Pub Date : 1900-01-01DOI: 10.1080/10556789908805726
Manfred Laumen
Several optimal shape design problems are defined as a minimization problem with an equality constraint that is given by a boundary value problem. The mapping method transforms this to a specific control problem on a fixed domain. Discretizing this optimal control problem normally leads to a large scale optimization formulation where the corresponding solution methods are characterized by the requirement of solving many boundary value problems. In spite of this interesting numerical challenge, until now less research has been done on comparing different numerical optimization approaches including second order methods for optimal shape design problems. In this paper, Newton's and several variants of quasi-Newton methods are derived for a class of optimal shape design problems and compared to the commonly used gradient method. The pros and cons of these methods plus their nested iteration versions are discussed in detail. Various numerical experiences underline the importance of choosing the right optimizat...
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Pub Date : 1900-01-01DOI: 10.1201/9781420035315.ch31
P. Gritzmann, V. Klee
The subject of Computational Convexity draws its methods from discrete mathematics and convex geometry, and many of its problems from operations research, computer science, data analysis, physics, material science, and other applied areas. In essence, it is the study of the computational and algorithmic aspects of high-dimensional convex sets (especially polytopes), with a view to applying the knowledge gained to convex bodies that arise in other mathematical disciplines or in the mathematical modeling of problems from outside mathematics. The name Computational Convexity is of more recent origin, having first appeared in print in 1989. However, results that retrospectively belong to this area go back a long way. In particular, many of the basic ideas of Linear Programming have an essentially geometric character and fit very well into the conception of Computational Convexity. The same is true of the subject of Polyhedral Combinatorics and of the Algorithmic Theory of Polytopes and Convex Bodies. The emphasis in Computational Convexity is on problems whose underlying structure is the convex geometry of normed vector spaces of finite but generally not restricted dimension, rather than of fixed dimension. This leads to closer connections with the optimization problems that arise in a wide variety of disciplines. Further, in the study of Computational Convexity, the underlying model of computation is mainly the binary (Turing machine) model that is common in studies of computational complexity. This requirement is imposed by prospective applications, particularly in mathematical programming. For the study of algorithmic aspects of convex bodies that are not polytopes, the binary model is often augmented by additional devices called “oracles.” Some cases of interest involve other models of computation, but the present discussion focuses on aspects of computational convexity for which binary models seem most natural. Many of the results stated in this chapter are qualitative, in the sense that they classify certain problems as being solvable in polynomial time, or show that certain problems are NP-hard or harder. Typically, the tasks remain to find optimal exact algorithms for the problems that are polynomially solvable, and to find useful approximation algorithms or heuristics for those that are NP-hard. In many cases, the known algorithms, even when they run in polynomial time, appear to be far from optimal from the viewpoint of practical application. Hence, the qualitative complexity results should in many cases be regarded as a guide to future efforts but not as final words on the problems with which they deal. Some of the important areas of computational convexity, such as linear and convex programming, packing and covering, and geometric reconstructions, are covered in other chapters of this Handbook. Hence, after some remarks on presentations of polytopes in Section 36.1, the present discussion concentrates on the following areas that are not c
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Pub Date : 1900-01-01DOI: 10.1080/10556789708805646
H. Jäger, E. Sachs
In this paper we consider infinite dimensional optimization problems with equality constraints. The basic underlying algorithm is a reduced SQP method. We give a global convergence proof in Hilbert space and extend this analysis to inexact reduced SQP methods. These methods are useful when discretizing the infinite dimensional problems and solving the resulting large scale discretized optimization problems. The convergence analysis takes into account the Maratos effect which occurs for nonsmooth norms. The inexact reduced SQP method is applied to a discretized parabolic control problem
{"title":"Global Convergence of Inexact Reduced SQP Methods","authors":"H. Jäger, E. Sachs","doi":"10.1080/10556789708805646","DOIUrl":"https://doi.org/10.1080/10556789708805646","url":null,"abstract":"In this paper we consider infinite dimensional optimization problems with equality constraints. The basic underlying algorithm is a reduced SQP method. We give a global convergence proof in Hilbert space and extend this analysis to inexact reduced SQP methods. These methods are useful when discretizing the infinite dimensional problems and solving the resulting large scale discretized optimization problems. The convergence analysis takes into account the Maratos effect which occurs for nonsmooth norms. The inexact reduced SQP method is applied to a discretized parabolic control problem","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127383398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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