Nonlinear Optimization: Algorithms, Software, Applications

非线性规划(光滑的(可能的)非线性约束下的光滑非线性函数的最小化)是最优化中的试金石问题。由于许多原因，这个问题继续引起优化研究人员的强烈兴趣，其中包括算法和软件(特别是内点变化)的新发展，带平衡约束的数学规划(MPEC)的研究，以及新应用领域的发现。非线性规划范式的广泛范围允许问题的几何性质和代数规范中的许多病理。设计出在所有情况下都可靠收敛的算法是不可能的。大多数算法的局部收敛理论依赖于正则性和严格的互补性假设，而这些假设不被满足的问题(退化问题)是大多数算法和代码中不期望和尴尬行为的原因。本项目将研究一些与当前使用的方法相关的非线性规划方法，这些方法在退化和大规模问题上具有改善性能的潜力。这些技术在理论上将是严谨和实用的，因为处理退化的边际计算成本不会太大，而且当它们以.将特别注意MPEC，它表现出一种特殊的c类型的简并。此外，还将研究退化互补问题和变分不等式的推广。将特别注意实现这些算法的数值方面，这是一个重要的问题，因为在每次迭代的子问题中可能存在病态条件。所有这些研究将以协调的方式进行，将理论进步与使用原型软件和修改生产软件的计算实验相结合。与领域科学家和工程师合作开展的非线性规划应用工作将是该提案的第二个关键贡献。这种类型的跨学科研究在算法优化的任何研究计划中都发挥着关键作用。PI在工程控制、统计学和癌症治疗计划等领域进行着合作。拟议工作的智力价值在于提高了我们对非线性程序和解决这些问题的算法的理解，在于构建更好的算法和软件，在面临退化时是健壮的，以及这些进步对许多应用领域的影响，包括提案中描述的那些。这项工作将对优化技术的用户产生更广泛的影响，最终使其受益。许多使用优化软件包在非常广泛的应用中解决问题的人将受益。在提案中强调的领域的科学家和工程师将特别受益。

Nonlinear programming (the minimization of a smooth nonlinear function subject to smooth(possibly) nonlinear constraints) is a touchstone problem in optimization. The problem continuesto excite strong interest among optimization researchers for many reasons, among them newdevelopments in algorithms and software (especially of the interior-point variety), the study ofmathematical programs with equilibrium constraints (MPEC), and the discovery of new applicationareas.The broad scope of the nonlinear programming paradigm admits a great many pathologies inthe geometric nature of the problem and in its algebraic speci.cation. It is impossible to designalgorithms that converge reliably in all circumstances. Local convergence theory for most algorithmsdepends on regularity and strict complementarity assumptions, and problems for whichthese assumptions are not satis.ed (degenerate problems) are the cause of undesirable and awkwardbehavior in most algorithms and codes.This project will investigate some approaches to nonlinear programming, related to approachescurrently in use, that have the potential for improved performance on degenerate and large-scaleproblems. These techniques will be theoretically rigorous and practical, in that the marginalcomputational cost of handling degeneracy will not be too great and in that they will behave welleven when implemented in .oating-point arithmetic. Particular attention will be given to MPECs,which exhibit degeneracy of a speci.c type. Extensions to degenerate complementary problemsand variational inequalities will also be investigated. Special attention will be paid to the numericalaspects of implementing these algorithms, an important issue because of the ill conditioning thatmay be present in the subproblems at each iteration. All this research will be carried out in acoordinated manner, coupling theoretical advances with computational experiments using bothprototype software and modi.cations to production software.Work on applications of nonlinear programming, performed in collaboration with domain scientistsand engineers, will be the second key contribution of the proposal. Interdisciplinary research ofthis type plays a key role in any research program in algorithmic optimization. The PI has ongoingcollaborations in such areas as engineering control, statistics, and cancer treatment planning.The intellectual merit of the proposed work lies in improvements to our understanding ofnonlinear programs and of the algorithms that solve these problems, in the construction of betteralgorithms and software that are robust in the face of degeneracy, and in the impact of theseadvances on many application areas including those described in the proposal.The work will have a wider impact on users of optimization technology, ultimately bene.tingmany of those who use optimization software packages to solve problems in an extremely widerange of applications. Domain scientists and engineers in the areas highlighted in the proposal willbene.t especially.