# Questions tagged [set-valued-analysis]

Questions about maps into the powerset of a set (called set-valued maps, multivalued maps, or relations), corresponding concepts of continuity (like upper and lower semicontinuity), inclusion problems (like differential inclusions), maximal monotone maps, hyperspaces (families of subsets of a set, endowed with the Hausdorff distance or Vietoris topology), etc.

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• 5,059
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### Convex optimization over compact sets defined as Aumann set-valued integrals

Let $(X,P)$ be a probability measure space. Let $K$ be a convex compact subset of $\mathbb R^d$ and let $F:X \to 2^{K}$ be a set-valued map. Assume that $F$ is: closed (i.e $F(x)$ is closed for ...
• 6,726
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• 6,726
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### Existence of a global solution to a differential inclusion that does not blow up

Let $\dot{x}(t) \in F(x(t))$ be a differential inclusion, with $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ an uppersemicontinuous, convex and compact valued set-valued map. On Wikipedia it is ...
• 95
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• 5,059
1 vote
71 views

### Closeness of the product of closed convex processes

I asked this question to the math.stackexchange but couldn't get an answer. Let $A$ be a closed convex process from $R^n$ to $R^n$, $I$ be the identity map, $\lambda$ be a real number, and $k$ be a ...
• 173
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### Upper semicontinuity of set-valued maps with open values

Let $X$ and $Y$ be metric spaces. The $(\varepsilon,\delta)$-definition of continuity of single-valued maps can be rephrased as: Let $f$ be a single-valued map from $X$ to $Y$. $f$ is continuous at ...
• 173
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### Random processes with smooth paths

Is there any prototypical example of a Random process with smooth paths? I imagine one can simple integrate each path of a Brownian motion and get a $C^{\frac32-\epsilon}$ path. It's easy to ...
193 views

### Is there a name for this property in set-valued analysis?

Consider a set-valued, finite-valued map $F$ from a set $X$ to subsets of $X$. Consider the following property: $|F(x)| \geq |F(y)|$ for all $x,y$ such that $y \in F(x)$. I have defined this property ...
• 61
172 views

### Differential inclusions for distributions

Given a set valued function $F$ such that for every $x\in M$ (a manifold) we have that $F(x)\subset T_xM$, a differential inclusion is the "equation", $\dot{x} \in F(x)$. I was wondering if someone ...
• 3,878
We are interested in the following question (definitions and references are given below): Main Question: Given an upper-semicontinuous polyhedral multifunction $F:R^n \rightarrow R^m$, is there ...