We study the problem of minimizing a convex function on a nonempty, finite subset of the integer lattice when the function cannot be evaluated at noninteger points. We propose a new underestimator that does not require access to (sub)gradients of the objective; such information is unavailable when the objective is a blackbox function. Rather, our underestimator uses secant linear functions that interpolate the objective function at previously evaluated points. These linear mappings are shown to underestimate the objective in disconnected portions of the domain. Therefore, the union of these conditional cuts provides a nonconvex underestimator of the objective. We propose an algorithm that alternates between updating the underestimator and evaluating the objective function. We prove that the algorithm converges to a global minimum of the objective function on the feasible set. We present two approaches for representing the underestimator and compare their computational effectiveness. We also compare implementations of our algorithm with existing methods for minimizing functions on a subset of the integer lattice. We discuss the difficulty of this problem class and provide insights into why a computational proof of optimality is challenging even for moderate problem sizes.

当无法在非整数点求值时，我们研究最小化整数格子的非空有限子集上的凸函数的问题。我们提出了一个新的低估器，它不需要访问目标的（子）梯度；当目标是黑盒功能时，此类信息将不可用。相反，我们的低估器使用割线线性函数，该函数在先前求值的点处插值目标函数。这些线性映射显示为低估了域的不连续部分中的目标。因此，这些有条件切割的并集提供了物镜的非凸低估量。我们提出了一种在更新低估器和评估目标函数之间交替的算法。我们证明算法在可行集上收敛到目标函数的全局最小值。我们提出了两种表示低估的方法，并比较了它们的计算效果。我们还将算法的实现方式与现有方法（用于最小化整数格子集上的函数）进行比较。我们讨论了此类问题的难点，并提供了对为什么即使对于中等规模的问题，最优性的计算证明也具有挑战性的见解。