197 research outputs found
The proximal point method for nonmonotone variational inequalities
We consider an application of the proximal point method to variational inequality problems subject to box constraints, whose cost mappings possess order monotonicity properties instead of the usual monotonicity ones. Usually, convergence results of such methods require the additional boundedness assumption of the solutions set. We suggest another approach to obtaining convergence results for proximal point methods which is based on the assumption that the dual variational inequality is solvable. Then the solutions set may be unbounded. We present classes of economic equilibrium problems which satisfy such assumptions
Proximal Gradient Methods Beyond Monotony
We address composite optimization problems, which consist in minimizing the
sum of a smooth and a merely lower semicontinuous function, without any
convexity assumptions. Numerical solutions of these problems can be obtained by
proximal gradient methods, which often rely on a line search procedure as
globalization mechanism. We consider an adaptive nonmonotone proximal gradient
scheme based on an averaged merit function and establish asymptotic convergence
guarantees under weak assumptions, delivering results on par with the monotone
strategy. Finally, we derive global worst-case rates for the iterates and a
stationarity measure.Comment: 16 pages, 1 algorith
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
Advances in variational and hemivariational inequalities : theory, numerical analysis, and applications
Highlighting recent advances in variational and hemivariational inequalities with an emphasis on theory, numerical analysis and applications, this volume serves as an indispensable resource to graduate students and researchers interested in the latest results from recognized scholars in this relatively young and rapidly-growing field. Particularly, readers will find that the volume’s results and analysis present valuable insights into the fields of pure and applied mathematics, as well as civil, aeronautical, and mechanical engineering. Researchers and students will find new results on well posedness to stationary and evolutionary inequalities and their rigorous proofs. In addition to results on modeling and abstract problems, the book contains new results on the numerical methods for variational and hemivariational inequalities. Finally, the applications presented illustrate the use of these results in the study of miscellaneous mathematical models which describe the contact between deformable bodies and a foundation. This includes the modelling, the variational and the numerical analysis of the corresponding contact processes. Furthermore, it can be used as supplementary reading material for advanced specialized courses in mathematical modeling for students with a strong background knowledge on nonlinear analysis, numerical analysis, partial differential equations, and mechanics of continua
Convergence Properties of Monotone and Nonmonotone Proximal Gradient Methods Revisited
Composite optimization problems, where the sum of a smooth and a merely lower
semicontinuous function has to be minimized, are often tackled numerically by
means of proximal gradient methods as soon as the lower semicontinuous part of
the objective function is of simple enough structure. The available convergence
theory associated with these methods (mostly) requires the derivative of the
smooth part of the objective function to be (globally) Lipschitz continuous,
and this might be a restrictive assumption in some practically relevant
scenarios. In this paper, we readdress this classical topic and provide
convergence results for the classical (monotone) proximal gradient method and
one of its nonmonotone extensions which are applicable in the absence of
(strong) Lipschitz assumptions. This is possible since, for the price of
forgoing convergence rates, we omit the use of descent-type lemmas in our
analysis.Comment: 23 page
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