Solving absolute value equations via complementarity and interior-point methods

Abstract

In this paper, an infeasible path-following interior-point algorithm is proposed for solving the NP-hard absolute value equations (AVE) of the type Ax − B|x| = b. Under the condition that the minimal singular value of A is strictly greater than the maximal singular value of B, the unique solvability theorem of AVE is presented by formulating the AVE as a monotone horizontal linear complementary problem (HLCP). We also propose an infeasible primal-dual interior-point algorithm for solving the AVE across the HLCP. Some numerical results are provided to show the efficiency of the proposed algorithm.

A Note on Unique Solvability of the Generalized Absolute Value Matrix Equation

“…In 2010, Hu et al [8] rewrite AVE as a standard LCP form without any condition. In 2018, Achache et al [1] provided the unique solvability conditions for GAVE (5.3) by transforming the GAVE into monotone HLCP.…”

Preconditioned conjugate gradient methods for absolute value equations

“…In this paper, by reformulating the AVE (1) into an equivalent unconstrained quadratic optimization problem, we prove first under the condition that the smallest singular value of A is greater than the largest singular value of B, the AVE (1) is uniquely solvable for any b. Secondly, we show that the unique minimum of the corresponding unconstrained quadratic problem is the unique solution of the AVE (1). Then across the latter, we apply the conjugate gradient algorithms to approximate numerically the solution of the AVE (1). In the presence of the ill-conditioned, preconditioned conjugate gradient methods can be used to ensure and to accelerate the convergence of the basic CG algorithms.…”

“…where A and B are given matrices in R n×n , b ∈ R n and |x| denotes the vector with absolute values of each components of the vector x, was investigated by [1], [11], [12], [15]. A special case of (1) when B = I (I denotes the identity matrix) is the AVE of the type:…”

“…As is known in [11], the general NP-hard linear complementarity can be formulated as the AVE (1), then it is an NP-hard in its general form. Furthermore, much research has been devoted to achieve their numerical solutions efficiently (see, e.g., [1], [2], [7], [8], [9], [12], [15]). In this paper, by reformulating the AVE (1) into an equivalent unconstrained quadratic optimization problem, we prove first under the condition that the smallest singular value of A is greater than the largest singular value of B, the AVE (1) is uniquely solvable for any b. Secondly, we show that the unique minimum of the corresponding unconstrained quadratic problem is the unique solution of the AVE (1).…”

A Two-steps Fixed-point Method for the Simplicial Cone Constrained Convex Quadratic Optimization

“…In particular, Mangasarian in [14] proposed a semi-smooth Newton's method for solving the AVE, and under suitable conditions he showed the finite and linear convergence to a solution of the AVE. However, other numerical approaches focus on reformulating the AVE as an horizontal linear complementarity problems (HLCP) (see [4]), where they introduce an infeasible path-following interior-point method for solving the AVE by using is equivalent reformulations as an HLCP. In this paper, we propose a new two-steps fixed point iterative method for solving the AVE (1.2) which is introduced in [11], and under a new mild assumption we show that this method is always well-defined and the generated sequence converges globally and linearly to the unique solution of the AVE from any starting initial point.…”