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

Abstract: In this work, based on the ideas of alternating direction method with multipliers (ADMM) and sequential quadratic programming (SQP), as well as Armijo line search technology, monotone splitting SQP algorithms for two-block nonconvex optimization problems with linear equality, inequality and box constraints are discussed. Firstly, the discussed problem is transformed into an optimization problem with only linear equality and box constraints by introducing slack variables. Secondly, we use the idea of ADMM to decompose the quadratic programming (QP) subproblem. Especially, the QP subproblem corresponding to the introducing slack variable is simple, and it has an explicit optimal solution without increasing computational cost. Thirdly, the search direction is generated by the optimal solutions of the subproblems, and the new iteration point is yielded by Armijo line search with augmented Lagrange function. And the global convergence of the algorithm is analyzed under weaker assumptions. In addition, box constraints are extended to general nonempty closed convex sets, moreover, the global convergence of the corresponding algorithm is also proved. Finally, some preliminary numerical experiments and applications in the mid-to-large-scale economic dispatch problems for power systems are reported, and these show that our proposed algorithm is promising.