On unique solvability of linear complementarity problems, horizontal linear complementarity problems and an n-absolute value equations

Abstract

The complementarity problems is getting a lot of attention because it is connected to real-world problems in scientific computing and engineering. It shows up in various situations like linear and quadratic programming, two person games, circuit simulation, optimal stopping in Markov chains, contact problems with friction, finding a Nash-equilibrium in bimatrix games. The linear complementarity problems (LCP) and absolute value equations (AVE) have an equivalence relation; that is, the AVE can be transformed into an LCP and vice versa. The relationship between LCP and AVE enables the conversion of one problem into another, offering different perspectives for analysis and solution. This equivalence aids in theoretical understanding and the development of numerical methods applicable to both mathematical formulations. In the present study, we discuss the unique solvability of the LCP and the horizontal linear complementarity problems (HLCP). Some superior unique solvability conditions are obtained for LCP and HLCP. The unique solvability of the n-absolute value equations 𝐴𝑛𝑥−𝐵𝑛|𝑥| = 𝑏 is also discussed. Some examples are highlighted for improving the current conditions of unique solutions for absolute value equations.

Keywords

• Absolute Value Equations
• Horizontal Linear Complementarity Problems
• Linear Complementarity Problems
• Unique Solvability

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