Nonempty Closed Convex Set Research Articles

Controllability for Problems with Mixed Constraints

For a controlled system with mixed equality- and inequality-type constraints anda geometric constraint that is a nonempty closed convex set, sufficient conditions for the existenceof feasible positional controls are obtained in terms of first derivatives of mappings that define themixed constraints. In addition, in terms of the first and second derivatives of these mappings,sufficient conditions for the existence of feasible positional controls are found that are alsoapplicable in the case of degeneration of the first derivatives of the mappings.

Viability for Itô stochastic systems with non-Lipschitzian coefficients and its application

In this article, we study the viability for the multi-dimensional stochastic system described by Itô stochastic differential equations under the non-Lipschitzian assumption on the coefficients. We establish a necessary and sufficient condition for viability property with respect to a given nonempty closed convex set. And with this result, we explore the multi-dimensional comparison theorems about stochastic differential equations.

Best approximation mappings in Hilbert spaces

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional spaces was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of a BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspaces to nonempty closed convex sets and from $${\mathbb {R}}^{n}$$ to general Hilbert spaces. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al. proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $${\mathbb {R}}^{n}$$ . We generalize their result from $${\mathbb {R}}^{n}$$ to general Hilbert spaces and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAMs with circumcenter mappings in Hilbert spaces and we report on Cegielski’s results on relationships with strongly quasinonexpansive mappings.

The difference vectors for convex sets and a resolution of the geometry conjecture

The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets.

Lagrange Multiplier Characterizations of Constrained Best Approximation with Infinite Constraints

In this paper, we first employ the subdifferential closedness condition and Guignard’s constraint qualification to present “dual cone characterizations” of the constraint set $$ \varOmega $$ with infinite nonconvex inequality constraints, where the constraint functions are Frechet differentiable that are not necessarily convex. We next provide sufficient conditions for which the “strong conical hull intersection property” (strong CHIP) holds, and moreover, we establish necessary and sufficient conditions for characterizing “perturbation property” of the best approximation to any $$x \in {\mathcal {H}}$$ from the convex set $$ \tilde{\varOmega }:=C \cap \varOmega $$ by using the strong CHIP of $$\lbrace C,\varOmega \rbrace ,$$ where C is a non-empty closed convex set in the Hilbert space $${\mathcal {H}}.$$ Finally, we derive the “Lagrange multiplier characterizations” of constrained best approximation under the subdifferential closedness condition and Guignard’s constraint qualification. Several illustrative examples are presented to clarify our results.

Fractional model of stem cell population dynamics

Abstract We develop the fractional model of stem cell population dynamics with state-dependent and time-dependent delays. In this model, the stem cell division rate and self-renewal rate are controlled by an external signal, depending on the effects of the environment’s heterogeneity. We quantify a relationship between the fractional derivative order, which shows the effects of the environment’s heterogeneity and stem cell division and stem cell self-renewal rate. We consider a general form of the fractional neutral delay differential equations with state-dependent and time-dependent delay to study this relationship. First, we show the solution’s existence and uniqueness using the fixed point theorem on the Banach space. We define a completely continuous operator on the non-empty closed convex set to use the fixed point theorem on the Banach space and show this operator has a uniquely defined fixed point. Also, we proof the Ulam–Hyers stability to make sure a close exact solution could be reached using the numerical approximation. Then, we develop a new numerical method based on Jacobi polynomials for solving the fractional neutral delay differential equations with state-dependent and time-dependent delay. We use the least-squares approximation of the candidate function to reduce the solution of fractional neutral delay differential equations to a set of algebraic equations and compare the results obtained by using different collocation points. We evaluate the accuracy of the numerical method, theoretically and numerically. We have used the numerical method to evaluate the fractional model’s behavior of stem cell population dynamics and quantify the relationship between the effects of the environment’s heterogeneity and the rate of stem cell division and stem cell self-renewal.

Viability property for multi-dimensional stochastic differential equation and its applications to comparison theorem

In this paper, we focus on the problem of viability property for the multi-dimensional stochastic differential equations, where the coefficients of the stochastic differential equations may be random and discontinuous in the time variable. First, we establish a necessary and sufficient condition for viability property with respect to a given non-empty closed convex set. And furthermore, with the help of this result, we also study the multi-dimensional comparison theorems about stochastic differential equations.

A Levenberg–Marquardt Method for Solving the Tensor Split Feasibility Problem

This paper considers the tensor split feasibility problem. Let C and Q be non-empty closed convex set and $${\mathcal {A}}$$ be a semi-symmetric tensor. The tensor split feasibility problem is to find $$x\in C $$ such that $${\mathcal {A}} x^{m-1} \in Q$$ . If we simply take this problem as a special case of the nonlinear split feasibility problem, then we can directly get a projection method to solve it. However, applying this kind of projection method to solve the tensor split feasibility problem is not so efficient. So we propose a Levenberg–Marquardt method to achieve higher efficiency. Theoretical analyses are conducted, and some preliminary numerical results show that the Levenberg–Marquardt method has advantage over the common projection method.

On the asymptotic behavior of the Douglas\u2013Rachford and proximal-point algorithms for convex optimization

Banjac et al. (J Optim Theory Appl 183(2):490–519, 2019) recently showed that the Douglas–Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. In particular, they showed that the difference between consecutive iterates generated by the algorithm converges to certificates of primal and dual strong infeasibility. Their result was shown in a finite-dimensional Euclidean setting and for a particular structure of the constraint set. In this paper, we extend the result to real Hilbert spaces and a general nonempty closed convex set. Moreover, we show that the proximal-point algorithm applied to the set of optimality conditions of the problem generates similar infeasibility certificates.

Some Notes on the Differentiability of the Support Function

Let X be a Banach space, C be a nonempty closed convex set and $$\sigma _C$$ be the support function of the set C. In this work, we give some necessary and sufficient conditions for the set C to satisfy $$\mathrm {int}(\mathrm {dom}\sigma _C)\ne \emptyset $$ , which enable us to study the Frechet and Gateaux differentiability of support function on the set C.

Solving k-center problems involving sets based on optimization techniques

The continuous k-center problem aims at finding k balls with the smallest radius to cover a finite number of given points in $$\mathbb {R}^n$$. In this paper, we propose and study the following generalized version of the k-center problem: Given a finite number of nonempty closed convex sets in $$\mathbb {R}^n$$, find k balls with the smallest radius such that their union intersects all of the sets. Because of its nonsmoothness and nonconvexity, this problem is very challenging. Based on nonsmooth optimization techniques, we first derive some qualitative properties of the problem and then propose new algorithms to solve the problem. Numerical experiments are also provided to show the effectiveness of the proposed algorithms.

Distributed Projection Subgradient Algorithm Over Time-Varying General Unbalanced Directed Graphs

This paper is concerned with a general class of distributed constrained optimization problems over a multiagent network, where the global objective function is represented by the sum of all local objective functions. Each agent in the network only knows its own local objective function, and is restricted to a global nonempty closed convex set. We discuss the scenario where the communication of the whole multiagent network is expressed as a sequence of time-varying general unbalanced directed graphs. The directed graphs are required to be uniformly jointly strongly connected and the weight matrices are only row-stochastic. To collaboratively deal with the optimization problems, existing distributed methods mostly require the communication graph to be fixed or balanced, which is impractical and hardly inevitable. In contrast, we propose a new distributed projection subgradient algorithm which is applicable to the time-varying general unbalanced directed graphs and does not need each agent to know its in-neighbors’ out-degree. When the objective functions are convex and Lipschitz continuous, it is proved that the proposed algorithm exactly converges to the optimal solution. Simulation results on a numerical experiment are shown to substantiate feasibility of the proposed algorithm and correctness of the theoretical findings.

Space of Continuous Set-Valued Mappings with Closed Unbounded Values

We consider a space of continuous set-valued mappings defined on a locally compact space T with a countable base. The values of these mappings are closed (not necessarily bounded) sets in a metric space (X, d(·)) whose closed balls are compact. The space (X, d(·)) is locally compact and separable. Let Y be a countable dense set in X. The distance ρ(A,B) between sets A and B belonging to the family CL(X) of all nonempty closed subsets of X is defined as follows: $$\rho (A, B) = \sum_{i=1}^\infty\frac{1}{2^i}\frac{|d(y_i, A) - d(y_i, B)|}{1+|d(y_i, A) - d(y_i, B)|},$$ where d(yi,A) is the distance from the point yi ∈ Y to the set A. This distance is independent of the choice of the set Y, and the function ρ(A,B) is a metric on the space CL(X). The convergence of a sequence of sets An, n ≥ 1, in the metric space (CL(X), ρ(·)) is equivalent to the Kuratowski convergence of this sequence. We prove the completeness and separability of the space (CL(X), ρ(·)) and give necessary and sufficient conditions for the compactness of sets in this space. The space C(T,CL(X)) of all continuous mappings from T to (CL(X), ρ(·)) is endowed with the topology of uniform convergence on compact sets in T. We prove the completeness and separability of the space C(T,CL(X)) and give necessary and sufficient conditions for the compactness of sets in this space. These results are reformulated for the space C(T,CCL(X)), where T = [0, 1], X is a finite-dimensional Euclidean space, and CCL(X) is the space of all nonempty closed convex sets in X with metric ρ(·). This space plays a crucial role in the study of sweeping processes. We give a counterexample showing the significance of the assumption of compactness of closed balls in X.

On a convex set with nondifferentiable metric projection

A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with \(C^{1,1}\) boundary which possesses the same property.

Strong convergence of iterative algorithms for the split equality problem

Abstract Let H 1 , H 2 , H 3 be real Hilbert spaces, C ⊆ H 1 , Q ⊆ H 2 be two nonempty closed convex sets, and let A : H 1 → H 3 , B : H 2 → H 3 be two bounded linear operators. The split equality problem (SEP) is finding x ∈ C , y ∈ Q such that A x = B y . Recently, Moudafi has presented the ACQA algorithm and the RACQA algorithm to solve SEP. However, the two algorithms are weakly convergent. It is therefore the aim of this paper to construct new algorithms for SEP so that strong convergence is guaranteed. Firstly, we define the concept of the minimal norm solution of SEP. Using Tychonov regularization, we introduce two methods to get such a minimal norm solution. And then, we introduce two algorithms which are viewed as modifications of Moudafi’s ACQA, RACQA algorithms and KM-CQ algorithm, respectively, and converge strongly to a solution of SEP. More importantly, the modifications of Moudafi’s ACQA, RACQA algorithms converge strongly to the minimal norm solution of SEP. At last, we introduce some other algorithms which converge strongly to a solution of SEP.

Iterative algorithms for finding the zeroes of sums of operators

Abstract Let H 1 , H 2 be real Hilbert spaces, C ⊆ H 1 be a nonempty closed convex set, and 0 ∉ C . Let A : H 1 → H 2 , B : H 1 → H 2 be two bounded linear operators. We consider the problem to find x ∈ C such that A x = − B x ( 0 = A x + B x ). Recently, Eckstein and Svaiter presented some splitting methods for finding a zero of the sum of monotone operator A and B. However, the algorithms are largely dependent on the maximal monotonicity of A and B. In this paper, we describe some algorithms for finding a zero of the sum of A and B which ignore the conditions of the maximal monotonicity of A and B.

Strong Convergence Theorems for Variational Inequalities and Split Equality Problem

LetH1, H2,andH3be real Hilbert spaces, letC⊆H1, Q⊆H2be two nonempty closed convex sets, and letA:H1→H3, B:H2→H3be two bounded linear operators. The split equality problem (SEP) is to findx∈C, y∈Qsuch thatAx=By. LetH=H1×H2; considerf:H→Ha contraction with coefficient0<α<1, a strongly positive linear bounded operatorT:H→Hwith coefficientγ̅>0, andM:H→His aβ-inverse strongly monotone mapping. Let0<γ<γ̅/α,S=C×QandG:H→H3be defined by restricting toH1isAand restricting toH2is-B, that is,Ghas the matrix formG=[A,-B]. It is proved that the sequence{wn}={(xn,yn)}⊆Hgenerated by the iterative methodwn+1=PS[αnγf(wn)+(I-αnT)PS(I-γnG*G)PS(wn-λnMwn)]converges strongly tow̃which solves the SEP and the following variational inequality:〈(T-λf)w̃,w-w̃〉≥0and〈Mw̃,w-w̃〉≥0for allw∈S. Moreover, if we takeM=G*G:H→H, γn=0, thenMis aβ-inverse strongly monotone mapping, and the sequence{wn}generated by the iterative methodwn+1=αnγf(wn)+(I-αnT)PS(wn-λnG*Gwn)converges strongly tow̃which solves the SEP and the following variational inequality:〈(T-λf)w̃,w-w̃〉≥0for allw∈S.

Regularization Method for the Approximate Split Equality Problem in Infinite-Dimensional Hilbert Spaces

We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let , , and   be infinite-dimensional real Hilbert spaces, let and   be two nonempty closed convex sets, and let and   be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function over and . Recently, Moudafi and Byrne had proposed several algorithms for solving the split equality problem and proved their convergence. Note that their algorithms have only weak convergence in infinite-dimensional Hilbert spaces. In this paper, we used the regularization method to establish a single-step iterative for solving the ASEP in infinite-dimensional Hilbert spaces and showed that the sequence generated by such algorithm strongly converges to the minimum-norm solution of the ASEP. Note that, by taking in the ASEP, we recover the approximate split feasibility problem (ASFP).

Approximating solutions of generalized pseudocontractive variational inequalities by admissible perturbation type iterative methods

In this paper we give the solvability class of generalized strongly nonlinear variational inequalities modified by the use of the new concept of admissible perturbation operator on nonempty closed convex sets in Hilbert spaces.

On the Levenberg-Marquardt methods for convex constrained nonlinear equations

In this paper, both the constrained Levenberg-Marquardt method and the projected Levenberg-Marquardt method are presented for nonlinear equations $F(x)=0$ subject to $x\in X$, where $X$ is a nonempty closed convex set. The Levenberg-Marquardt parameter is taken as $\| F(x_k) \|_2^\delta$ with $\delta\in (0, 2]$. Under the local error bound condition which is weaker than nonsingularity, the methods are shown to have the same convergence rate, which includes not only the convergence results obtained in [12] for $\delta=2$ but also the results given in [7] for unconstrained nonlinear equations.