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Brno University of Technology

1. Mrázková, Eva. Approximations in Stochastic Optimization and Their Applications: Approximations in Stochastic Optimization and Their Applications.

Degree: 2019, Brno University of Technology

Many optimum design problems in engineering areas lead to optimization models constrained by ordinary (ODE) or partial (PDE) differential equations, and furthermore, several elements of the problems may be uncertain in practice. Three engineering problems concerning the optimization of vibrations and an optimal design of beam dimensions are considered. The uncertainty in the form of random load or random Young's modulus is involved. It is shown that two-stage stochastic programming offers a promising approach in solving such problems. Corresponding mathematical models involving ODE or PDE type constraints, uncertain parameters and multiple criteria are formulated and lead to (multi-objective) stochastic nonlinear optimization models. It is also proved for which type of problems stochastic programming approach (EO reformulation) should be used and when it is sufficient to solve simpler deterministic problem (EV reformulation). This fact has the big importance in practice in term of computational intensity of large scale problems. Computational schemes for this type of problems are proposed, including discretization methods for random elements and ODE or PDE constraints. By means of derived approximations the mathematical models are implemented and solved in GAMS. The solution quality is determined by an interval estimate of the optimality gap computed via Monte Carlo bounding technique. Parametric analysis of multi-criteria model results in efficient frontier computation. The alternatives of approximations of the model with reliability-related probabilistic terms including mixed-integer nonlinear programming and penalty reformulations are discussed. Furthermore, the progressive hedging algorithm is implemented and tested for the selected problems with respect to future possibilities of parallel computing of large engineering problems. The results show that it can be used even when the mathematical conditions for convergence are not fulfilled. Finite difference method and finite element method are compared for deterministic version of ODE constrained problem by using GAMS and ANSYS with quite comparable results. Advisors/Committee Members: Karpíšek, Zdeněk (advisor), Horová, Ivana (referee), Štěpánek, Petr (referee).

Subjects/Keywords: optimální inženýrský návrh; ODR a PDR omezení; stochastické programování; optimalizace s pravděpodobnostními omezeními; vícekriteriální optimalizace; metoda Monte Carlo; PHA algoritmus; optimum engineering design; ODE and PDE constraints; stochastic programming; chance constrained programming; multi-objective programming; Monte Carlo method; progressive hedging algorithm

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Mrázková, E. (2019). Approximations in Stochastic Optimization and Their Applications: Approximations in Stochastic Optimization and Their Applications. (Thesis). Brno University of Technology. Retrieved from http://hdl.handle.net/11012/1571

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Mrázková, Eva. “Approximations in Stochastic Optimization and Their Applications: Approximations in Stochastic Optimization and Their Applications.” 2019. Thesis, Brno University of Technology. Accessed April 13, 2021. http://hdl.handle.net/11012/1571.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Mrázková, Eva. “Approximations in Stochastic Optimization and Their Applications: Approximations in Stochastic Optimization and Their Applications.” 2019. Web. 13 Apr 2021.

Vancouver:

Mrázková E. Approximations in Stochastic Optimization and Their Applications: Approximations in Stochastic Optimization and Their Applications. [Internet] [Thesis]. Brno University of Technology; 2019. [cited 2021 Apr 13]. Available from: http://hdl.handle.net/11012/1571.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Mrázková E. Approximations in Stochastic Optimization and Their Applications: Approximations in Stochastic Optimization and Their Applications. [Thesis]. Brno University of Technology; 2019. Available from: http://hdl.handle.net/11012/1571

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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