Non-linear semi-infinite programming

Optimisation problems occur in many branches of science, engineering, and economics, as well as in other areas. The diversity of the various types of optimisation problems is extremely large, and so a unified approach is not attempted here. This thesis concentrates on a specific type of problem: non-linear semi-infinite programming

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Mathematical Programming manuscript No. (will be inserted by the editor)

On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions Dedicated to Professor Stephen Robinson on the occasion of his 65th birthday. The second author remembers, with a sense of privilege, the courses and advice he received from Professor Robinson during his stay at UW-Madison

COMPUTATIONAL ADVANCES IN CONTINUUM TOPOLOGY OPTIMIZATION ALGORITHMS

Topology optimization is a fascinating area of research with numerous unsolved computational challenges. In this thesis, the author aims to advance the research on improving the computational efficiency of common topology algorithms for practical real life problems. Beside the research contributions in this thesis, the introduction (chapter 1) is written to cover much of the theory behind the algorithms and formulations used in topology optimization including some details that often get ignored in most papers and texts in the field of topology optimization. A lot of the details presented in the introduction is scattered in multiple resources between computational mechanics books, optimization theory books and papers, and topology optimization literature. This makes it difficult for people starting to learn topology optimization to easily cover the theory needed to do advanced research in the field. An attempt is made to give a reasonably comprehensive coverage of the theory of the finite element method with an emphasis on linear elasticity as well as the theory behind common nonlinear programming algorithms used in topology optimization. Additionally, a presentation of all the common paradigms for decision-making under uncertainty is presented. Topology optimization under uncertainty is a field of research with many unsolved computational problems. This presentation will hopefully help more researchers get started in this field of research more easily. In chapter 2, the first research contribution of this thesis is presented. In par- ticular, a flexible and theoretically sound way to adapt penalties in the continuation solid isotropic material with penalization (CSIMP) method is proposed which gives significant speedups in the experiments run. Four common test problems from literature, three 2D and one 3D, are used to test the efficacy of the penalty adap- tation with different parameter settings. The main factors affecting the efficacy of the penalty adaptation in the CSIMP algorithm in reducing the number of fi- nite element analysis (FEA) simulations needed to converge to the final solution are identified. The experimental results demonstrate a significant reduction in the number of FEA simulations required to reach the optimal solution in the decreasing tolerance CSIMP algorithm, with exponentially decaying tolerance, with little to no detriment in the objective value and the other metrics used. Finally, a mathematical and experimental treatment of the effect of the minimum pseudo-density parameter on the convergence of the CSIMP algorithm is given with some recommendations for choosing a suitable value. These results appear in the Computer Methods in Applied Mechanics and Engineering journal (Tarek & Ray 2020). In chapter 3, the problem of handling load uncertainty efficiently in compliance- based topology optimization problems is tackled. A comprehensive review of all the literature on handling uncertainty in compliance-based problems is presented. And a number of exact methods are proposed to handle load uncertainty in compliance- based topology optimization problems where the uncertainty is described in the form of a set of finitely many loading scenarios. This includes mean compliance minimization or constraining the mean compliance, minimizing or constraining a weighted sum of the mean and standard deviation of the load compliances as well as minimizing or constraining the maximum load compliance for all the loading scenarios. By detecting and exploiting low rank structures in the loading scenar- ios, significant performance improvements are achieved using some novel methods. The computational complexities of the algorithms proposed are demonstrated and experiments are run to verify the efficacy of the proposed algorithms at reducing the computational cost of these classes of topology optimization problems. The meth- ods presented here are fundamentally data-driven in the sense that no probability distributions or continuous domains are assumed for the loading scenarios. This sets this work apart from most of the literature in the domain of stochastic and robust topology optimization where a distribution or domain is assumed. Additionally, the methods proposed here are shown to be particularly suitable with the augmented Lagrangian algorithm when dealing with maximum compliance constraints. This work appears in the Structural and Multidisciplinary Optimization journal. In chapter 4, approximate methods for handling many loading scenarios with a high rank loading matrix are developed. In particular, approximation schemes for the mean compliance and a class of scalar-valued functions of the load compliances are developed. The approximation schemes are based on a reformulation of the function approximated as a trace or diagonal estimation problem, opening the door to using many of the available methods for trace or diagonal estimation. The approximation methods are tested on a number of standard 2D and 3D benchmark problems using low and high rank loading scenarios to solve mean compliance minimization as well as minimizing the weighted sum of the mean compliance and its standard deviation. Significant speedups are achieved compared to the exact methods when the rank of the load matrix is high. This work is submitted to the Structural and Multidisciplinary Optimization journal as of the time of the writing of this thesis. In chapter 5, a summary of all the findings in this thesis and some potential future work for the author here or for aspiring researchers in topology optimization is presented