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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 ..."
Abstract

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
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 ..."
Abstract

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.