scholarly journals Well-Posedness and Porosity for Symmetric Optimization Problems

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1253
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Metric Space ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Complete Metric Space ◽  
Well Posedness ◽  
Porous Set ◽  
Minimization Problems ◽  
Symmetric Optimization ◽  
Lower Semi ◽  
Well Posed

In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric optimization problem possesses two solutions. In this paper, we strengthen this result using a porosity notion. We investigate the collection of all functions such that the corresponding optimization problem is well-posed and prove that its complement is a σ-porous set.

scholarly journals A density result in vector optimization

2006 ◽  
Vol 2006 ◽  
pp. 1-10
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Vector Optimization ◽  
Metric Space ◽  
Complete Metric Space ◽  
Density Result ◽  
Minimization Problems ◽  
Whole Class ◽  
Vector Minimization Problems

We study a class of vector minimization problems on a complete metric space such that all its bounded closed subsets are compact. We show that a subclass of minimization problems with a nonclosed set of minimal values is dense in the whole class of minimization problems.

scholarly journals A porosity result in convex minimization

2005 ◽  
Vol 2005 (3) ◽  
pp. 319-326
Author(s):  
P. G. Howlett ◽  
A. J. Zaslavski
Keyword(s):  
Banach Space ◽  
Minimization Problem ◽  
Metric Space ◽  
Convex Functions ◽  
Convex Sets ◽  
Reflexive Banach Space ◽  
Complete Metric Space ◽  
Baire Category ◽  
Countable Intersection ◽  
Porous Set

We study the minimization problemf(x)→min,x∈C, wherefbelongs to a complete metric spaceℳof convex functions and the setCis a countable intersection of a decreasing sequence of closed convex setsCiin a reflexive Banach space. Letℱbe the set of allf∈ℳfor which the solutions of the minimization problem over the setCiconverge strongly asi→∞to the solution over the setC. In our recent work we show that the setℱcontains an everywhere denseGδsubset ofℳ. In this paper, we show that the complementℳ\ℱis not only of the first Baire category but also aσ-porous set.

scholarly journals Generic well-posedness in minimization problems

2005 ◽  
Vol 2005 (4) ◽  
pp. 343-360 ◽  
Author(s):  
A. Ioffe ◽  
R. E. Lucchetti
Keyword(s):  
Uniform Convergence ◽  
Well Posedness ◽  
Minimization Problems ◽  
Bounded Sets ◽  
Well Posed ◽  
Class Of Functions

The goal of this paper is to provide an overview of results concerning, roughly speaking, the following issue: given a (topologized) class of minimum problems, “how many” of them are well-posed? We will consider several ways to define the concept of “how many,” and also several types of well-posedness concepts. We will concentrate our attention on results related to uniform convergence on bounded sets, or similar convergence notions, as far as the topology on the class of functions under investigation is concerned.

scholarly journals An admissible Hybrid contraction with an Ulam type stability

2019 ◽  
Vol 52 (1) ◽  
pp. 428-436 ◽  
Author(s):  
Erdal Karapınar ◽  
Andreea Fulga
Keyword(s):  
Metric Space ◽  
Complete Metric Space ◽  
Well Posedness ◽  
Set Up ◽  
Ulam Type Stability

AbstractIn this manuscript, we introduce a new hybrid contraction that unify several nonlinear and linear contractions in the set-up of a complete metric space. We present an example to indicate the genuine of the proved result. In addition, we consider Ulam type stability and well-posedness for this new hybrid contraction.

scholarly journals Generic Existence of Solutions of Symmetric Optimization Problems

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2004
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Objective Function ◽  
Optimization Problem ◽  
Existence Of Solutions ◽  
Optimization Problems ◽  
Baire Category ◽  
Objective Functions ◽  
Countable Intersection ◽  
Symmetric Optimization ◽  
Dense Sets ◽  
Generic Existence

In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution.

scholarly journals A Shape Optimization Problem on Planar Sets with Prescribed Topology

2021 ◽  
Author(s):  
Luca Briani ◽  
Giuseppe Buttazzo ◽  
Francesca Prinari
Keyword(s):  
Shape Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Torsional Rigidity ◽  
Connected Components ◽  
Topological Constraints ◽  
Ill Posed ◽  
Complementary Set ◽  
Cost Functionals ◽  
Well Posed

AbstractWe consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $$P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}$$ P ( Ω ) T q ( Ω ) | Ω | - 2 q - 1 / 2 , and the class of admissible domains consists of two-dimensional open sets $$\Omega $$ Ω satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when $$q<1/2$$ q < 1 / 2 an optimal relaxed domain exists. When $$q>1/2$$ q > 1 / 2 , the problem is ill-posed, and for $$q=1/2$$ q = 1 / 2 , the explicit value of the infimum is provided in the cases $$k=0$$ k = 0 and $$k=1$$ k = 1 .

scholarly journals Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs

Processes ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 838 ◽  
Author(s):  
Georgia Kouyialis ◽  
Xiaoyu Wang ◽  
Ruth Misener
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Mathematical Optimization ◽  
Computation Time ◽  
Packing Problem ◽  
Quadratic Optimization ◽  
Detection Methods ◽  
Symmetric Optimization ◽  
Adjacency Matrices ◽  
Constrained Quadratic Optimization

Symmetry in mathematical optimization may create multiple, equivalent solutions. In nonconvex optimization, symmetry can negatively affect algorithm performance, e.g., of branch-and-bound when symmetry induces many equivalent branches. This paper develops detection methods for symmetry groups in quadratically-constrained quadratic optimization problems. Representing the optimization problem with adjacency matrices, we use graph theory to transform the adjacency matrices into binary layered graphs. We enter the binary layered graphs into the software package nauty that generates important symmetric properties of the original problem. Symmetry pattern knowledge motivates a discretization pattern that we use to reduce computation time for an approximation of the point packing problem. This paper highlights the importance of detecting and classifying symmetry and shows that knowledge of this symmetry enables quick approximation of a highly symmetric optimization problem.

On Fixed Point Theorems in Complete Metric Space

2019 ◽  
Vol 10 (7) ◽  
pp. 1419-1425
Author(s):  
Jayashree Patil ◽  
Basel Hardan
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Complete Metric Space
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