Notes on Lipschitz estimates for the stop and play operator in plasticity

We present a generalisation of existing Lipschitz estimates for the stop and play operator for an arbitrary convex and closed characteristic, which contains the origin, in a separable Hilbert space. We are especially concerned with the dependence of stop and play on different scalar products.     

A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems

We propose a strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions.     

Convergence rates for forward–backward dynamical systems associated with strongly monotone inclusions

We investigate the convergence rates of the trajectories generated by implicit first and second-order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space. We show that these trajectories strongly converge with exponential rate to a zero of the sum, provided the latter is strongly monotone. We derive from here convergence rates for the trajectories generated by dynamical systems associated to the minimization of the sum of a proper, convex and lower semicontinuous function with a smooth convex one provided the objective function fulfills a strong convexity assumption. In the particular case of minimizing a smooth and strongly convex function, we prove that its values converge along the trajectory to its minimum value with exponential rate, too.     

Outer approximation methods for solving variational inequalities in Hilbert space

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions on F. We provide several examples for the case where C is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.     

Relations between Lipschitz functions and convex functions

We discuss the relationship between Lipschitz functions and convex functions. By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differentiate to be residual.     

An Example of a Banach Space with Non-Lipschitzian Metric Projection on Any Straight Line

Mathematical Notes - We construct an example of a three-dimensional strictly convex normed space on which the operator of metric projection onto any straight line does not satisfy the Lipschitz...     

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.     

Weakly Convex Sets and Their Properties

In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game.     

Continuous Gradient Projection Method in Hilbert Spaces

This paper is concerned with the asymptotic analysis of the trajectories of some dynamical systems built upon the gradient projection method in Hilbert spaces. For a convex function with locally Lipschitz gradient, it is proved that the orbits converge weakly to a constrained minimizer whenever it exists. This result remains valid even if the initial condition is chosen out of the feasible set and it can be extended in some sense to quasiconvex functions. An asymptotic control result, involving a Tykhonov-like regularization, shows that the orbits can be forced to converge strongly toward a well-specified minimizer. In the finite-dimensional framework, we study the differential inclusion obtained by replacing the classical gradient by the subdifferential of a continuous convex function. We prove the existence of a solution whose asymptotic properties are the same as in the smooth case.     

Gradient projection method for stable approximation of quasisolutions to irregular nonlinear operator equations

An iterative process of the gradient projection type is constructed and examined as a tool for approximating quasisolutions to irregular nonlinear operator equations in a Hilbert space. One step of this process combines a gradient descent step in a finite-dimensional affine subspace and the Fejrér operator with respect to the convex closed set to which the quasisolution belongs. It is proved that the approximations generated by the proposed method stabilize in a small neighborhood of the desired quasisolution, and the diameter of this neighborhood is estimated.     

Iterative algorithms with the regularization for the constrained convex minimization problem and maximal monotone operators

In this paper, we prove a strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the set of solutions of a finite family of variational inclusion problems in Hilbert space. A strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the solution sets of a finite family of zero points of the maximal monotone operator problem in Hilbert space is also obtained. Using our main result, we have some additional results for various types of non-linear problems in Hilbert space.     

Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces

In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

     

The composition of projections onto closed convex sets in Hilbert space is asymptotically regular

The composition of finitely many projections onto closed convex sets in Hilbert space arises naturally in the area of projection algorithms. We show that this composition is asymptotically regular, thus proving the so-called ``zero displacement conjecture' of Bauschke, Borwein and Lewis. The proof relies on a rich mix of results from monotone operator theory, fixed point theory, and convex analysis.

     

Evolution Differential Inclusion with Projection for Solving Constrained Nonsmooth Convex Optimization in Hilbert Space

This paper introduces a projection subgradient system modeled by an evolution differential inclusion to solve a class of hierarchical optimization problems in Hilbert space. Basing on the Moreau–Yosida approximation, we prove the global existence and uniqueness of the solution of the proposed evolution differential inclusion with projection and the unique solution of the proposed system is just its “slow solution” when the constrained set is defined by the affine equalities. When the outer layer objective function ψ is strongly convex, any solution of the proposed system is strongly convergent to the unique minimizer of the constrained optimization problem, while, the strongly convergence is also given when the inner layer objective function ϕ is strongly convex. Furthermore, we present some other optimization problem models, which can be solved by the proposed system. All the results obtained are new not only in the infinite dimensional Hilbert space framework but also in the finite dimensional space.     

Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces

In this paper, we introduce two new numerical methods for solving a variational inequality problem involving a monotone and Lipschitz continuous operator in a Hilbert space. We describe how to incorporate a regularization term depending on a parameter in the projection method and then establish the strong convergence of the resulting iterative regularization projection methods. Unlike known hybrid methods, the strong convergence of the new methods comes from the regularization technique. The first method is designed to work in the case where the Lipschitz constant of cost operator is known, whereas the second one is more easily implemented without this requirement. The reason is because the second method has used a simple computable stepsize rule. The variable stepsizes are generated by the second method at each iteration and based on the previous iterates. These stepsizes are found with only one cheap computation without line-search procedure. Several numerical experiments are implemented to show the computational effectiveness of the new methods over existing methods.     

Lipschitz and Hölder continuity results for some functions of cones

We prove the Lipschitz continuity of the maximal angle function on the set of closed convex cones in a Hilbert space. A similar result is obtained for the minimal angle function. On the other hand, we prove that the incenter of a solid cone and the circumcenter of a sharp cone behave in a locally Hölderian manner.     

一类多值算子方程的迭代算法

本讨论Hilbert空间上具有强单调、上半Lipschitz连续性质的多值算子方程的求解方法,构造了迭代格式并证明迭代过程是收敛的;同时给出它在求解多值微分方程边值问题中的应用。     

Regularization of ill-posed constrained minimization problems by iteration methods

The ill-posed minimization problems in Hilbert space with quadratic objective function and closed convex constraint set are considered. For the compact set the regularization methods for such problems are well understood [1, 2] The regularizing properties of some Iteration projection methods for noncompact constraint set are the main issues of this paper. We are looking the gradient projection method for the sphere.     

On the stability of iterative approximations to fixed points of nonexpansive mappings

We study iterative retraction approximations to fixed points of the nonexpansive self-mapping given on the closed convex set G in a Banach space B. The conditions which guarantee weak and strong convergence and stability of these approximations with respect to perturbations of both operator A and constraint set G are considered. The results of this paper are new even in a Hilbert space for the iterative projection approximations.     

Hilbert geometry for convex polygonal domains

We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to the Euclidean plane.