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Algorithmic Optimization
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
Variational inequality theory was introduced by Hartman and Stampacchia (1966) as a tool for the study of partial differential equations with applications arising from mechanics. Such variational inequalities were infinite-dimensional rather than finite-dimensional as we describe in this section. The breakthrough in finite-dimensional theory of variational inequalities occurred in 1980 when Dafermos established that the traffic network equilibrium conditions of Smith (1979) contained a formulation of a variational inequality. This gave birth to a new methodlogy for the study of problems in economics, management science, operation research, and also in engineering with a focus on transportation. Variational inequality theory is a powerful tool for formulating a variety of equilibrium problems. It contains, as special cases, such well-known problems in mathematical programming as systems of non-linear equations, optimization problems, complementarity problems.
Necessary conditions of extremum for functionals
Published in Simon Serovajsky, Optimization and Differentiation, 2017
The easiest result of extremum theory is the necessary condition of extremum for smooth functions of one variable. This is the equality to zero of its derivative at the point of extremum. Minimization problems for functionals are more difficult. The relevant stationary condition includes the derivative of the given functional. This result can be generalized to minimization problems on subspaces or on affine varieties. The variational inequality is the general necessary condition of minimum for functionals on convex sets. These results are the basis of the optimal control theory. The minimized functional depends on the state function that depends on the control here. The analysis of the minimization problems for functionals will be continued in the next chapter. We shall consider the sufficiency of extremum conditions, the existence of optimization problems, and some others there.
Mixed vector equilibrium-like problems on Hadamard manifolds: error bound analysis
Published in Applicable Analysis, 2023
Nguyen Van Hung, Vo Minh Tam, Donal O'Regan
In 1976, Auslender [1] introduced the notion of (non-differentiable) gap function for finite-dimensional variational inequalities. It is known as a bridge between variational inequalities and optimization problems, i.e. a variational inequality can be reformulated as an optimization problem. To overcome the non-differentiability of gap functions considered in [1], Fukushima [2] developed the concept of a differentiable gap function which is called the regularized gap function. Furthermore, Yamashita et al. [3] investigated the regularized function of Moreau–Yosida type for variational inequalities and in terms of regularized gap functions in [2] and regularized functions of Moreau–Yosida type, they also provided error bounds for variational inequalities. An error bound is an expression which proposes an upper estimate of the distance of an arbitrary feasible point to the solution set of a certain problem. It plays a major role in convergence analysis of iterative algorithms to a solution of the variational inequality. Moreover, the theory of equilibrium problems and variational inequalities has applications in the fields of mechanics, economics, transportation, optimization and operations research. For details, we refer the reader to [4–13] and the references therein.
A class of differential hemivariational inequalities constrained on nonconvex star-shaped sets
Published in Optimization, 2023
Liang Lu, Lijie Li, Jiangfeng Han
The notion of differential variational inequalities was introduced in [20] by Aubin and Cellina. Differential variational inequalities are systems which couple differential or partial differential equations with a time-dependent variational inequality. Various mathematical models arising in the study of contact and impact problems lead to differential variational inequalities. Since a systematic study was carried out by Pang and Stewart [21], there is a number of papers have been dedicated to the development of the theory of differential variational inequalities and their applications [1,22–27]. Furthermore, differential hemivariational inequalities were firstly introduced by Liu et al. [2]. Interest in differential hemivariational inequalities and, more generally, differential variational–hemivariational inequalities, represent an important extension, originated, similarly as in differential variational inequalities. Over the past decade, the theory of this topic grew rapidly. We refer the reader to some recent references [2,3,28–32] and their references.
A cyclic iterative method for solving a class of variational inequalities in Hilbert spaces
Published in Optimization, 2018
Let C be a non-empty closed convex subset of a real Hilbert space H. The classical variational inequality problem initially studied by Stampacchia [1] for a nonlinear operator is the problem of finding an element such that where D is a non-empty closed convex subset of C. We denote by VI the problem (1) and if we denote by VI this problem. The problem VI is equivalent to the problem of finding a fixed point of the mapping , for all , where is the metric projection from H onto D. We know that if A is Lipschitzian and strongly monotone, then for small , the mapping is a strict contraction. So, by the Banach contraction principle, the problem VI has a unique solution and the Picard iterations defined by converge strongly to . The equivalence relation between the variational inequality problems and fixed point problems plays an important role in developing some efficient methods for solving variational inequality problems and related optimization problems. The problem of finding the fixed points of a non-expansive mapping is the subject of the current interest related to variational inequality problems in functional analysis.