A smoothing and regularization Broyden-like method for nonlinear inequalities

In this paper, we consider the smoothing and regularization Broyden-like algorithm for the system of nonlinear inequalities. By constructing a new smoothing function $\phi(\mu,a)=\frac{1}{2}(a+\mu(\ln2+\ln(1+\cosh\frac{a}{\mu})))$ , the problem is approximated via a family of parameterized smooth equations H(μ,ε,x)=0. A smoothing and regularization Broyden-like algorithm with a non-monotone linear search is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The global convergence of the algorithm is established under suitable assumptions. In addition, the smoothing parameter μ and the regularization parameter ε in our algorithm are viewed as two different independent variables. Preliminary numerical results show the efficiency of the algorithm and reveal that the regularization parameter ε in our algorithm plays an important role in numerical improvement, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods.