# Questions tagged [convex-analysis]

Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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### A Slight Difference Between the Definition of Sub-gradient in Different Literatures

To start, I would like to state both definitions for sub-gradient and where they came from. Definition 1 Let $f$ be $\mathbb E \mapsto \mathbb {\bar R}$ with $x\in \text{dom}(f)$ then $v\in \mathbb E$...
1 vote
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### Strong convergence of subgradients of a convex Fréchet differentiable function on a normed space

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
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### Weak$^*$ convergence of subgradients of a convex continuous function on a normed space

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
• 1,425
23 views

### Is this function strongly convex? approaching boundary case

The definition for strongly convex is: A differentiable function $f$ is strongly convex if $$f(y) \geq f(x)+\nabla f(x)^{T}(y-x)+\frac{\mu}{2}\|y-x\|^{2}$$ for some $\mu>0$ and all $x, y$. And ...
1 vote
24 views

### Assume $A$ is open convex and $f$ convex continuous. Then $f$ is Gâteaux differentiable at $a \in A$ if and only if $\partial f (a)$ is a singleton

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
• 1,425
12 views

### Characterize directional derivative of a convex continuous function by subdifferential

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
• 1,425
17 views

### Characterize subdifferential of a convex function by directional derivative

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
• 1,425
17 views

### Assume $A$ is open convex and $f$ convex continuous. Then $f$ is $L$-lipschitz on $A$ if and only if $\partial f(A) \subset L B_{X^*}$

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
• 1,425
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### Random convex functions

I am collaborating with a couple of machine learning specialists on a project related to phase retrieval. The problem we are investigating has solutions which are convex 2D functions on a rectangular ...
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1 vote
23 views

### Showing that there always exists an optimal mixed strategy for two player zero sum games

Exercise Let there be a game of two players $A,B$ and $x',y'$ optimal strategies such that: \begin{align} f(x') = \max_{x \in A} \min_{y \in B} E(x,y) = \min_{y \in B} \max_{x \in A} E(x,y) = g(y'). \...
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1 vote
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### Approximate a convex proper lower semi-continuous function by an increasing sequence of Lipchitz-continuous convex ones

I'm trying to prove this well-known approximation of proper l.s.c. convex function. Could you have a check on my attempt? Let $(X, | \cdot|)$ be a normed space and $f:X \to \mathbb R \cup \{+\infty\}$...
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32 views

### must max of sum lie in convex subspace? [closed]

$f$ and $g$ are two concave functions defined on a convex subset $A$ of a vector space. Each of $f$ and $g$ achieves its maximum on a convex subset $B$ of $A$. Does it imply that the maximum of $f+g$ ...
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### The relation between convex conjugate and subdifferential

I'm reading about convex conjugate and its relation to subdifferential. In order to characterise subgradients we will use the convex conjugate defined below. This is essentially a special case of the ...
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### An example of convex function that is not lower semi-continuous [duplicate]

Let $f:\mathbb R^n \to \mathbb R \cup \{+\infty\}$ be convex. Let $D := \{x\in \mathbb R^n \mid f(x) \in \mathbb R\}$ be the domain of $f$. Then $f$ is continuous on $\operatorname{int} D$. Could you ...
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31 views

### Krein Milman theorem, when the closure is not needed

Krein-Milman theorem states that If ${K}$ is a non-empty compact convex subset of a locally convex space ${X}$, then ${\text{ext}\;K\neq\emptyset}$ and ${K=\overline{\text{co}}(\text{ext}\;K)}.$ Note ...
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27 views

### Continuous convex function on a normed vector space has non-empty subdifferential at every point

I'm trying to prove this well-known result. Could you have a check on my attempt? Theorem: Let $X$ be a normed vector space and $A$ an open convex subset of $X$. Let $f:A \to \mathbb R$ be convex ...
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38 views

### Quasiconcavity/ Quasiconvexity of $f: \mathbb R \to \mathbb R\ldots$ [closed]

The function $f: \mathbb{R} \to \mathbb{R}$ given by $$f(x)=\begin{cases}\frac{x}{|x|}, &\text{if x\neq 0}\\\\1, &\text{if x=0}\end{cases}$$ is Concave Convex Neither Concave nor Convex ...
1 vote
### A Clarkson-type inequality $|a+b+c|^q + \sum\limits_{\mathrm{cyc}} |a + b - c|^q \ge \sum\limits_{\mathrm{cyc}} (|a + b|^q + |a - b|^q)$
Show that, for $a,b,c \in \mathbb{R}$ and $q \geq 2$: $$|a+b|^{q} + |a-b|^{q} + |a+c|^q + |a-c|^q + |b+c|^q + |b-c|^q \\\leq \\|a+b+c|^q + |a+b-c|^q + |a-b+c|^q + |a-b-c|^q$$ When $c=0$, this ...