A basic requirement for a mathematical model is often that its solution (output) shouldn't change much if the model's parameters (input) are perturbed. This is important because the exact values of parameters may not be known and one would like to avoid being mislead by an output obtained using incorrect values. Thus, it's rarely enough to address an application by formulating a model, solving the resulting optimization problem and presenting the solution as the answer. One would need to confirm that the model is suitable, i.e., "good," and this can, at least in part, be achieved by considering a family of optimization problems constructed by perturbing parameters of concern. The resulting sensitivity analysis uncovers troubling situations with unstable solutions, which we referred to as "bad" models, and indicates better model formulations. Embedding an actual problem of interest within a family of problems is also a primary path to optimality conditions as well as computationally attractive, alternative problems, which under ideal circumstances, and when properly tuned, may even furnish the minimum value of the actual problem. The tuning of these alternative problems turns out to be intimately tied to finding multipliers in optimality conditions and thus emerges as a main component of several optimization algorithms. In fact, the tuning amounts to solving certain dual optimization problems. In this tutorial, we'll discuss the opportunities and insights afforded by this broad perspective.

翻译：数学模型的基本要求往往是,如果模型参数(投入)受到干扰,其解决方案(产出)不应发生很大变化。这很重要,因为参数的确切值可能不为人所知,人们希望避免被使用不正确值获得的产出误导。因此,它很少足以通过制定模型、解决由此产生的优化问题和提出解决方案作为答案来解决应用问题。人们需要确认模型是否合适,即“良好”,这至少可以部分地通过考虑通过渗透性关切参数构建的优化问题组合来实现。由此产生的敏感度分析揭示出不稳定的解决方案的麻烦情况,我们称之为“坏”模型,并指明更好的模型配方。在问题组合中嵌入实际利益问题,也是实现最佳性条件的主要途径,以及具有计算吸引力的替代问题,在理想情况下,当适当调整时,甚至可以通过考虑实际问题的最低价值来实现这一点。这些替代问题的调整结果与寻找广义的解决方案密切相关,我们称之为“坏”模型,并指明更好的模型配方。 将实际利益问题嵌入到一个家庭里,也是实现最佳性条件和双轨问题,从而通过优化的双重方法来解决我们最佳化的双重问题。