Further Investigation of the Nonlinear Rescaling Principle in Constrained Optimization

摘要本文的主要目的是发展非线性重标度（NR）理论的新方面和基于该理论的求解约束优化问题的新方法。我们的第二个目标是开发软件，该软件将考虑到NR方法中典型的无约束优化问题的特殊结构。解决现实世界的大规模非线性优化问题是我们的第三个主要目标。解决大规模非线性优化问题的能力在力学、结构工程、最优控制、断层扫描、图像识别、电力系统优化和经济学等领域都是至关重要的。近年来，NR型方法在求解非常复杂的桁架拓扑设计（TTD）问题上取得了巨大的成功。TTD问题出现在结构优化中，特别是在创建桥梁、悬臂和能够在不同载荷情况下承受外部载荷的飞机机翼内骨架时。构造一个既能承受外力，又具有一定刚度特性，同时成本最小的结构是一个典型的大规模非线性约束优化问题。我们的研究旨在找到解决这些问题的有效方法。

9705672 Polyak ABSTRACT The main purpose of this proposal is to develop new aspects of the Nonlinear Rescaling (NR) theory and new methods, that are based on this theory, for solving constrained optimization problems. Developing software, which will take into account the special structure of unconstrained optimization problems typical for the NR approach is our second goal. Solving real world large scale nonlinear optimization problems is our third main target. The ability to solve large scale nonlinear optimization problems is critical in mechanics, structural engineering, optimal control, tomography, image recognition, power system optimization, and economics, to mention a few. Recently the NR type methods have been used with great success for solving very complex Truss Topology Design (TTD) problems. The TTD problems arise in structural optimization, in particular, in creating bridges, cantilevers, and inner skeletons of airplane wings capable of carrying external loads under different loading scenarios. Funding a structure, which is able to withstand external forces, have certain characteristics of rigidity and at the same time to have minimal cost is a typical large scale nonlinear constrained optimization problem. Our research is aimed to find efficient methods for solving such problems.