In this article, we discuss an exact algorithm for solving mixed integer concave minimization problems. A piecewise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program, which provides a lower bound to the original problem. The bilevel program is reduced to a single level formulation with the help of Karush-Kuhn-Tucker (KKT) conditions. Incorporating the KKT conditions lead to complementary slackness conditions that are linearized using BigM, for which we identify a tight value for general problems. Multiple bilevel programs, when solved over iterations, guarantee convergence to the exact optimum of the original problem. Though the algorithm is general and can be applied to any optimization problem with concave function(s), in this paper, we solve two common classes of operations and supply chain problems; namely, the concave knapsack problem, and the concave production-transportation problem. The computational experiments indicate that our proposed approach outperforms the customized methods that have been used in the literature to solve the two classes of problems by an order of magnitude in most of the test cases.

翻译：在文章中,我们讨论解决混合整数最小化问题的精确算法。 使用一个辅助线性程序实现对组合函数的细微内部偏近, 从而导致一个双级程序, 它为最初的问题提供了较低的约束。 双级程序在Karush- Kuhn- Tucker (KKT) 帮助下, 将双级程序降低到一个单一层次的配方。 纳入 KKT 条件导致使用 BigM 线性化的补齐性松懈状态, 我们为此确定了对一般问题的严格价值。 多个双级程序, 当通过迭代解决时, 保证与原问题的精确最佳结合。 尽管算法是一般性的, 并且可以适用于与组合函数( s) 有关的任何优化问题。 在本文中, 我们解决了两种共同的操作和供应链问题, 即 concave knapsack 问题, 和 concave 生产- 运输问题。 计算实验表明, 我们提出的方法超过了文献中使用的定制方法, 在大多数情况下用数量级解决两个问题。