A polynomial-time algorithm for a class of linear complementarity problems

Complexity analysis of a linear complementarity algorithm based on a Lyapunov function

“…Hence if we fix [3 and choose co such that Ilcll < 2• L and g(co) = O(4-m), where L denotes the size of the problem encoding in binary (defined as in [9], [12]- [14], [18], [19], [26]) and X is a positive scalar, then the homotopy algorithm with 6 = 2 -2 hL would terminate in O(/-mL) steps with an approximate solution of (LCP) whose error is less than 2 -X L . For X sufficiently large (independent of L), a solution for (LCP) can be recovered by using, say, the techniques described in [12].…”

“…For X sufficiently large (independent of L), a solution for (LCP) can be recovered by using, say, the techniques described in [12]. Since the amount of computation per step is at most O(m 3 ) arithmetic operations (not counting Step 0), the homotopy algorithm has a complexity of O(m 3 SL) arithmetic operations.…”

“…The first polynomial-time algorithm for solving (LCP) was given by Kojima, Mizuno and Yoshise [12], based on path following. Subsequently, Kojima, Megiddo, Yoshise [13] and Kojima, Mizuno, Ye [14] developed polynomial-time algorithms for solving (LCP), using a different notion of potential reduction.…”

“…For the special case of convex quadratic programs (QP), the first polynomial-time algorithm was given by Kozlov, Tarasov and Khachiyan [15] based on the ellipsoid method [23], [27]. This was followed by a number of algorithms of the interior point variety (see [9], [12], [18], [19], [26]). In this paper we consider a polynomial-time algorithm for solving (LCP) that is motivated by Karmarkar's method [11] and its interpretation as a projected Newton method based on the logarithmic barrier function [8].…”

“…In this paper we consider a polynomial-time algorithm for solving (LCP) that is motivated by Karmarkar's method [11] and its interpretation as a projected Newton method based on the logarithmic barrier function [8]. Our approach, which is similar to that taken in [9], [12], [18], [19], [26] is to solve approximately a sequence of nonlinear equations ((PWt) } over Ex9i n , where {w t } is a geometrically decreasing sequence of positive vectors. Each (Pwt) is solved (approximately) by taking a Newton step for (Pwt-1) starting at the previous solution.…”