Strong Stationarity Conditions for a Class of Optimization Problems Governed by Variational Inequalities of the Second Kind

2015 ◽  
Vol 168 (2) ◽  
pp. 375-409 ◽  
Author(s):  
J. C. De los Reyes ◽  
C. Meyer
Keyword(s):  
Variational Inequalities ◽  
Optimization Problems ◽  
Strong Stationarity ◽  
Stationarity Conditions

scholarly journals On implicit variables in optimization theory

2021 ◽  
Vol Volume 2 (Original research articles) ◽  
Author(s):  
Matúš Benko ◽  
Patrick Mehlitz
Keyword(s):  
Optimization Problems ◽  
Constraint Qualifications ◽  
Optimization Theory ◽  
Mathematical Optimization ◽  
General Setting ◽  
Necessary Optimality Conditions ◽  
Detailed Comparison ◽  
Mordukhovich Stationarity ◽  
Stationarity Conditions ◽  
The One

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory. Comment: 34 pages

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.

scholarly journals On a class of multivalued variational inequalities

1998 ◽  
Vol 11 (1) ◽  
pp. 79-93 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Optimization Problems ◽  
Iterative Algorithms ◽  
Auxiliary Principle ◽  
Existence Of A Solution ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Projection Techniques ◽  
Multivalued Variational Inequalities

In this paper, we introduce and study a new class of variational inequalities, which are called multivalued variational inequalities. These variational inequalities include as special cases, the previously known classes of variational inequalities. Using projection techniques, we show that multivalued variational inequalities are equivalent to fixed point problems and Wiener-Hopf equations. These alternate formulations are used to suggest a number of iterative algorithms for solving multivalued variational inequalities. We also consider the auxiliary principle technique to study the existence of a solution of multivalued variational inequalities and suggest a novel iterative algorithm. In addition, we have shown that the auxiliary principle technique can be used to find the equivalent differentiable optimization problems for multivalued variational inequalities. Convergence analysis is also discussed.

scholarly journals Vector and Ordered Variational Inequalities and Applications to Order-Optimization Problems on Banach Lattices

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Linsen Xie ◽  
Jinlu Li ◽  
Wenshan Yang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Fixed Point Theorems ◽  
Optimization Problems ◽  
Banach Lattices ◽  
Variational Inequality Problems ◽  
Vector Spaces ◽  
Real Vector ◽  
Finite Dimensional

We investigate the connections between vector variational inequalities and ordered variational inequalities in finite dimensional real vector spaces. We also use some fixed point theorems to prove the solvability of ordered variational inequality problems and their application to some order-optimization problems on the Banach lattices.

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