A new bundle trust region method based on linear subproblems for nonsmooth and nonconvex semi-infinite programming

In this paper, a new bundle trust region method for nonsmooth and nonconvex seminfinite programming (SIP) problems is presented. This newly developed method imposes weaker assumptions on the structure of the objective function in comparison with the existing numerical methods for solving SIP. Furthermore, in contrast to most of the existing bundle methods, such as proximal bundle methods, level bundle methods, and bundle trust region methods, which all necessitate the solution of quadratic programming subproblems, the new algorithm proposed herein only requires the solution of linear programming subproblems. This substitution can reduce the computational cost. The new bundle trust region method addresses the challenge of SIP having infinitely many constraints by leveraging a relaxation approach to derive an upper bound problem of SIP. Based on this upper bound problem, a local approximation model is constructed. At each iteration, by virtue of the approximation model, the trust region technique, and the infinity norm, a linear subproblem is formulated. Under reasonable assumptions, the iteration sequence generated by the new algorithm converges to an ϵ-KKT point of the SIP. The numerical results confirm the effectiveness of the new algorithm.