scholarly journals On the upper C-continuity on the solution set of general vector alpha optimization problems in infinite-dimensional spaces

2018 ◽  
Vol 2018 (1) ◽  
pp. 1-10
Keyword(s):  
Optimization Problems ◽  
Solution Set ◽  
Infinite Dimensional ◽  
General Vector

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 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.

scholarly journals Bézier Simplex Fitting: Describing Pareto Fronts of´ Simplicial Problems with Small Samples in Multi-Objective Optimization

2019 ◽  
Vol 33 ◽  
pp. 2304-2313 ◽  
Author(s):  
Ken Kobayashi ◽  
Naoki Hamada ◽  
Akiyoshi Sannai ◽  
Akinori Tanaka ◽  
Kenichi Bannai ◽  
...  
Keyword(s):  
Sample Size ◽  
Optimization Problems ◽  
Solution Set ◽  
Small Samples ◽  
High Dimensional ◽  
Large Sample Size ◽  
Multi Objective Optimization ◽  
Dimensional Solution ◽  
Multi Objective ◽  
Simplex Model

Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M-objective optimization problem often forms an (M − 1)-dimensional topological simplex (a curved line for M = 2, a curved triangle for M = 3, a curved tetrahedron for M = 4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bézier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bézier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the postoptimization process and enable a better trade-off analysis.

scholarly journals Quasi-regularity in Optimization

1986 ◽  
Vol 41 (2) ◽  
pp. 188-192 ◽  
Author(s):  
Kung-Fu Ng ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Lagrange Multiplier ◽  
Optimization Problems ◽  
General Situation ◽  
Lagrange Multiplier Rule ◽  
Infinite Dimensional ◽  
Space Setting ◽  
Multiplier Rule ◽  
Dimensional Version ◽  
Definition Of

AbstractThe notion of quasi-regularity, defined for optimization problems in Rn, is extended to the Banach space setting. Examples are given to show that our definition of quasi-regularity is more natural than several other possibilities in the general situation. An infinite dimensional version of the Lagrange multiplier rule is established.

Definition of the feasible solution set in multicriteria optimization problems with continuous, discrete, and mixed design variables

2009 ◽  
Vol 71 (12) ◽  
pp. e109-e117 ◽  
Author(s):  
Roman Statnikov ◽  
Alex Bordetsky ◽  
Josef Matusov ◽  
Il’ya Sobol’ ◽  
Alexander Statnikov
Keyword(s):  
Multicriteria Optimization ◽  
Feasible Solution ◽  
Optimization Problems ◽  
Solution Set ◽  
Mixed Design ◽  
Design Variables ◽  
Definition Of ◽  
Mixed Design Variables

Properties of the Solution Set for Generalized Equilibrium Problems with Lower and Upper Bounds in FC-Metric Spaces

2013 ◽  
Vol 671-674 ◽  
pp. 1557-1560
Author(s):  
Kai Ting Wen
Keyword(s):  
Metric Spaces ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Upper Bounds ◽  
Solution Set ◽  
Lower And Upper Bounds ◽  
Paper Properties ◽  
Special Cases ◽  
Mathematical Problems ◽  
Generalized Equilibrium

The equilibrium problem includes many fundamental mathematical problems, e.g., optimization problems, saddle point problems, fixed point problems, economics problems, comple- mentarity problems, variational inequality problems, mechanics, engineering, and others as special cases. In this paper, properties of the solution set for generalized equilibrium problems with lower and upper bounds in FC-metric spaces are studied. In noncompact setting, we obtain that the solution set for generalized equilibrium problems with lower and upper bounds is nonempty and compact. Our results improve and generalize some recent results in the reference therein.

Sign in / Sign up

Export Citation Format

Share Document