在信息不完全情况下高效计算拟凹包络
Efficiently Computing the Quasiconcave Envelope with Incomplete Information
AI总结：
研究基于有限部分信息近似未知拟凹函数的问题，提出一种两步法高效计算满足附加函数性质的拟凹包络，将赋值与插值问题所需的线性规划求解数量从最坏情况下的指数级分别降至多项式级和对数级。
摘要：
在本文中，我们研究基于有限的部分信息来近似未知的拟凹函数。可用信息包括目标函数在指定点集上的值的下界，以及一些函数性质，包括单调性、Lipschitz连续性、排序和置换不变性。我们考虑满足这些下界并满足这些函数性质的容许拟凹函数类。然后，我们计算容许拟凹函数类中最小的拟凹函数。具体来说，我们展示了如何在附加函数性质的约束下，高效地计算点集数据样本的拟凹包络（QCoE）。求解过程分为两步。首先，求解一个赋值问题，以确定QCoE在给定数据样本上的值。其次，求解一个插值问题，以计算QCoE在其他点上的值。赋值问题和插值问题都带来了一些理论和计算上的挑战，因为它们是非凸且大规模的。这两个问题的MILP重构在最坏情况下需要求解指数级数量的线性规划（LP）。作为我们的主要贡献，我们仅用多项式数量的LP求解赋值问题，然后仅用对数数量的LP求解任意候选点的插值问题。一些初步的数值测试表明所提出的方法是高效且恰当的。
英文摘要：
In this paper, we study the approximation of an unknown quasiconcave function based on limited partial information. Available information includes lower bounds on the values of the target function at a specified set of points, as well as some functional properties including monotonicity, Lipschitz continuity, ranking, and permutation invariance. We consider the class of admissible quasiconcave functions that dominate these lower bounds and satisfy these functional properties. We then compute the smallest quasiconcave function among the class of admissible quasiconcave functions. Specifically, we show how to efficiently compute the quasiconcave envelope (QCoE) of a data sample of points, subject to the additional functional properties. The solution procedure takes two steps. First, a value problem is solved to determine the values of the QCoE on the given data sample. Second, an interpolation problem is solved to compute the values of the QCoE on other points. Both the value problem and the interpolation problem introduce some theoretical and computational challenges, as they are non-convex and large-scale. The MILP reformulations of both problems require an exponential number of linear programs (LPs) to be solved in the worst-case. As our main contribution, we solve the value problem with only a polynomial number of LPs, and then solve the interpolation problem for any candidate point with only a logarithmic number of LPs. Some preliminary numerical tests show that the proposed approach is efficient and proper.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
