tutorial on proof techniques that have been useful in smoothed analyses.
Progress in Linear Programming: InteriorPoint Algorithms
, 1994
"... Abstract: According to current estimates, more than $100 million in human and computer time is invested yearly in the formulation and solution of linear programming problems. Businesses, large and small, use linear programming models to optimize communication systems, to schedule transportation netw ..."
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Abstract: According to current estimates, more than $100 million in human and computer time is invested yearly in the formulation and solution of linear programming problems. Businesses, large and small, use linear programming models to optimize communication systems, to schedule transportation networks, to control inventories, to plan investments, and to maximize production.... In this article we describe some recent developments in linear programming. We highlight progress in interiorpoint algorithms during the last ten years.
A Lower Bound on the Number of Iterations of LongStep PrimalDual Linear Programming Algorithms
, 1994
"... Abstract Recently, Todd has analyzed in detail the primaldual affinescaling method for linear programming, which is close to what is implemented in practice, and proved that it may take at least n1=3 iterations to improve the initial duality gap by a constant factor. He has also showed that this l ..."
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Abstract Recently, Todd has analyzed in detail the primaldual affinescaling method for linear programming, which is close to what is implemented in practice, and proved that it may take at least n1=3 iterations to improve the initial duality gap by a constant factor. He has also showed that this lower bound holds for some polynomial variants of primaldual interiorpoint methods, which restrict all iterates to certain neighborhoods of the central path. In this paper, we further extend his result to longstep primaldual variants that restrict the iterates to a wider neighborhood. This neighborhood seems the least restrictive one to guarantee polynomiality for primaldual pathfollowing methods, and the variants are also even closer to what is implemented in practice.
On the probabilistic complexity of finding an approximate solution for linear programming
"... We consider the problem of finding an −optimal solution of a standard linear program with real data, i.e., of finding a feasible point at which the objective function value differs by at most from the optimal value. In the worstcase scenario the best complexity result to date guarantees that such ..."
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We consider the problem of finding an −optimal solution of a standard linear program with real data, i.e., of finding a feasible point at which the objective function value differs by at most from the optimal value. In the worstcase scenario the best complexity result to date guarantees that such a point is obtained in at most O( n  ln ) steps of an interior point method. We show that the expected value of the number of steps required to obtain an −optimal solution for a probabilistic linear programming model is at most O(min{n1.5,m√n ln(n)}) + log2(  ln ).
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 ∥ ∥