论文标题
变分问题的粘度近似方法
Viscosity approximation method for a variational problem
论文作者
论文摘要
Let $Q$ be a nonempty closed and convex subset of a real Hilbert space $% \mathcal{H}$, $S:Q\rightarrow Q$ a nonexpansive mapping, $A:Q\rightarrow Q$ an inverse strongly monotone operator, and $f:Q\rightarrow Q$ a contraction mapping.我们证明，在适当的条件下，在真实序列$％\ {α_{α_{n} \} $和$ \ {λ_{n} \}中，$对于任何起点$ x_ {1} $ in $ q，in $ q，$ q，$ x_ {n+1} =α_{n} f（x_ {n}）+（1-α_{n}）sp_ {q}（x_ {n}-λ_{n} ax_ {ax_ {n} ax_ {n} n}） S_{VI(A,Q)}$ which we suppose that it is nonempty, where $F_{ix}(S)$ is the set of fixed point of the mapping $% S$ and $S_{VI(A,Q)}$ is the set of $q\in Q$ such that $\langle Aq,x-q\rangle\geq0$ for every $x\in Q.$ Moreover, we研究上述过程产生的算法的扰动版本的强收敛。最后，我们将主要结果应用于构建与约束凸优化问题相关的算法，并提供了一个数值实验，以强调参数$ \ {α__{n} \} $对该算法收敛速率的影响。
Let $Q$ be a nonempty closed and convex subset of a real Hilbert space $% \mathcal{H}$, $S:Q\rightarrow Q$ a nonexpansive mapping, $A:Q\rightarrow Q$ an inverse strongly monotone operator, and $f:Q\rightarrow Q$ a contraction mapping. We prove, under appropriate conditions on the real sequences $% \{α_{n}\}$ and $\{λ_{n}\},$ that for any starting point $x_{1}$ in $Q,$ the sequence $\{x_{n}\}$ generated by the iterative process \begin{equation} x_{n+1}=α_{n}f(x_{n})+(1-α_{n})SP_{Q}(x_{n}-λ_{n}Ax_{n}) \label{Alg} \end{equation} converges strongly to a particular element of the set $F_{ix}(S)\cap S_{VI(A,Q)}$ which we suppose that it is nonempty, where $F_{ix}(S)$ is the set of fixed point of the mapping $% S$ and $S_{VI(A,Q)}$ is the set of $q\in Q$ such that $\langle Aq,x-q\rangle\geq0$ for every $x\in Q.$ Moreover, we study the strong convergence of a perturbed version of the algorithm generated by the above process. Finally, we apply the main result to construct an algorithm associated to a constrained convex optimization problem and we provide a numerical experiment to emphasize the effect of the parameter $\{α_{n}\}$ on the convergence rate of this algorithm.
