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13
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 ..."
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Cited by 146 (14 self)
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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
A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension
 JOURNAL OF THE ACM
, 1985
"... It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplextype algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the socalled selfdual method, is analyzed. The algorithm is not started ..."
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Cited by 30 (2 self)
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It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplextype algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the socalled selfdual method, is analyzed. The algorithm is not started at the traditional point (1,..., but points of the form (1, e, e2,...)T, with t sufficiently small, are used. The result is better, in two respects, than those of the previous analyses. First, it is shown that the expected number of steps is bounded between two quadratic functions cl(min(m, n))' and cz(min(m, n)) ' of the smaller dimension of the problem. This should be compared with the previous two major results in the field. Borgwardt proves an upper bound of 0(n4m1'(n1') under a model that implies that the zero vector satisfies all the constraints, and also the algorithm under his consideration solves only problems from that particular subclass. Smale analyzes the selfdual algorithm starting at (1,..., He shows that for any fixed m there is a constant c(m) such the expected number of steps is less than ~(m)(lnn)"'("+~); Megiddo has shown that, under Smale's model, an upper bound C(m) exists. Thus, for the first time, a polynomial upper bound with no restrictions (except for nondegeneracy) on the problem is proved, and, for the first time, a nontrivial lower bound of precisely the same order of magnitude is established. Both Borgwardt and Smale require the input vectors to be drawn from
The Many Facets of Linear Programming
, 2000
"... . We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction A ..."
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Cited by 25 (1 self)
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. We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction At the last Mathematical Programming Symposium in Lausanne, we celebrated the 50th anniversary of the simplex method. Here, we are at or close to several other anniversaries relating to linear programming: the sixtieth of Kantorovich's 1939 paper on "Mathematical Methods in the Organization and Planning of Production" (and the fortieth of its appearance in the Western literature) [55]; the fiftieth of the historic 0th Mathematical Programming Symposium that took place in Chicago in 1949 on Activity Analysis of Production and Allocation [64]; the fortyfifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twentyfifth of the awarding of the 1975 Nobe...
Beyond Hirsch conjecture: Walks on random polytopes and smoothed complexity of the simplex method
 In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
, 2006
"... Abstract. The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadowvertex simplex method has polynomial smoothed complexity. On a slight random perturbation of an ar ..."
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Cited by 19 (4 self)
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Abstract. The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadowvertex simplex method has polynomial smoothed complexity. On a slight random perturbation of an arbitrary linear program, the simplex method finds the solution after a walk on polytope(s) with expected length polynomial in the number of constraints n, the number of variables d and the inverse standard deviation of the perturbation 1/σ. We show that the length of walk in the simplex method is actually polylogarithmic in the number of constraints n. SpielmanTeng’s bound on the walk was O ∗ (n 86 d 55 σ −30), up to logarithmic factors. We improve this to O(log 7 n(d 9 + d 3 σ −4)). This shows that the tight Hirsch conjecture n − d on the length of walk on polytopes is not a limitation for the smoothed Linear Programming. Random perturbations create short paths between vertices. We propose a randomized phaseI for solving arbitrary linear programs, which is of independent interest. Instead of finding a vertex of a feasible set, we add a vertex at
On the curvature of the central path of linear programming theory
 Foundations of Computational Mathematics
, 2003
"... Abstract. We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of variables. 1 Introduction. In this paper we study the curvature of the central path of linear programming theory. We establish that for a linear programming pro ..."
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Cited by 13 (2 self)
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Abstract. We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of variables. 1 Introduction. In this paper we study the curvature of the central path of linear programming theory. We establish that for a linear programming problem defined on a compact polytope contained in R n, the total curvature of the central path is less than or
IMPROVED ASYMPTOTIC ANALYSIS OF THE AVERAGE NUMBER OF STEPS PERFORMED BY THE SELFDUAL SIMPLEX ALGORITHM
, 1986
"... In this paper we analyze the average number of steps performed by the selfdual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem of n variables with m constraints. Smale established that for every numbe ..."
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Cited by 9 (1 self)
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In this paper we analyze the average number of steps performed by the selfdual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem of n variables with m constraints. Smale established that for every number of constraints m, there is a constant c(m) such that the number of pivot steps of the selfdual algorithm, p(m, n), is less than c(m)(ln n)"""'+". We improve upon this estimate by showing that p(m, n) is bounded by a function of m only. The symmetry of the function in m and n implies that p(m, n) is in fact bounded by a function of the smaller of m and n.
ON THE EXPECTED NUMBER OF LINEAR COMPLEMENTARITY CONES INTERSECTED BY RANDOM AND SEMIRANDOM RAYS
, 1986
"... Lemke's algorithm for the linear complementarity problem follows a ray which leads from a certain fixed point (traditionally, the point (1,..., I)~) to the point given in the problem. The problem also induces a set of 2 " cones, and a question which is relevant to the probabilistic analysis of ..."
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Cited by 4 (1 self)
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Lemke's algorithm for the linear complementarity problem follows a ray which leads from a certain fixed point (traditionally, the point (1,..., I)~) to the point given in the problem. The problem also induces a set of 2 " cones, and a question which is relevant to the probabilistic analysis of Lemke's algorithm is to estimate the expected number of times a (semirandom) ray intersects the boundary between two adjacent cones. When the problem is sampled from a spherically symmetric distribution this number turns out to be exponential. For an ndimensional problem the natural logarithm of this number is equal to ln(r)n + o(n), where T is approximately 1.151222. This number stands in sharp contrast with the expected number of cones intersected by a ray which is determined by two random points (call it random). The latter is only (n/2)+ 1. The discrepancy between linear behavior (under the 'random ' assumption) and exponential behavior (under the 'semirandom ' assumption) has implications with respect to recent analyses of the average complexity of the linear programming problem. Surprisingly, the semirandom case is very sensitive to the fixed point of the ray, even when that point is confined to the positive orthant. We show that for points of the form (E, E',..., E ") ~ the expected number of facets of cones cut by a semirandom ray tends to in2+2n when E tends to zero.
Parametric linear programming and anticycling pivoting rules
 MATHEMATICAL PROGRAMMING
, 1988
"... The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore, restrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the choice of ..."
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Cited by 2 (0 self)
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The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore, restrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the choice of exiting variables. Using ideas from parametric linear programming, we develop anticycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule. That is, any variable that ties for the ratio rule can leave the basis. A similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables.
Some problems in asymptotic convex geometry and random matrices motivated by numerical algorithms
 Proceedings of the conference on Banach Spaces and their applications in analysis (in honor of N. Kalton’s 60th birthday
"... Abstract. The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions – computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their ..."
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Cited by 2 (2 self)
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Abstract. The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions – computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses. 1. Asyptotic convex geometry and Linear Programming Linear Programming studies the problem of maximizing a linear functional subject to linear constraints. Given an objective vector z ∈ R d and constraint vectors a1,...,an ∈ R d, we consider the linear program (LP) maximize 〈z, x〉 subject to 〈ai, x 〉 ≤ 1, i = 1,...,n. This linear program has d unknowns, represented by x, and n constraints. Every linear program can be reduced to this form by a simple interpolation argument [36]. The feasible set of the linear program is the polytope P: = {x ∈ R d: 〈ai, x 〉 ≤ 1, i = 1,..., n}. The solution of (LP) is then a vertex of P. We can thus look at (LP) from a geometric viewpoint: for a polytope P in R d given by n faces, and for a vector z, find the vertex that maximizes the linear functional 〈z, x〉. The oldest and still the most popular method to solve this problem is the simplex method. It starts at some vertex of P and generates a walk on the edges of P toward the solution vertex. At each step, a pivot rule determines a choice of the next vertex; so there are many variants of the simplex method with different pivot rules. (We are not concerned here with how to find the initial vertex, which is a nontrivial problem in itself).
A Geometric Theory of Outliers and Perturbation
, 2002
"... We develop a new understanding of outliers and the behavior of linear programs under perturbation. Outliers are ubiquitous in scientific theory and practice. We analyze a simple algorithm for removal of outliers from a highdimensional data set and show the algorithm to be asymptotically good. We ex ..."
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Cited by 1 (0 self)
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We develop a new understanding of outliers and the behavior of linear programs under perturbation. Outliers are ubiquitous in scientific theory and practice. We analyze a simple algorithm for removal of outliers from a highdimensional data set and show the algorithm to be asymptotically good. We extend this result to distributions that we can access only by sampling, and also to the optimization version of the problem. Our results cover both the discrete and continuous cases. This is joint work with Santosh Vempala. The complexity