A Sparsity Preserving Convexification Procedure for Indefinite Quadratic Programs Arising in Direct Optimal Control
针对直接最优控制中出现的不定Hessian矩阵二次规划，提出一种新的正则化方法，保持原问题的稀疏结构，且计算复杂度随控制时域长度线性增长，适用于高效求解器。
Quadratic programs (QP) with an indefinite Hessian matrix arise naturally in some direct optimal control methods, e.g., as subproblems in a sequential quadratic programming scheme. Typically, the Hessian is approximated with a positive definite matrix to ensure having a unique solution; such a procedure is called regularization. We present a novel regularization method tailored for QPs with optimal control structure. Our approach exhibits three main advantages. First, when the QP satisfies a second order sufficient condition for optimality, the primal solution of the original and the regularized problem are equal. In addition, the algorithm recovers the dual solution in a convenient way. Second, and more importantly, the regularized Hessian bears the same sparsity structure as the original one. This allows for the use of efficient structure-exploiting QP solvers. As a third advantage, the regularization can be performed with a computational complexity that scales linearly in the length of the control horizon. We showcase the properties of our regularization algorithm on a numerical example for nonlinear optimal control. The results are compared to other sparsity preserving regularization methods.