Hierarchical least-squares programs with linear constraints (HLSP) are a type of optimization problem very common in robotics. Each priority level contains an objective in least-squares form which is subject to the linear constraints of the higher priority levels. Active-set methods are a popular choice for solving them. However, they can perform poorly in terms of computational time if there are large changes of the active set. We therefore propose a computationally efficient primal-dual interior-point method (IPM) for dense HLSP's which is able to maintain constant numbers of solver iterations in these situations. We base our IPM on the computationally efficient nullspace method as it requires only a single matrix factorization per solver iteration instead of two as it is the case for other IPM formulations. We show that the resulting normal equations can be expressed in least-squares form. This avoids the formation of the quadratic Lagrangian Hessian and can possibly maintain high levels of sparsity. Our solver reliably solves ill-posed instantaneous hierarchical robot control problems without exhibiting the large variations in computation time seen in active-set methods.

翻译：具有线性约束的层级最小二乘规划（HLSP）是机器人领域中非常常见的一类优化问题。每个优先级层级包含一个最小二乘形式的目标函数，该目标函数受到更高优先级层级线性约束的制约。活动集方法是求解此类问题的常用方法，但当活动集发生较大变化时，其计算时间可能表现不佳。因此，我们针对稠密HLSP问题提出了一种计算高效的原始-对偶内点法（IPM），该方法能够在这些情况下保持恒定的求解器迭代次数。我们将IPM建立在计算高效的空空间方法基础上，因为该方法每次求解器迭代仅需一次矩阵分解，而非其他IPM公式中的两次。我们证明了由此产生的法方程可以表示为最小二乘形式，这避免了二次拉格朗日海森矩阵的构建，并可能保持高度稀疏性。我们的求解器能够可靠地求解不适定的瞬时层级机器人控制问题，且不会出现活动集方法中常见的计算时间大幅波动现象。