On the expected number of linear complementarity cones intersected by random and semi-random rays

نویسنده

  • Nimrod Megiddo
چکیده

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 (semi-random) 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 n-dimensional 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 'semi-random' assumption) has implications with respect to recent analyses of the average complexity of the linear programming problem. Surprisingly, the semi-random 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 semi-random ray tends to in2+2n when E tends to zero.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A full NT-step O(n) infeasible interior-point method for Cartesian P_*(k) –HLCP over symmetric cones using exponential convexity

In this paper, by using the exponential convexity property of a barrier function, we propose an infeasible interior-point method for Cartesian P_*(k) horizontal linear complementarity problem over symmetric cones. The method uses Nesterov and Todd full steps, and we prove that the proposed algorithm is well define. The iteration bound coincides with the currently best iteration bound for the Ca...

متن کامل

A full Nesterov-Todd step infeasible interior-point algorithm for symmetric cone linear complementarity problem

‎A full Nesterov-Todd (NT) step infeasible interior-point algorithm‎ ‎is proposed for solving monotone linear complementarity problems‎ ‎over symmetric cones by using Euclidean Jordan algebra‎. ‎Two types of‎ ‎full NT-steps are used‎, ‎feasibility steps and centering steps‎. ‎The‎ ‎algorithm starts from strictly feasible iterates of a perturbed‎ ‎problem‎, ‎and, using the central path and feasi...

متن کامل

An Algorithm for Finding All Extremal Rays of Polyhedral Convex Cones with Some Complementarity Conditions

In this paper, we show a method for finding all extremal rays of polyhedral convex cones with some complementarity conditions. The polyhedral convex cone is defined as the intersection of half-spaces expressed by linear inequalities. By a complementarity extremal ray, we mean an extremal ray vector that satisfies some complementarity conditions among its elen~nts. Our method is iterative in the...

متن کامل

Robust solution of monotone stochastic linear complementarity problems

We consider the stochastic linear complementarity problem (SLCP) involving a random matrix whose expectation matrix is positive semi-definite. We show that the expected residual minimization (ERM) formulation of this problem has a nonempty and bounded solution set if the expected value (EV) formulation, which reduces to the LCP with the positive semi-definite expectation matrix, has a nonempty ...

متن کامل

An improved infeasible‎ ‎interior-point method for symmetric cone linear complementarity‎ ‎problem

We present an improved version of a full Nesterov-Todd step infeasible interior-point method for linear complementarityproblem over symmetric cone (Bull. Iranian Math. Soc., 40(3), 541-564, (2014)). In the earlier version, each iteration consisted of one so-called feasibility step and a few -at most three - centering steps. Here, each iteration consists of only a feasibility step. Thus, the new...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:
  • Math. Program.

دوره 35  شماره 

صفحات  -

تاریخ انتشار 1986