Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.

中文翻译：

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何时具有约束不确定性等价的静态和可调稳健优化问题？

可调稳健优化 (ARO) 通常比静态稳健优化 (RO) 产生更好的最坏情况解决方案。然而，ARO 在计算上比 RO 更困难。在本文中，我们提供了 ARO 和 RO 问题的最坏情况目标值相等的条件。我们证明了当不确定性是constraint-wise时，问题对于可调变量是凸的，对于不确定参数是凹的，可调变量位于凸紧集，不确定集是凸紧集，那么稳健的解决方案对于相应的 ARO 问题也是最佳的。此外，我们证明，如果一些不确定参数是受约束的，而其余的不是，那么在一组类似的假设下，对于 ARO 问题有一个最佳决策规则，它不依赖于约束方式的不确定参数。此外，我们针对一类问题表明，使用依赖于所有不确定参数的仿射决策规则产生与规则仅依赖于非约束不确定参数时相同的最佳目标值。最后，我们通过将这些结果应用于凸二次和二次二次问题来说明这些结果的有用性。