Characterization of the feasible set mapping in one class of semi-infinite optimization problems

Top ◽  
2004 ◽  
Vol 12 (1) ◽  
pp. 135-147 ◽  
Author(s):  
Estela L. Juárez ◽  
Maxim I. Todorov
Keyword(s):  
Optimization Problems ◽  
Feasible Set ◽  
Feasible Set Mapping

scholarly journals A Characterization of Polynomial Density on Curves via Matrix Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1231
Author(s):  
Carmen Escribano ◽  
Raquel Gonzalo ◽  
Emilio Torrano
Keyword(s):  
Matrix Algebra ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Point Of View ◽  
Alternative Proof ◽  
General Context ◽  
Positive Semidefinite Matrices ◽  
Jordan Curves ◽  
Point Evaluations

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

ADAPTIVE OPERATOR EXTRAPOLATION METHOD FOR VARIATIONAL INEQUALITIES IN BANACH SPACES

2021 ◽  
Vol 5 ◽  
pp. 82-92
Author(s):  
Sergei Denisov ◽  
◽  
Vladimir Semenov ◽  
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Operations Research ◽  
Optimization Problems ◽  
Metric Projection ◽  
Feasible Set ◽  
Uniformly Smooth ◽  
Developing Area ◽  
Lipschitz Operators

Many problems of operations research and mathematical physics can be formulated in the form of variational inequalities. The development and research of algorithms for solving variational inequalities is an actively developing area of applied nonlinear analysis. Note that often nonsmooth optimization problems can be effectively solved if they are reformulated in the form of saddle point problems and algorithms for solving variational inequalities are applied. Recently, there has been progress in the study of algorithms for problems in Banach spaces. This is due to the wide involvement of the results and constructions of the geometry of Banach spaces. A new algorithm for solving variational inequalities in a Banach space is proposed and studied. In addition, the Alber generalized projection is used instead of the metric projection onto the feasible set. An attractive feature of the algorithm is only one computation at the iterative step of the projection onto the feasible set. For variational inequalities with monotone Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, a theorem on the weak convergence of the method is proved.

HYBRID SEARCH STRATEGIES FOR HETEROGENEOUS SEARCH SPACES

2000 ◽  
Vol 09 (01) ◽  
pp. 45-57 ◽  
Author(s):  
CARLA GOMES ◽  
BART SELMAN
Keyword(s):  
Operations Research ◽  
Resource Constraints ◽  
Optimization Problems ◽  
Search Strategies ◽  
Mixed Integer ◽  
Hybrid Search ◽  
Search Spaces ◽  
Probabilistic Information ◽  
Numeric Information

Recently, there has been much interest in enhancing purely combinatorial formalisms with numerical information. For example, planning formalisms can be enriched by taking resource constraints and probabilistic information into account. The Mixed Integer Programming (MIP) paradigm from operations research provides a natural tool for solving optimization problems that combine such numeric and non-numeric information. The MIP approach relies heavily on linear program relaxations and branch-and-bound search. This is in contrast with depth-first or iterative deepening strategies generally used in artificial intelligence. We provide a detailed characterization of the structure of the underlying search spaces as explored by these search strategies. Our analysis shows that much can be gained by combining different search strategies for solving hard MIP problems, thereby leveraging each strategy's strength in terms of the combinatorial and numeric information.

scholarly journals Lower Semicontinuity of the Feasible Set Mapping of Linear Systems Relative to Their Domains

2012 ◽  
Vol 21 (1) ◽  
pp. 67-92 ◽  
Author(s):  
Aris Daniilidis ◽  
Miguel A. Goberna ◽  
Marco A. López ◽  
Roberto Lucchetti
Keyword(s):  
Linear Systems ◽  
Lower Semicontinuity ◽  
Feasible Set ◽  
Feasible Set Mapping
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