Pseudotransient continuation for solving systems of nonsmooth equations with inequality constraints (Q613599)

Let \(f:\mathbb{R}^{n}\longrightarrow \mathbb{R}^{n}\) and \(g:\mathbb{R} ^{n}\longrightarrow \mathbb{R}^{m}\) be semismooth functions. The authors introduce a slack variable \(\gamma \in \mathbb{R}^{m}\) to reformulate the system \(f(x)=0,\) \(g(x)\leq 0\) as \(F(u)=0,\) \(\gamma \geq 0\), with \( u=(x,\gamma )\) and \(F(u)=(f(x),g(x)+\gamma )\). Among other assumptions, they impose on \(F\) a BD-regularity condition in the sense of \textit{J. M. Martinez} and the second author [J. Comput. Appl. Math. 60, 127--145 (1995; Zbl 0833.65045)], which allows for the applicability of the projected pseudotransient continuation algorithm of \textit{C. T. Kelley, Li-Zhi Liao, Liqun Qi, Moody T. Chu, J. P. Reese} and \textit{C. Winton} [SIAM J. Numer. Anal. 46, No.~6, 3071--3083 (2008; Zbl 1180.65060)] to solve the reformulated system. A global convergence theorem is proved and the results of some numerical tests are reported.