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Most numerical methods for conic problems use the homogenous primal-dualembedding, which yields a primal-dual solution or a certificate establishingprimal or dual infeasibility. Following Patrinos (and others, 2018), we expressthe embedding as the problem of finding a zero of a mapping containing askew-symmetric linear function and projections onto cones and their duals. Wefocus on the special case when this mapping is regular, i.e., differentiablewith nonsingular derivative matrix, at a solution point. While this is notalways the case, it is a very common occurrence in practice. We propose asimple method that uses LSQR, a variant of conjugate gradients for leastsquares problems, and the derivative of the residual mapping to refine anapproximate solution, i.e., to increase its accuracy. LSQR is a matrix-freemethod, i.e., requires only the evaluation of the derivative mapping and itsadjoint, and so avoids forming or storing large matrices, which makes itefficient even for cone problems in which the data matrices are given anddense, and also allows the method to extend to cone programs in which the dataare given as abstract linear operators. Numerical examples show that the methodalmost always improves an approximate solution of a conic program, and oftendramatically, at a computational cost that is typically small compared to thecost of obtaining the original approximate solution. For completeness wedescribe methods for computing the derivative of the projection onto the conescommonly used in practice: nonnegative, second-order, semidefinite, andexponential cones. The paper is accompanied by an open source implementation.

Solution Refinement at Regular Points of Conic Problems