Results 1  10
of
12
Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time
, 2003
"... We introduce the smoothed analysis of algorithms, which continuously interpolates between the worstcase and averagecase analyses of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We me ..."
Abstract

Cited by 146 (14 self)
 Add to MetaCart
We introduce the smoothed analysis of algorithms, which continuously interpolates between the worstcase and averagecase analyses of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has smoothed complexity polynomial in the input size and the standard deviation of
Deformed Products and Maximal Shadows of Polytopes
 ADVANCES IN DISCRETE AND COMPUTATIONAL GEOMETRY, AMER. MATH. SOC., PROVIDENCE, CONTEMPORARY MATHEMATICS 223
, 1996
"... We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with "many pivots," starting with the famous KleeMinty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for ddimensional simpl ..."
Abstract

Cited by 30 (1 self)
 Add to MetaCart
We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with "many pivots," starting with the famous KleeMinty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for ddimensional simple polytopes with at most n facets: ffl the maximal number of vertices on an increasing path, ffl the maximal number of vertices on a "greedy" greatest increase path, and ffl the maximal number of vertices of a 2dimensional projection. This, equivalently, provides good estimates for the worstcase behaviour of the simplex algorithm on linear programs with these parameters with the worstpossible, the greatest increase, and the shadow vertex pivot rules. The bounds on the maximal number of vertices on an increasing path or a greatest increase path unify and slightly improve a number of known results. One bound on the maximal number of vertices of a 2dimensional projection is new: we show ...
Randomized Simplex Algorithms on KleeMinty Cubes
 COMBINATORICA
, 1994
"... We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the randomedge simplex algorithm on KleeMinty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a KleeMinty cube is exponential when all paths are taken with equal probability.
CrissCross Methods: A Fresh View on Pivot Algorithms
 Mathematical Programming
, 1997
"... this paper is to present mathematical ideas and ..."
Construction and analysis of projected deformed products
, 2007
"... We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the kfaces) are “strictly p ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the kfaces) are “strictly preserved ” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)polytope Q on n−1 vertices we construct a deformed ncube, whose projection to the last d coordinates yields a neighborly cubical dpolytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs), which have a projection to dspace that retains the complete ( ⌊ d 2 ⌋ − 1)skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the projected deformed products of polygons that were announced by Ziegler (2004), a family of 4polytopes whose “fatness ” gets arbitrarily close to 9. 1
A Survey on Pivot Rules for Linear Programming
 ANNALS OF OPERATIONS RESEARCH. (SUBMITTED
, 1991
"... The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. Th ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...
An exponential lower bound on the complexity of regularization paths. arXiv:0903.4817v2 [cs.LG
, 2009
"... For a variety of regularization methods, algorithms computing the entire solution path have been developed recently. Solution path algorithms do not only compute the solution for one particular value of the regularization parameter but the entire path of solutions, making the selection of an optimal ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
For a variety of regularization methods, algorithms computing the entire solution path have been developed recently. Solution path algorithms do not only compute the solution for one particular value of the regularization parameter but the entire path of solutions, making the selection of an optimal parameter much easier. It has been assumed that these piecewise linear solution paths have only linear complexity, i.e. linearly many bends. We prove that for the support vector machine this complexity can indeed be exponential in the number of training points in the worst case.
A Monotonic BuildUp Simplex Algorithm for Linear Programming
, 1991
"... We devise a new simplex pivot rule which has interesting theoretical properties. Beginning with a basic feasible solution, and any nonbasic variable having a negative reduced cost, the pivot rule produces a sequence of pivots such that ultimately the originally chosen nonbasic variable enters the ba ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We devise a new simplex pivot rule which has interesting theoretical properties. Beginning with a basic feasible solution, and any nonbasic variable having a negative reduced cost, the pivot rule produces a sequence of pivots such that ultimately the originally chosen nonbasic variable enters the basis, and all reduced costs which were originally nonnegative remain nonnegative. The pivot rule thus monotonically builds up to a dual feasible, and hence optimal, basis. A surprising property of the pivot rule is that the pivot sequence results in intermediate bases which are neither primal nor dual feasible. We prove correctness of the procedure, give a geometric interpretation, and relate it to other pivoting rules for linear programming.
On the Hardness and Smoothed Complexity of QuasiConcave Minimization
"... In this paper, we resolve the smoothed and approximative complexity of lowrank quasiconcave minimization, providing both upper and lower bounds. As an upper bound, we provide the first smoothed analysis of quasiconcave minimization. The analysis is based on a smoothed bound for the number of extr ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
In this paper, we resolve the smoothed and approximative complexity of lowrank quasiconcave minimization, providing both upper and lower bounds. As an upper bound, we provide the first smoothed analysis of quasiconcave minimization. The analysis is based on a smoothed bound for the number of extreme points of the projection of the feasible polytope onto a kdimensional subspace, where k is the rank (informally, the dimension of nonconvexity) of the quasiconcave function. Our smoothed bound is polynomial in the original dimension of the problem n and the perturbation size ρ, and it is exponential in the rank of the function k. From this, we obtain the first randomized fully polynomialtime approximation scheme for lowrank quasiconcave minimization under broad conditions. In contrast with this, we prove log nhardness of approximation for general quasiconcave minimization. This shows that our smoothed bound is essentially tight, in that no polynomial smoothed bound is possible for quasiconcave functions of general rank k. The tools that we introduce for the smoothed analysis may be of independent interest. All previous smoothed analyses of polytopes analyzed projections onto twodimensional subspaces and studied them using trigonometry to examine the angles between vectors and 2planes in R n. In this paper, we provide what is, to our knowledge, the first smoothed analysis of the projection of polytopes onto higherdimensional subspaces. To do this, we replace the trigonometry with tools from random matrix theory and differential geometry on the Grassmannian. Our hardness reduction is based on entirely different proofs that may also be of independent interest: we show that the stochastic 2stage minimum spanning tree problem has a supermodular objective and that su
A New Approach to Strongly Polynomial Linear Programming
"... Abstract: We present an affineinvariant approach for solving linear programs. Unlike previous approaches, the potential strong polynomiality of the new approach does not require that graphs of polytopes have polynomial diameter (the Hirsch conjecture or weaker versions). We prove that two natural r ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract: We present an affineinvariant approach for solving linear programs. Unlike previous approaches, the potential strong polynomiality of the new approach does not require that graphs of polytopes have polynomial diameter (the Hirsch conjecture or weaker versions). We prove that two natural realizations of the approach work efficiently for deformed products [AZ99], a class of polytopes that generalizes all known difficult examples for variants of the simplex method, e.g., the KleeMinty [KM72] and GoldfarbSit [GS79] cubes.