Feasible Interior Point Method for Absolute Value Equation

Abstract

A feasible interior point method is proposed for solving the NP-hard absolute value equation (AVE) when the singular values of A exceed one. We formulate the NP-hard AVE as linear complementary problem, and prove that the solution to AVE is existent and unique under suitable assumptions. Then we present a feasible interior point algorithm for AVE based on the Newton direction and centering direction. We show that this algorithm has the polynomial complexity. Preliminary numerical results show that this method is promising.

Solving absolute value equations via complementarity and interior-point methods

“…In the next example, we compare Algorithm 3.2 with a feasible primal-dual IPMs used to solve the AVE (1.2). This latter is developed by Long ( [16]).…”

“…Problem 3 [16]. The data (A, B, b) of the AVE is given by rand( state , 0); R = rand(n, n); A = R * R + n * eye(n); b = rand(n, 1); B = I; q = ((A + eye(n)) * (inv(A − eye(n))) − eye(n)) * b.…”

“…The numerical results are summarized in Table 3, where Algorithm 3.2 is denoted by Algo1 and the algorithm used in [16] is denoted by Algo2. From [16], we use the same tolerance ε = 10 −4 . 3.…”

“…In the literatures, most numerical methods directly tackled the AVE based on smoothing and non-smoothing nonlinear optimization techniques. However, other numerical approaches focus on reformulating the AVE (1.1)-(1.2) as complementarity problems; see e.g., Yong et al [16].…”