Mixed integer nonlinear multiobjective optimization by outer approximations

计划研究的目的是开发多目标混合整数非线性优化问题（MOMIPs）的确定性算法求解程序。因此，我们的目标是同时解决具有多个目标和整数变量和连续变量的问题。这些类型的优化问题出现在许多应用领域，如位置或生产计划、化学工程、金融和制造业。多目标非线性规划问题结合了多目标优化和混合整数非线性规划的所有困难，是一类理论上困难的问题（np完全）。这些问题本质上是非凸的，因此需要全局优化技术。因此，设计有效的求解方法是一个很大的挑战。对于同时出现多目标和混合整数变量的问题，迄今为止只有非常有限的方法存在，而且几乎没有任何方法可以为找到的解决方案提供数学保证。其原因可能是单目标混合整数非线性优化的典型技术不容易转移到多目标：典型的解决方法使用减小上界和增大下界的计算，直到差值小于规定的容差。在多目标优化中，通常存在无穷多个最优值。此外，这些值是高维向量空间的元素，并且谈论最佳（最大）下界和最佳（最小）上界不再是可能的，因为在图像空间中只有部分排序是可能的。多目标优化问题可以通过所谓的标量化方法重新表述为参数相关的单目标问题。然后利用单目标优化技术求解得到的单目标混合整数非线性问题。然而，为了得到MOMIP的不同解，必须对不同的参数进行重新表述和求解。这是一个耗费时间的弯路，并没有利用已获得的信息。如何选择合适的参数也是一个悬而未决的问题。我们将避免这种绕弯路的标量化。相反，我们将假设MOMIP中出现的所有函数至少是凸的，我们将检查多面体松弛以获得下界。这些边界现在是点的集合。结合合适的上界（也将是点的集合）和一种新开发的终止策略，我们的目标是开发一种以预定义质量停止的数值算法。

The aim of the planned research is to develop a deterministic algorithmic solution procedure for multiobjective mixed integer nonlinear optimization problems (MOMIPs). Thus, we aim to solve problems which have multiple objectives and integer and continuous variables at the same time. These types of optimization problems arise in many application fields such as location or production planning, chemical engineering, finance, and manufacturing. MOMIPs combine all the difficulties of both, multiobjective optimization and mixed integer nonlinear programming, which are among the class of theoretically difficult problems (NP-complete). These problems are intrinsically nonconvex and thus require global optimization techniques. Therefore, the design of efficient solution methods is a big challenge.For problems in which both challenges occur, multiple objectives and mixed integer variables, only a very limited number of approaches exist so far and hardly any of those deliver a mathematical guarantee for the found solutions. A reason for that might be that the typical techniques from single-objective mixed integer nonlinear optimization cannot be transferred to multiple objectives easily: the typical solution approaches use the computation of decreasing upper bounds and increasing lower bounds till the difference is less than a prescribed tolerance. In multiobjective optimization, in general an infinite number of optimal values exist. Moreover, the values are elements of a higher dimensional vector space, and speaking of the best (largest) lower bound and the best (smallest) upper bound is no longer possible, as only a partial ordering in the image space is possible. Multiobjective optimization problems can be reformulated as parameter-dependent single-objective problems by so called scalarization approaches. Then the obtained single-objective mixed-integer nonlinear problems can be solved by techniques from single-objective optimization. However, this reformulation and solution would have to be done for various parameters to obtain various solutions of the MOMIP. This is time a consuming detour and does not make use of gained information. It is also an open question on how to choose the parameters suitable. We will avoid this detour of scalarization. Instead, as we will assume that all functions appearing in the MOMIP are at least convex, we will examine polyhedral relaxations to obtain lower bounds. These bounds are now sets of points. Combined with suitable upper bounds, which will also be sets of points, and a new-to-develop termination strategy, we aim for developing a numerical algorithm, which stops with a predefined quality.