  An optimization algorithm for a group of nonsmooth nonconvex problems
inspired by two-stage stochastic programming problems is proposed. The main
challenges for these problems include (1) the problems lack the popular
lower-type properties such as prox-regularity assumed in many nonsmooth
nonconvex optimization algorithms, (2) the objective can not be analytically
expressed and (3) the evaluation of function values and subgradients are
computationally expensive. To address these challenges, this report first
examines the properties that exist in many two-stage problems, specifically
upper-C^2 objectives. Then, we show that quadratic penalty method for
security-constrained alternating current optimal power flow (SCACOPF)
contingency problems can make the contingency solution functions upper-C^2 .
Based on these observations, a simplified bundle algorithm that bears
similarity to sequential quadratic programming (SQP) method is proposed. It is
more efficient in implementation and computation compared to conventional
bundle methods. Global convergence analysis of the algorithm is presented under
novel and reasonable assumptions. The proposed algorithm therefore fills the
gap of theoretical convergence for some smoothed SCACOPF problems. The
inconsistency that might arise in our treatment of the constraints are
addressed through a penalty algorithm whose convergence analysis is also
provided. Finally, theoretical capabilities and numerical performance of the
algorithm are demonstrated through numerical examples.
