Solution refinement at regular points of conic problems
Solution refinement at regular points of conic problems
复制标题
圆锥曲线问题常规点的解细化
DOI：
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发表时间：
2018
影响因子：
2.2
通讯作者：
Stephen P. Boyd
中科院分区：
文献类型：
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作者：
Enzo Busseti;W. M. Moursi;Stephen P. Boyd
许多求解圆锥曲线问题的数值方法使用齐次原始-对偶嵌入，这会产生原始-对偶解或证明原始或对偶不可行性。根据Themelis和Patrinos（IEEE Trans Autom Control，2019），我们将嵌入表示为找到包含反对称线性函数的映射的零点以及到锥及其顶点的投影的问题。我们专注于特殊情况下，当这个映射是正规的，即，可微的非奇异导数矩阵，在一个解决方案点。虽然这并不总是如此，但在实践中非常常见。在本文中，我们不追求新的理论结果。相反，我们提出了一种简单的方法，该方法使用LSQR，最小二乘问题的共轭梯度的变体，以及残差映射的导数来细化近似解，即，以提高其准确性。LSQR是一种无矩阵方法，即，只需要评估的衍生物映射和它的伴随，并因此避免形成或存储大型矩阵，这使得它有效的，甚至锥问题中的数据矩阵是给定的和密集的，也允许该方法扩展到锥程序中的数据是作为抽象的线性算子。数值例子表明，该方法提高了一个圆锥规划的近似解，往往显着，在计算成本通常是小的成本相比，获得原始的近似解。为了完整性，我们描述的方法来计算导数的投影到锥上常用的实践中：非负的，二阶，半定的，指数锥。该文件是伴随着一个开放源代码的实现。
Many numerical methods for conic problems use the homogenous primal–dual embedding, which yields a primal–dual solution or a certificate establishing primal or dual infeasibility. Following Themelis and Patrinos (IEEE Trans Autom Control, 2019), we express the embedding as the problem of finding a zero of a mapping containing a skew-symmetric linear function and projections onto cones and their duals. We focus on the special case when this mapping is regular, i.e., differentiable with nonsingular derivative matrix, at a solution point. While this is not always the case, it is a very common occurrence in practice. In this paper we do not aim for new theorerical results. We rather propose a simple method that uses LSQR, a variant of conjugate gradients for least squares problems, and the derivative of the residual mapping to refine an approximate solution, i.e., to increase its accuracy. LSQR is a matrix-free method, i.e., requires only the evaluation of the derivative mapping and its adjoint, and so avoids forming or storing large matrices, which makes it efficient even for cone problems in which the data matrices are given and dense, and also allows the method to extend to cone programs in which the data are given as abstract linear operators. Numerical examples show that the method improves an approximate solution of a conic program, and often dramatically, at a computational cost that is typically small compared to the cost of obtaining the original approximate solution. For completeness we describe methods for computing the derivative of the projection onto the cones commonly used in practice: nonnegative, second-order, semidefinite, and exponential cones. The paper is accompanied by an open source implementation.