61 research outputs found
Existence and solution methods for equilibria
Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algorithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them
The Forward-Backward-Forward Method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces
Tseng's forward-backward-forward algorithm is a valuable alternative for
Korpelevich's extragradient method when solving variational inequalities over a
convex and closed set governed by monotone and Lipschitz continuous operators,
as it requires in every step only one projection operation. However, it is
well-known that Korpelevich's method converges and can therefore be used also
for solving variational inequalities governed by pseudo-monotone and Lipschitz
continuous operators. In this paper, we first associate to a pseudo-monotone
variational inequality a forward-backward-forward dynamical system and carry
out an asymptotic analysis for the generated trajectories. The explicit time
discretization of this system results into Tseng's forward-backward-forward
algorithm with relaxation parameters, which we prove to converge also when it
is applied to pseudo-monotone variational inequalities. In addition, we show
that linear convergence is guaranteed under strong pseudo-monotonicity.
Numerical experiments are carried out for pseudo-monotone variational
inequalities over polyhedral sets and fractional programming problems
Fixed-Time Stable Proximal Dynamical System for Solving MVIPs
In this paper, a novel modified proximal dynamical system is proposed to
compute the solution of a mixed variational inequality problem (MVIP) within a
fixed time, where the time of convergence is finite, and is uniformly bounded
for all initial conditions. Under the assumptions of strong monotonicity and
Lipschitz continuity, it is shown that a solution of the modified proximal
dynamical system exists, is uniquely determined and converges to the unique
solution of the associated MVIP within a fixed time. As a special case for
solving variational inequality problems, the modified proximal dynamical system
reduces to a fixed-time stable projected dynamical system. Furthermore, the
fixed-time stability of the modified projected dynamical system continues to
hold, even if the assumption of strong monotonicity is relaxed to that of
strong pseudomonotonicity. Connections to convex optimization problems are
discussed, and commonly studied dynamical systems in the continuous-time
optimization literature follow as special limiting cases of the modified
proximal dynamical system proposed in this paper. Finally, it is shown that the
solution obtained using the forward-Euler discretization of the proposed
modified proximal dynamical system converges to an arbitrarily small
neighborhood of the solution of the associated MVIP within a fixed number of
time steps, independent of the initial conditions. Two numerical examples are
presented to substantiate the theoretical convergence guarantees.Comment: 12 pages, 5 figure
Characterizations of -well-posedness for parametric quasivariational inequalities defined by bifunctions
The purpose of this paper is to investigate the
well-posedness issue of parametric quasivariational inequalities
defined by bifunctions. We generalize the concept of
-well-posedness to parametric quasivariational inequalities
having a unique solution and derive some characterizations of
-well-posedness. The corresponding concepts of
-well-posedness in the generalized sense are also introduced
and investigated for the problems having more than one solution.
Finally, we give some sufficient conditions for
-well-posedness of parametric quasivariational inequalities
Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators
The purpose of this work is to explore some notions of monotonicity for operators between Banach spaces and the applications to the study of boundary value problems (BVPs) and initial boundary value problems (IBVPs) for partial differential equations (PDEs), with the possibility in the end to examine new problems and provide some solutions. Variational approach will be used to reformulate these problems into stationary equations (in the case of BVPs) and evolution equations (in the case of IBVPs), where the underlined operators constructed as realizations of those problems in appropriate function spaces. This is known as weak formulation, which allows us to find weak solutions of the problems in a larger functions space rather than classical solutions that are sufficiently smooth. The theory of monotone and pseudomonotone operators will be applied to find existence theorems for stationary equations and evolution equations. In addition, the existence theorem for evolution equations with locally monotone operator will also be presented as a generalisation of the one with monotone operators. Another type of monotonicity so-called strict p-quasimonotonicity, which is defined in term of Young measures. This type of weaker, integrated version of monotonicity is directly applied in the study of elliptic and parabolic system of PDEs, the difficulty arises from dealing with this monotonicity is overcome by the theory of Young measures. The application of these monotonicity in the study of variational inequality will also be discussed. In particular, there is a new setting for strict p-quasimonotonicity in a particular type of elliptic variational inequalities, the proof of the new existence theorem will also be presented. Some open problems on the application of strict p-quasimonotonicity in the study of parabolic variational inequalities will also be discussed. Finally, we mention the theory of monotone and pseudomonotone operators in the study of second order evolution equations. A new setting of the local monotonicity in the second order evolution equations will be presented as well as the new existence theorem
Existence results and optimal control for a class of quasi mixed equilibrium problems involving the (f, g, h)-quasimonotonicity
In this paper, by introducing a new concept of the (f, g, h)-quasimonotonicity and applying the maximal monotonicity of bifunctions and KKM technique, we show the existence results of solutions for quasi mixed equilibrium problems when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. Next, we also obtain a result of optimal control to a minimization problem. Our main results can be applied to the problems of evolution equations, differential inclusions and hemivariational inequalities
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