Monotone Splitting SQP Algorithms for Two-block Nonconvex Optimization Problems with General Linear Constraints and Applications

This work discusses a class of two-block nonconvex optimization problems with linear equality, inequality and box constraints. Based on the ideas of alternating direction method with multipliers (ADMM), sequential quadratic programming (SQP) and Armijo line search technique, we propose a novel monotone splitting SQP algorithm. First, the discussed problem is transformed into an optimization problem with only linear equality and box constraints by introduction of slack variables. Second, the idea of ADMM is used to decompose the traditional quadratic programming (QP) subproblem. In particular, the QP subproblem corresponding to the introduction of the slack variable is simple, and it has an explicit optimal solution without increasing the computational cost. Third, the search direction is generated by the optimal solutions of the subproblems, and the new iteration point is yielded by an Armijo line search with augmented Lagrange function. Fourth, the multiplier is updated by a novel approach that is different from the ADMM. Furthermore, the algorithm is extended to the associated optimization problem where the box constraints can be replaced by general nonempty closed convex sets. The global convergence of the two proposed algorithms is analyzed under weaker assumptions. Finally, some preliminary numerical experiments and applications in mid-to-large-scale economic dispatch problems for power systems are reported, and these show that the proposed algorithms are promising.