Improved McCormick Relaxations for the efficient Global Optimization in the Space of Degrees of Freedom

该项目的目标是为确定性全局优化问题开发一种先进的程序。我们的目标是扩展McCormicks构造凹凸弛豫的技术，以便直接构造给定问题的有效仿射弛豫。我们将新开发的方法与其他放松技术进行比较。McCormicks方法保留了原有的变量空间，而其他松弛技术则通过在松弛过程中引入辅助变量来增加自由度空间。我们期望在全局优化领域中求解原变量空间中的仿射松弛可以提高计算效率，从而减少基于构造凹凸松弛的算法的总体计算时间。为了获得有效的仿射弛豫，将开发四种方法，并与当前最先进的弛豫方法进行比较。为了进行比较，将在数值库中找到的用于全局优化的给定基准案例上实施和测试所提出的想法。此外，这些想法将在创新的过程系统工程公式中进行测试。

The objective of the project is to develop an advanced procedure for deterministic global optimization problems. We aim to extend McCormicks technique for construction of convex and concave relaxations in order to directly construct valid affine relaxations of the given problem. We compare the newly developed methods to other relaxation techniques. McCormicks procedure retains the original variable space while other relaxation techniques increase the space of degrees of freedom by introducing auxiliary variables during the relaxation process. We expect that solving the affine relaxations in the original variable space will result in higher computational efficiency and thus decrease the overall computation time of algorithms based on the construction of convex and concave relaxations in the field of global optimization. Four approaches in order to obtain valid affine relaxations will be developed and compared to current state-of-the-art relaxation methods. To enable the comparison, the proposed ideas will be implemented and tested on given benchmark cases found in numerical libraries for global optimization. Furthermore, the ideas will be tested in innovative process systems engineering formulations.