Occupational measures and averaged shape optimization

2018 ◽  
Vol 24 (3) ◽  
pp. 1141-1165
Author(s):  
Ido Bright ◽  
Qinfeng Li ◽  
Monica Torres
Keyword(s):  
Shape Optimization ◽  
Linear Problem ◽  
Optimization Problems ◽  
Convex Polytopes ◽  
Probability Measures ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter ◽  
Existence Of Minimizers ◽  
Optimal Value ◽  
Occupational Measures

We consider the minimization of averaged shape optimization problems over the class of sets of finite perimeter. We use occupational measures, which are probability measures defined in terms of the reduced boundary of sets of finite perimeter, that allow to transform the minimization into a linear problem on a set of measures. The averaged nature of the problem allows the optimal value to be approximated with sets with unbounded perimeter. In this case, we show that we can also approximate the optimal value with convex polytopes with n+1 faces shrinking to a point. We derive conditions under which we show the existence of minimizers and we also analyze the appropriate spaces in which to study the problem.

scholarly journals On quantitative stability in infinite-dimensional optimization under uncertainty

2021 ◽  
Author(s):  
M. Hoffhues ◽  
W. Römisch ◽  
T. M. Surowiec
Keyword(s):  
Stochastic Optimization ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Optimization Under Uncertainty ◽  
Optimal Solutions ◽  
Probability Metrics ◽  
Pde Constrained Optimization ◽  
Quantitative Stability ◽  
Infinite Dimensional ◽  
Optimal Value

AbstractThe vast majority of stochastic optimization problems require the approximation of the underlying probability measure, e.g., by sampling or using observations. It is therefore crucial to understand the dependence of the optimal value and optimal solutions on these approximations as the sample size increases or more data becomes available. Due to the weak convergence properties of sequences of probability measures, there is no guarantee that these quantities will exhibit favorable asymptotic properties. We consider a class of infinite-dimensional stochastic optimization problems inspired by recent work on PDE-constrained optimization as well as functional data analysis. For this class of problems, we provide both qualitative and quantitative stability results on the optimal value and optimal solutions. In both cases, we make use of the method of probability metrics. The optimal values are shown to be Lipschitz continuous with respect to a minimal information metric and consequently, under further regularity assumptions, with respect to certain Fortet-Mourier and Wasserstein metrics. We prove that even in the most favorable setting, the solutions are at best Hölder continuous with respect to changes in the underlying measure. The theoretical results are tested in the context of Monte Carlo approximation for a numerical example involving PDE-constrained optimization under uncertainty.

scholarly journals Relaxed gradient projection algorithm for constrained node-based shape optimization

2021 ◽  
Author(s):  
Ihar Antonau ◽  
Majid Hojjat ◽  
Kai-Uwe Bletzinger
Keyword(s):  
Shape Optimization ◽  
Optimization Problems ◽  
Gradient Projection ◽  
Projection Algorithm ◽  
Design Parameters ◽  
Active Set ◽  
Physical Constraints ◽  
Original Algorithm ◽  
Gradient Projection Algorithm ◽  
Non Linear

AbstractIn node-based shape optimization, there are a vast amount of design parameters, and the objectives, as well as the physical constraints, are non-linear in state and design. Robust optimization algorithms are required. The methods of feasible directions are widely used in practical optimization problems and know to be quite robust. A subclass of these methods is the gradient projection method. It is an active-set method, it can be used with equality and non-equality constraints, and it has gained significant popularity for its intuitive implementation. One significant issue around efficiency is that the algorithm may suffer from zigzagging behavior while it follows non-linear design boundaries. In this work, we propose a modification to Rosen’s gradient projection algorithm. It includes the efficient techniques to damp the zigzagging behavior of the original algorithm while following the non-linear design boundaries, thus improving the performance of the method.

scholarly journals Structure of Pareto Solutions of Generalized Polyhedral-Valued Vector Optimization Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Qinghai He ◽  
Weili Kong
Keyword(s):  
Banach Spaces ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Solution Set ◽  
Pareto Optimal ◽  
Pareto Solution ◽  
Value Set ◽  
Set Valued Mapping ◽  
Optimal Value

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

The mine blast algorithm for the structural optimization of electrical vehicle components

2020 ◽  
Vol 62 (5) ◽  
pp. 497-502 ◽  
Author(s):  
B. S. Yıldız
Keyword(s):  
Shape Optimization ◽  
Optimization Problems ◽  
Element Analysis ◽  
Mine Blast Algorithm ◽  
Component Design ◽  
Derivative Free ◽  
Door Hinge ◽  
Electrical Vehicles ◽  
Electrical Vehicle ◽  
Vehicle Components

Abstract The shape optimization of mechanical and automotive component plays a crucial role in the development of automotive technology. Presently, the use of derivative-free metaheuristics in combination with finite element analysis for mechanical component design is one of the most focused on topics due to its simplicity and effectiveness. In this research paper, the mine blast algorithm (MBA) is used to solve the problem of shape optimization for a vehicle door hinge to prove how the MBA can be used for solving shape optimization problems in designing electrical vehicles. The results show the advantage of the MBA for designing optimal components in the automotive industry.

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