Supportedness in Multiobjective Optimization

虽然加权和标量化是一种常用的多目标优化方法，但众所周知，通常不是所有的解都可以用这种方法找到，而只能找到支持的解。保证所有解的凸性的一个标准假设是多目标问题的凸性。这个项目的第一个主要目的是确定一个显着更大的类比凸多目标问题，其中所有的解决方案都支持，因此可以计算加权和标量化。我们称之为多目标优化问题的支持。由于上述凸性假设的基本结果是所谓的上像集的凸性，因此隐藏凸问题也具有所需的性质，我们指的是尽管是非凸的，但具有凸上像集的问题。此外，一个多目标问题，甚至可以支持，虽然上图像集是非凸的。我们的目标是更好地理解支持的多目标优化问题，在上述意义上，是一类凸和一般非凸多目标优化。在这种情况下，我们还研究松弛技术，如共正或半定的重新制定，多目标优化。 我们的目的是了解哪些松弛可能是有前途的多目标优化和支持的解决方案是紧的，也可以通过第一次标量化与加权和，然后应用单目标松弛技术。该项目的第二个主要目的是基于观察到的多目标问题的解决方案的不确定性不是一个内在的属性，但可以通过一定的图像空间变换技术的一些甚至所有的解决方案。经过深入研究这种技术，我们的目标是设计算法上可行的结构，将多目标问题与非支持的解决方案到类的支持问题。我们称可能存在隐藏支持的问题。转换技术承认的基础上，如本森的方法和某些方法的混合整数线性多目标问题的重复性算法的推广。这特别导致了非线性切割的产生，这可能会促进分支定界框架中的更紧边界。该项目的主要目的是探索非线性的理论和算法方面，从隐凸性和隐不凸性的检测，到松弛的精确性的研究，以及支撑有效点的构造算法的推广，不支持的人。

Although weighted sum scalarization is a popular method to find solutions of multiobjective optimization problems, it is well-known that in general not all solutions can be found this way, but only the supported ones. A standard assumption for guaranteeing the supportedness of all solutions is the convexity of the multiobjective problem. The first main aim of this project is to identify a significantly larger class than convex multiobjective problems, for which all solutions are supported and can thus be computed by weighted sum scalarization. We call such multiobjective optimization problems supported. Since the essential consequence of the above convexity assumption is the convexity of the so-called upper image set, also hidden convex problems possess the desired property, by which we mean problems which possess a convex upper image set despite being nonconvex. Moreover, a multiobjective problem may even be supported although the upper image set is nonconvex. We aim at a better understanding of the class of supported multiobjective optimization problems which, in the above sense, is a category between convex and general nonconvex multiobjective optimization. In this context we also study relaxation techniques, like copositive or semidefinite reformulations, for multiobjective optimization. We aim on understanding which relaxations may be promising for multiobjective optimization and which are tight in the supported solutions only and can also be obtained by first scalarizing with a weighted sum and then applying single-objective relaxation techniques. The second main aim of this project bases on the observation that supportedness of solutions of multiobjective problems is not an intrinsic property but may be enforced by a certain image space transformation technique for some or even for all solutions. After a thorough study of this technique, we aim at designing algorithmically feasible constructions which move multiobjective problems with nonsupported solutions into the class of supported problems. We call problems for which this is possible hidden supported. The transformation technique admits generalizations of algorithms based on supportedness like Benson's method and certain approaches for mixed-integer linear multiobjective problems. This leads in particular to the generation of nonlinear cuts which may promote tighter bounds in branch-and-bound frameworks.This the main aims of the project are to explore theoretical and algorithmic aspects of supportedness, ranging from the detection of hidden convexity and of hidden supportedness to the investigation of exactness of relaxations as well as the generalization of algorithms for the construction of supported efficient points to the case of nonsupported ones.