Adjustable robust optimization (ARO) involves recourse decisions (i.e. reactive actions after the realization of the uncertainty, ‘wait-and-see’) as functions of the uncertainty, typically posed in a two-stage stochastic setting. Solving the general ARO problems is challenging, therefore ways to reduce the computational effort have been proposed, with the most popular being the affine decision rules, where ‘wait-and-see’ decisions are approximated as affine adjustments of the uncertainty. In this work we propose a novel method for the derivation of generalized affine decision rules for linear mixed-integer ARO problems through multi-parametric programming, that lead to the exact and global solution of the ARO problem. The problem is treated as a multi-level programming problem and it is then solved using a novel algorithm for the exact and global solution of multi-level mixed-integer linear programming problems. The main idea behind the proposed approach is to solve the lower optimization level of the ARO problem parametrically, by considering ‘here-and-now’ variables and uncertainties as parameters. This will result in a set of affine decision rules for the ‘wait-and-see’ variables as a function of ‘here-and-now’ variables and uncertainties for their entire feasible space. A set of illustrative numerical examples are provided to demonstrate the potential of the proposed novel approach.

中文翻译：

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通过多参数编程进行可调整的鲁棒优化

可调鲁棒优化（ARO）涉及作为不确定性函数的追索性决策（即，实现不确定性后的反应性行为，“等待和看”），通常在两阶段随机环境中提出。解决一般的ARO问题具有挑战性，因此已经提出了减少计算量的方法，其中最流行的是仿射决策规则，其中“等待和看”决策近似为不确定性的仿射调整。在这项工作中，我们提出了一种新颖的方法，可以通过多参数编程来推导线性混合整数ARO问题的广义仿射决策规则，从而可以精确，全局地求解ARO问题。该问题被视为多级编程问题，然后使用一种新颖的算法对其进行了求解，以实现多级混合整数线性规划问题的精确和全局解。该方法背后的主要思想是通过考虑“此时此地”的变量和不确定性作为参数来解决ARO问题的较低优化级别。这将导致针对“等待和看”变量的一组仿射决策规则，这是“现在和现在”变量及其整个可行空间的不确定性的函数。提供了一组说明性的数值示例，以演示所提出的新颖方法的潜力。该方法背后的主要思想是通过考虑“此时此地”的变量和不确定性作为参数来解决ARO问题的较低优化级别。这将导致针对“等待和看”变量的一组仿射决策规则，这是“现在和现在”变量及其整个可行空间的不确定性的函数。提供了一组说明性的数值示例，以演示所提出的新颖方法的潜力。该方法背后的主要思想是通过考虑“此时此地”的变量和不确定性作为参数来解决ARO问题的较低优化级别。这将导致针对“等待和看”变量的一组仿射决策规则，这是“现在和现在”变量及其整个可行空间的不确定性的函数。提供了一组说明性的数值示例，以演示所提出的新颖方法的潜力。