Given $D\subset \mathbb{R}^n$. $f:D\longrightarrow\mathbb{R}^n$ is said to be a monotone operator if $$\left\langle f(x)-f(y),x-y\right\rangle\ge 0,\text{ for all }x,y\in D.$$ I wonder if this hypothesis is true about monotone operators. Given $D\subset \mathbb{R}^n$ and a differentiable operat...

I found the following theorem: Given $H$ Hilbert space and a monotone operator $A\colon H\rightarrow H$, then A is maximal monotone if and only if $\operatorname{Range}(A+I)=H$. Note that: $A$ monotone (multivalued) means that $\forall u,v \in H$ and $\forall f\in Au, g \in Av$, then $(u-v,f-g)...

I trying to prove the following assestment Every linear monotone operator on $L^2 (0, \infty)$ is bounded Any ideas? Thank you

I'm looking for an example of a (multi-valued) maximal monotone operator $A$ mapping a Banach space $X$ into its dual $X^*$ such that the domain $D(A)=\{x\in X: Ax\neq\emptyset\}$ is not convex. Preferably, the example should be simple (maybe with $X=\mathbb{R}^2$). Thanks a lot in advance for s...

I'm reading Barbus & Precupanu's 'Convexity and optimization in Banach spaces'. The authors define what I think is the resolvent for an operator $A: X \to X^*$ (or subset $A \subset X \times X^*$) where $X^*$ is the dual space: Let $ X,X^*$ be reflexive and strictly convex and $A$ maximal mon...

I'm learning monotone operator these days and can't figure out the meaning of the symbol. If you know the symbol please tell me. Thanks for your time!

I am looking for a general definition of monotone condition for a function $G: \mathbb{R}^m \to \mathbb{R}^m$, and since I did not find a unique definition of monotone condition for multivariable function (see Monotonicity of function of two variables), I was wondering if the definition of monoto...

In the paper Ernest K. Yyu, Stephen Noyd - A Primer on Monotone Operator Methods - Survey, the authors frame Iterative Refinement (of an approximate solution to a linear system $Ax=b$) in the context of monotone operators, Resolvents, and Cayley operators. They require $A+A^T \succeq 0$ to ensure...

Definition: Let $H$ a Hilbert space. An unbounded linear operator $A: D(A) \subseteq H \to H$ is said to be monotone if it satisfies $$\forall v \in D(A),\ (Av, v) \ge 0 $$ It is called maximal monotone if, in addition, $R(I + A) = H$, i.e., $\forall f \in H,\ \exists u \in D(A) \tex...

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let $A, B : H \rightarrow H$ be monotone operators, that is (for both $A$ and $B$) $$ \langle A x - ...

Consider a convex optimization problem. $$\min_{u\in\Re^k} f(u)$$ s.t. $g_i(u)\leq0,\ i=1,\ldots,m$ Let $F(x)=F(u,\lambda)=(f'(u)+\sum_{i=1}^m\lambda_ig_i'(u),-g_1(u),\ldots,-g_m(u)):\Re^n\rightarrow\Re^n$, ($n=k+m$) $G=\{x=(u,\lambda)\in\Re^n:\lambda\geq0\}$. $f$ and $g_i$ are all convex. ...

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