1,169 research outputs found

    A System of Generalized Variational Inclusions Involving a New Monotone Mapping in Banach Spaces

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    We introduce a new monotone mapping in Banach spaces, which is an extension of the -monotone mapping studied by Nazemi (2012), and we generalize the variational inclusion involving the -monotone mapping. Based on the new monotone mapping, we propose a new proximal mapping which combines the proximal mapping studied by Nazemi (2012) with the mapping studied by Lan et al. (2011) and show its Lipschitz continuity. Based on the new proximal mapping, we give an iterative algorithm. Furthermore, we prove the convergence of iterative sequences generated by the algorithm under some appropriate conditions. Our results improve and extend corresponding ones announced by many others

    A first-order stochastic primal-dual algorithm with correction step

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    We investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in {\em Y. Drori, S. Sabach, and M. Teboulle, A simple algorithm for a class of nonsmooth convex-concave saddle-point problems, 2015} and {\em I. Loris and C. Verhoeven, On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty, 2011} for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi Fej\'er-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models

    Evolutionary Oseen model for generalized Newtonian fluid with Multivalued Nonmonotone Friction Law

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    The paper deals with the non-stationary Oseen system of equations for the generalized Newtonian incompressible fluid with multivalued and nonmonotone frictional slip boundary conditions. First, we provide a result on existence of a unique solution to an abstract evolutionary inclusion involving the Clarke subdifferential term for a nonconvex function. We employ a method based on a surjectivity theorem for multivalued L-pseudomonotone operators. Then, we exploit the abstract result to prove the weak unique solvability of the Oseen system

    A class of differential hemivariational inequalities in Banach spaces

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    In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function u↦f(t,x,u) and compactness of C0-semigroup eA(t)

    Existence results and optimal control for a class of quasi mixed equilibrium problems involving the (f, g, h)-quasimonotonicity

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    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

    Optimal Control of Semilinear Evolution Inclusions via Discrete Approximations

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    This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions for constrained continuous-time evolution inclusions by passing to the limit from discrete approximations

    Controllability and Observability of Control Systems under Uncertainty

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    This report surveys the results of nonlinear systems theory (controllability and observability) obtained at IIASA during the last three summers. Classical methods based on differential geometry require some regularity and fail as soon as state-dependent constraints are brought to bear on the controls, or uncertainty and disturbances are involved in the system. Since these important features appear in most realistic control problems, new methods had to be devised, which encompass the classical ones, and allow the presence of a priori feedback into the control systems. This is now possible thanks to new tools, in the development of which IIASA played an important role: differential inclusions and set-valued analysis

    Constraint Qualifications and Optimality Conditions for Nonconvex Semi-Infinite and Infinite Programs

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    The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian-Fromovitz and Farkas-Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for efficient computing the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.Comment: 28 page
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