Mixed integer nonlinear multiobjective optimization by outer approximations

计划的研究的目的是开发一个确定性的算法求解程序的多目标混合整数非线性优化问题（MOMIPs）。因此，我们的目标是解决同时具有多个目标和整数和连续变量的问题。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。MOMIP联合收割机结合了多目标优化和混合整数非线性规划的所有困难，这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。对于同时存在多个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，而且几乎没有任何一种方法可以提供所找到的解的数学保证。其原因可能是单目标混合整数非线性优化的典型技术不能轻易地转移到多目标：典型的解决方案方法使用减小上界和增加下界的计算，直到差值小于规定的公差。在多目标优化中，通常存在无限多个最优值。此外，这些值是更高维向量空间的元素，并且说最佳（最大）下界和最佳（最小）上界不再是可能的，因为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标量化方法转化为参数依赖的单目标优化问题。然后将得到的单目标混合整数非线性问题用单目标优化技术求解。然而，这种重新制定和解决方案将不得不对各种参数进行，以获得MOMIP的各种解决方案。这是一个消耗时间的弯路，并没有利用获得的信息。如何选择合适的参数也是一个悬而未决的问题。 我们将避免这种标量化的迂回。相反，我们将假设所有的功能出现在MOMIP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。结合适当的上限，这也将是一组点，和一个新的开发终止策略，我们的目标是开发一个数值算法，停止与预定义的质量。

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.