It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex-type algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the so-called self-dual 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'(n-1') 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 self-dual 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

In this paper we analyze the average number of steps performed by the self-dual 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 self-dual 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.

this paper. Of course, the first question we must answer is: what do we mean by a typical instance of a given size? Sometimes, there is a natural answer to this question. For example, in developing an algorithm which is typically efficent for an NP-complete optimization problems on graphs, we might assume that an n vertex input is equally likely to be any of the 2 2 ) labelled graphs with n vertices. This allows us to exploit any property which holds on almost all such graphs when developing the algorithm

this paper a simple classification of several known algorithms for finding the maxima, together with several new algorithms; among these are two efficient algorithms---one with expected complexity n +O( # nlogn) when the point samples are issued from some planar regions, and another more efficient than existing ones

We prove that the number of maximal points in a random sample taken uniformly and independently from a convex polygon is asymptotically normal in the sense of convergence in distribution. Many new results for other planar regions are also derived. In particular, precise Poisson approximation results are given for the number of maxima in regions bounded above by a nondecreasing curve. Keywords Maximal points, multicriterial optimization, central limit theorems, Poisson approximations, convex polygons. AMS subject classification Primary. 60D05; Secondary. 60C05 Submitted to EJP on September 29, 2000. Final version accepted on January 22, 2001.

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