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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.
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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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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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
The Work of Daniel A. Spielman
 PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2010
"... Dan Spielman has made groundbreaking contributions in theoretical computer science and mathematical programming and his work has profound connections to the study of polytopes and convex bodies, to errorcorrecting codes, expanders, and numerical analysis. Many of Spielman’s achievements came with a ..."
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Dan Spielman has made groundbreaking contributions in theoretical computer science and mathematical programming and his work has profound connections to the study of polytopes and convex bodies, to errorcorrecting codes, expanders, and numerical analysis. Many of Spielman’s achievements came with a beautiful collaboration spanned over two decades with ShangHua Teng. This paper describes some of Spielman’s main achievements. Section 1 describes smoothed analysis of algorithms, which is a new paradigm for the analysis of algorithms introduced by Spielman and Teng. Section 2 describes Spielman and Teng’s explanation for the excellent practical performance of the simplex algorithm via smoothed analysis. Spielman and Teng’s theorem asserts that the simplex algorithm takes a polynomial number of steps for a random Gaussian perturbation of every linear programming problem. Section 3 is devoted to Spielman’s works on errorcorrecting codes and in particular his construction of lineartime encodable and decodable highrate codes based
Denmark Worstcase Analysis of Strategy Iteration and the Simplex Method
, 2012
"... In this dissertation we study strategy iteration (also known as policy iteration) algorithms for solving Markov decision processes (MDPs) and twoplayer turnbased stochastic games (2TBSGs). MDPs provide a mathematical model for sequential decision making under uncertainty. They are widely used to mo ..."
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In this dissertation we study strategy iteration (also known as policy iteration) algorithms for solving Markov decision processes (MDPs) and twoplayer turnbased stochastic games (2TBSGs). MDPs provide a mathematical model for sequential decision making under uncertainty. They are widely used to model stochastic optimization problems in various areas ranging from operations research, machine learning, artificial intelligence, economics and game theory. The class of twoplayer turnbased stochastic games is a natural generalization of Markov decision processes that is obtained by introducing an adversary. 2TBSGs form an intriguing class of games whose status in many ways resembles that of linear programming 40 years ago. They can be solved efficiently with strategy iteration algorithms, resembling the simplex method for linear programming, but no polynomial time algorithm is known. Linear programming is an exceedingly important problem with numerous applications. The simplex method was introduced by
1.4 The Shadow Vertex Pivot Rule......................... 7
, 2008
"... We introduce the smoothed analysis of algorithms, which is a hybrid of the worstcase and averagecase analysis of algorithms. Essentially, we study the performance of algorithms under small random perturbations of their inputs. We show that the simplex algorithm has polynomial smoothed ..."
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We introduce the smoothed analysis of algorithms, which is a hybrid of the worstcase and averagecase analysis of algorithms. Essentially, we study the performance of algorithms under small random perturbations of their inputs. We show that the simplex algorithm has polynomial smoothed