Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist known impossibility results for approximating the optimal solution when constraints for covering SDPs arrive in an online fashion. In this paper, we study online covering linear and semidefinite programs in which the algorithm is augmented with advice from a possibly erroneous predictor. We show that if the predictor is accurate, we can efficiently bypass these impossibility results and achieve a constant-factor approximation to the optimal solution, i.e., consistency. On the other hand, if the predictor is inaccurate, under some technical conditions, we achieve results that match both the classical optimal upper bounds and the tight lower bounds up to constant factors, i.e., robustness. More broadly, we introduce a framework that extends both (1) the online set cover problem augmented with machine-learning predictors, studied by Bamas, Maggiori, and Svensson (NeurIPS 2020), and (2) the online covering SDP problem, initiated by Elad, Kale, and Naor (ICALP 2016). Specifically, we obtain general online learning-augmented algorithms for covering linear programs with fractional advice and constraints, and initiate the study of learning-augmented algorithms for covering SDP problems. Our techniques are based on the primal-dual framework of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and can be further adjusted to handle constraints where the variables lie in a bounded region, i.e., box constraints.
翻译:半精度编程( SDP) 是一个统一框架, 它既概括线性编程, 也概括了线性编程, 也概括了四进制的二次编程, 同时在理论和实践上产生了高效的解答器。 然而, 当覆盖 SDP 的制约以在线方式出现时, 存在已知无法达到最佳解决方案的最佳解决方案。 在本文中, 我们研究线性和半定量程序, 其中算法由可能错误的预测器提供的建议来增加算法。 我们显示, 如果预测器准确, 我们就能有效地绕过这些不可能的结果, 并实现与最佳解决方案( 即, 一致性。 ) 另一方面, 如果预测器不准确, 在某些技术条件下, 我们就能取得符合典型的最佳上限和紧紧紧的下限的解决方案。 在本文中, 我们推出一个框架, 将在线设置覆盖了基于机器的预测器的问题, 由Bamas、 Magggiori 和 Svensson ( NeurIPS 2020) 实现一个不变的近效近似点,, 而 覆盖Sal- Deal Exal Exal Proview Proview Produal Produstrational Produstrisl Produstrationalalal Produstrational Produstrationalalalal Produstrationalalalalal Prostrismal Prostrismalal roisal 。