Nonlinear Optimization: Algorithms, Software, Applications

非线性优化：算法、软件、应用

• 批准号: 0430504
• 负责人: Stephen Wright
• 金额: $ 25万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-12-15 至 2009-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0430504&HistoricalAwards=false
• 关键词: Nonlinear; Optimization; Algorithms; Software; Applications

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.

非线性编程（平滑的非线性函数的最小化受到平滑（可能）非线性约束）是优化的试金石问题。该问题连续群落在优化研究人员之间引起了强烈的兴趣，其原因是算法和软件的新开发（尤其是内部点种类），具有平衡约束（MPEC）的对数学程序的研究（MPEC）的研究，以及新的应用程序的发现。非线性编程范式的广泛范围，该范围具有许多病理学的范围。规格。在任何情况下，都无法设计可靠地收敛的词汇。大多数算法依赖于规律性和严格互补性假设的局部收敛理论，以及这些假设不是satis.ed（退化问题）的问题是导致大多数算法和代码中不可避免且尴尬的行为的原因。这些项目将调查对非线性编程的某些方法，并在范围内进行效率，并且在近似程序中，与方法相关的是，该方法的使用效果是，在近似范围内，该方法既可以使用，又可以使用，并在附近使用，以至于在附近使用，并在范围内努力。大问题。这些技术在理论上将是严格和实用的，因为处理堕落的边际计算成本不会太大，并且在以.oating点算术中实施时，它们将表现出来。将特别注意MPEC，该MPEC表现出特定类型的脱落性。还将研究互补问题和变分不平等的扩展。将特别注意实施这些算法的数值，这是一个重要的问题，因为条件不适，即每次迭代的子问题都存在。所有这些研究都将以敏捷的方式进行，将理论进步与使用Prototype软件和Modi的计算实验相连。与生产软件的cotations.非线性编程的应用，与Domain Scientists和Domain Scientors和Indeersssical一起进行的应用程序将是该提案的第二个关键贡献。对这种类型的跨学科研究在算法优化的任何研究计划中都起着关键作用。 PI在工程控制，统计和癌症治疗计划等领域进行了持续的策略。拟议工作的智力优点在于改善我们的理解non线性计划以及解决这些问题的算法，在构建更好地纳入和软件的构建方面，这些领域的构建以及在许多方面的构建方面都在努力，并在许多方面构成了既定的范围，并且在许多方面的影响都在范围内构成了众多影响。对优化技术的用户的更广泛影响，最终受益。那些使用优化软件包来解决应用程序中问题的人的tingmany。尤其是提案中强调的领域的领域科学家和工程师尤其是。