An adaptive competitive penalty method for nonsmooth constrained optimization (Q526728)

The article is a valuable contribution to nonsmooth constrained optimization. The authors develop a competitive algorithm to minimize a locally Lipschitz function constrained with locally Lipschitz constraints which is to use the \(l_I\) nonsmooth penalty function. By using second-order descent directions, the method minimizes the \(\widetilde H_k\) functions. A new adaptive approximation method for approximating Goldstein subdifferentials which lead to reduce functions and gradient evaluations significantly by using the gradient of the \(l_I\) penalty function. This article is well written, structured and explained, it contains seven sections: Section 1 on Introduction, Section 2 on Background and motivation and Section 3 on Computing descent directions, Section 4 on Boundedness of the matrices \(\widetilde H_k\) and \(\bar H_k\), Section 5 on Global convergence, Section 6 on Numerical experiments and Section 7 on Conclusion.