The Unique Solvability Conditions for the Generalized Absolute Value Equations

This paper investigates the conditions that guarantee unique solvability and unsolvability for the generalized absolute value equations (GAVE) given by $Ax - B \vert x \vert = b$. Further, these conditions are also valid to determine the unique solution of the generalized absolute value matrix equations (GAVME) $AX - B \vert X \vert =F$. Finally, certain aspects related to the solvability and unsolvability of the absolute value equations (AVE) have been deliberated upon

A Note on the Paper "The unique solution of the absolute value equations"

In this note, we give the possible revised version of the unique solvability conditions for the two incorrect results that appeared in the published paper by Wu et al. (Appl Math Lett 76:195-200, 2018)

Necessary and sufficient conditions for unique solvability of absolute value equations: A Survey

In this survey paper, we focus on the necessary and sufficient conditions for the unique solvability and unsolvability of the absolute value equations (AVEs) during the last twenty years (2004 to 2023). We discussed unique solvability conditions for various types of AVEs like standard absolute value equation (AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE) and a class of NGAVE based on interval matrix, P-matrix, singular value conditions, spectral radius and $\mathcal{W}$-property. Based on the unique solution of AVEs, we also discussed unique solvability conditions for linear complementarity problems (LCP) and horizontal linear complementarity problems (HLCP)

Preconditioned conjugate gradient methods for absolute value equations

We investigate the NP-hard absolute value equations (AVE), $Ax-B|x| =b$, where $A,B$ are given symmetric matrices in $\mathbb{R}^{n\times n}, \ b\in \mathbb{R}^{n}$. By reformulating the AVE as an equivalent unconstrained convex quadratic optimization, we prove that the unique solution of the AVE is the unique minimum of the corresponding quadratic optimization. Then across the latter, we adopt the preconditioned conjugate gradient methods to determining an approximate solution of the AVE. The computational results show the efficiency of these approaches in dealing with the AVE

On the unique solvability and numerical study of absolute value equations

The aim of this paper is twofold. Firstly, we consider the unique solvability of absolute value equations (AVE), $Ax-B\vert x\vert =b$, when the condition $\Vert A^{-1}\Vert <\frac{1}{\left\Vert B\right\Vert }$ holds. This is a generalization of an earlier result by Mangasarian and Meyer for the special case where $B=I$. Secondly, a generalized Newton method for solving the AVE is proposed. We show under the condition $\Vert A^{-1}\Vert <\frac{1}{4\Vert B\Vert }$, that the algorithm converges linearly global to the unique solution of the AVE. Numerical results are reported to show the efficiency of the proposed method and to compare with an available method

L∞ estimates on trajectories confined to a closed subset, for control systems with bounded time variation

The term ‘distance estimate’ for state constrained control systems refers to an estimate on the distance of an arbitrary state trajectory from the subset of state trajectories that satisfy a given state constraint. Distance estimates have found widespread application in state constrained optimal control. They have been used to establish regularity properties of the value function, to establish the non-degeneracy of first order conditions of optimality, and to validate the characterization of the value function as a unique solution of the HJB equation. The most extensively applied estimates of this nature are so-called linear L∞L∞ distance estimates. The earliest estimates of this nature were derived under hypotheses that required the multifunctions, or controlled differential equations, describing the dynamic constraint, to be locally Lipschitz continuous w.r.t. the time variable. Recently, it has been shown that the Lipschitz continuity hypothesis can be weakened to a one-sided absolute continuity hypothesis. This paper provides new, less restrictive, hypotheses on the time-dependence of the dynamic constraint, under which linear L∞L∞ estimates are valid. Here, one-sided absolute continuity is replaced by the requirement of one-sided bounded variation. This refinement of hypotheses is significant because it makes possible the application of analytical techniques based on distance estimates to important, new classes of discontinuous systems including some hybrid control systems. A number of examples are investigated showing that, for control systems that do not have bounded variation w.r.t. time, the desired estimates are not in general valid, and thereby illustrating the important role of the bounded variation hypothesis in distance estimate analysis