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首页> 外文期刊>SIAM Journal on Optimization: A Publication of the Society for Industrial and Applied Mathematics >A SPARSITY PRESERVING CONVEXIFICATION PROCEDURE FOR INDEFINITE QUADRATIC PROGRAMS ARISING IN DIRECT OPTIMAL CONTROL
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A SPARSITY PRESERVING CONVEXIFICATION PROCEDURE FOR INDEFINITE QUADRATIC PROGRAMS ARISING IN DIRECT OPTIMAL CONTROL

机译：在直接最佳控制中产生的无限二次方案的稀疏性凸起程序

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摘要

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 de finite 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.
机译：具有无限期的Hessian矩阵的二次程序（QP）在一些直接的最佳控制方法中，例如，作为顺序二次编程方案中的子问题。通常，Hessian用正DE有限矩阵近似以确保具有独特的解决方案;这样的程序被称为正规化。我们提出了一种针对QPS定制的新型正则化方法，具有最佳控制结构。我们的方法展现了三个主要优势。首先，当QP满足最优性的二阶足够条件时，原始问题和正则化问题的原始解决方案是相等的。此外，该算法以方便的方式恢复双解决方案。其次，更重要的是，正规化的奇森熊在与原始的相同的诽谤结构上。这允许使用有效的结构利用QP溶剂。作为第三个优点，可以以计算复杂度来执行正则化，该计算复杂度在控制范围的长度中线性缩放。我们在非线性最优控制的数值示例中展示了我们的正则化算法的属性。将结果与其他稀疏性保存的正则化方法进行比较。

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