Uncertainty Quantification in Variational Inequalities

Gwinner, Joachim; Jadamba, Baasansuren; Khan, Akhtar A.; Raciti, Fabio

doi:10.1201/9781315228969

Cited by 22 publications

(10 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For n∈N, let uεn∈K be the solution of the regularized variational inequality (2.5) with the regularization parameter εn. Since uεn=PKfalse[uεnfalse] and PK is non-expansive (see [17]), we have right left right left right left right left right left right left3pt0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0emtrue ‖un+1−uεnfalsefalse‖2≤‖PK[un−αn(F(un)+εnun+γn(un−un−1)+ωn)]−PK[uεn]falsefalse‖2 ≤‖un−uεn…”

Section: Stochastic Regularized Second-order Iterative Methodsmentioning

confidence: 99%

See 1 more Smart Citation

A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations

Dambrine

Khan

Sama

2022

Phil. Trans. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

Numerous applied models used in the study of optimal control problems, inverse problems, shape optimization, machine learning, fractional programming, neural networks, image registration and so on lead to stochastic optimization problems in Hilbert spaces. Under a suitable convexity assumption on the objective function, a necessary and sufficient optimality condition for stochastic optimization problems is a stochastic variational inequality. This article presents a new stochastic regularized second-order iterative scheme for solving a variational inequality in a stochastic environment where the primary operator is accessed by employing sampling techniques. The proposed iterative scheme, which fits within the general framework of the stochastic approximation approach, has its almost-sure convergence analysis given in a Hilbert space. We test the feasibility and the efficacy of the proposed stochastic approximation approach for a stochastic optimal control problem and a stochastic inverse problem, both associated with a second-order stochastic partial differential equation. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

show abstract

Section: Stochastic Regularized Second-order Iterative Methodsmentioning

confidence: 99%

“…For n ∈ N, let u ε n ∈ K be the solution of the regularized variational inequality (2.5) with the regularization parameter ε n . Since u ε n = P K [u ε n ] and P K is non-expansive (see [17]), we have…”

Section: 7) (A 4 )mentioning

confidence: 99%

A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations

Dambrine

Khan

Sama

2022

Phil. Trans. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then, in [ 17 ], the same authors carried out the theory of random variational inequalities to study a class of random equilibrium problems on networks in the linear case, and in [ 18 ] they studied the application to nonlinear random traffic equilibrium problem. A valuable additional contribution of the same authors is the book in [ 19 ]. In [ 11 ], the authors formulated the multicriteria spatial price network equilibrium problem as a random variational inequality, in which the consumers weight, using random fluctuations, transportation cost and the transportation time associated with the shipment of a given item.…”

Section: Introductionmentioning

confidence: 99%

A Stochastic Nash Equilibrium Problem for Medical Supply Competition

Fargetta

Maugeri

Scrimali

2022

J Optim Theory Appl

View full text Add to dashboard Cite

In this paper, we study the competition of healthcare institutions for medical supplies in emergencies caused by natural disasters. In particular, we develop a two-stage procurement planning model in a random environment. We consider a pre-event policy, in which each healthcare institution seeks to minimize the purchasing cost of medical items and the transportation time from the first stage, and a recourse decision process to optimize the expected overall costs and the penalty for the prior plan, in response to each disaster scenario. Thus, each institution deals with a two-stage stochastic programming model that takes into account the unmet demand at the first stage, and the consequent penalty. Then, the institutions simultaneously solve their own stochastic optimization problems and reach a stable state governed by the stochastic Nash equilibrium concept. Moreover, we formulate the problem as a variational inequality; both the discrete and the general probability distribution cases are described. We also present an alternative formulation using infinite-dimensional duality tools. Finally, we discuss some numerical illustrations applying the progressive hedging algorithm.

show abstract

“…An abstract theory of pseudo-monotone, hemi-, quasi- and conventional variational inequalities is outlined in the recent monograph by Gwinner et al [9] and the references therein. Our particular methods of non-smooth analysis stem from the variational approach to non-penetrating cracks in solids developed by Khludnev & Kovtunenko [10] and co-authors (e.g.…”

Section: Introductionmentioning

confidence: 99%

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model

Kovtunenko

2022

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

The Boussinesq problem, which describes quasi-static indentation of a rigid punch into a deformable body, is studied within the context of nonlinear constitutive equations. By this, the material response expresses the linearized strain in terms of the stress and cannot be inverted in general. A contact area between the punch and the body is unknown a priori , whereas the total contact force is prescribed and yields a non-local integral condition. Consequently, the unilateral indentation problem is stated as a quasi-variational inequality for unknown variables of displacement, stress and indentation depth. The Lagrange multiplier approach is applied in order to establish well-posedness to the underlying physically and geometrically nonlinear problem based on augmented penalty regularization and applying the minimax theorem of Ekeland and Témam. A sufficient solvability condition implies response functions that are bounded, hemi-continuous, coercive and obey a convex potential. A typical example is power-law hardening models for titanium alloys, Norton–Hoff and Ramberg–Osgood materials. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

show abstract

Uncertainty Quantification in Variational Inequalities

Cited by 22 publications

References 0 publications

A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations

A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations

A Stochastic Nash Equilibrium Problem for Medical Supply Competition

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model

Contact Info

Product

Resources

About