32,619 research outputs found
Variational inequalities in vector optimization
In this paper we investigate the links among generalized scalar variational inequalities of differential type, vector variational inequalities and vector optimization problems. The considered scalar variational inequalities are obtained through a nonlinear scalarization by means of the so called ”oriented distance” function [14, 15]. In the case of Stampacchia-type variational inequalities, the solutions of the proposed ones coincide with the solutions of the vector variational inequalities introduced by Giannessi [8]. For Minty-type variational inequalities, analogous coincidence happens under convexity hypotheses. Furthermore, the considered variational inequalities reveal useful in filling a gap between scalar and vector variational inequalities. Namely, in the scalar case Minty variational inequalities of differential type represent a sufficient optimality condition without additional assumptions, while in the vector case the convexity hypothesis was needed. Moreover it is shown that vector functions admitting a solution of the proposed Minty variational inequality enjoy some well-posedness properties, analogously to the scalar case [4].
Differential variational inequalities
International audienceThis paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems , and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensiona
Uniqueness of Viscosity Solutions for Optimal Multi-Modes Switching Problem with Risk of default
In this paper we study the optimal m-states switching problem in finite
horizon as well as infinite horizon with risk of default. We allow the
switching cost functionals and cost of default to be of polynomial growth and
arbitrary. We show uniqueness of a solution for a system of m variational
partial differential inequalities with inter-connected obstacles. This system
is the deterministic version of the Verification Theorem of the Markovian
optimal m-states switching problem with risk of default. This problem is
connected with the valuation of a power plant in the energy market.Comment: 25 pages; Real options, Backward stochastic differential equations,
Snell envelope, Stopping times, Switching, Viscosity solution of PDEs,
Variational inequalities. arXiv admin note: text overlap with arXiv:0805.1306
and arXiv:0904.070
Existence, iteration procedures and directional differentiability for parabolic QVIs
We study parabolic quasi-variational inequalities (QVIs) of obstacle type.
Under appropriate assumptions on the obstacle mapping, we prove the existence
of solutions of such QVIs by two methods: one by time discretisation through
elliptic QVIs and the second by iteration through parabolic variational
inequalities (VIs). Using these results, we show the directional
differentiability (in a certain sense) of the solution map which takes the
source term of a parabolic QVI into the set of solutions, and we relate this
result to the contingent derivative of the aforementioned map. We finish with
an example where the obstacle mapping is given by the inverse of a parabolic
differential operator.Comment: 41 page
Boundedness and pointwise differentiability of weak solutions to quasi-linear elliptic differential equations and variational inequalities
summary:The local boundedness of weak solutions to variational inequalities (obstacle problem) with the linear growth condition is obtained. Consequently, an analogue of a theorem by Reshetnyak about a.e\. differentiability of weak solutions to elliptic divergence type differential equations is proved for variational inequalities
Stability analysis of partial differential variational inequalities in Banach spaces
In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques
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