# Tagged Questions

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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### Is the function $f(x) = |x|$ convex?

I am asking because some Internet pages contradict. In Wikipedia, the definition I found of a convex function is: "Let $X$ be a convex set in a real vector space and let $f: X \to R$ be a function. •...
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### Minimum of biconjugate of a nonconvex $f(x)$ is the minimum of $f(x)$ also?

In Fazel (2002) Matrix rank minimization with applications, Ch. 5.1.4-5.1.5, the author finds an analytic expression for the convex biconjugate of their nonconvex function; however, they state that ...
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### Are convex functions enough to determine a measure?

Suppose we are talking about $\mathbb{R}^n$. We know that if $\mu$, $\nu$ are two finite Borel measures such that $$\int_{\mathbb{R}^n}f(x) \, d\mu(x)=\int_{\mathbb{R}^n}f(x) \, d\nu(x),$$ for all ...
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### $A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A$ is non-empty ?
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### Interior of polar cone and self-concordant function

Suppose $f$ is a self-concordant(see 9.6.2) barrier of a proper cone $K$ ( solid,convex, closed and pointed) in $\mathbb{R}^n$. It looks like the value of all $\nabla f$ is just the interior of the ...
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### Convex Optimization Closed Form Solution

I'm currently studying for my exame in convex optimization. It covers problem formulation, first and second order conditions of optimality, unconstrained optimization, constrained optimization with ...
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### Prove if C is midpoint convex and closed then its a convex set [duplicate]

Midpoint convexity. A set C is midpoint convex if whenever two points a,b are in C, the average or midpoint (a + b)/2 is in C.Prove that if C is closed and midpoint convex, then C is convex. Convex ...
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### Does this relation imply convexity?

I'm trying to figure out wheter the following condition inplies convexity or not. Let $\cal{X}$ be an inner product space with inner product $\langle \cdot, \cdot \rangle$ and a norm $\|\cdot\|$ (not ...
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### Prove that the sum of convex functions is again convex.

I must to prove that the sum of convex functions is again convex. I know the definition of convex function: $f(tx_1+(1-t)x_2)\leq f(x_1)+(1-t)f(x_2)$ - this the first convex function, then I have the ...
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### Convex analysis of a vectorial function

In the context of equilibrium equations on structural concrete study, I deal with an system (3 non linear equations) that relates an entry of variables $(\epsilon_0,\kappa_x,\kappa_y)$ (which ...
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### On convexity of $\frac{1}{x}$

I would like to prove convexity of $\frac{1}{x}$. It can be proved by using second derivative but I want without using second derivative. Can someone help me?
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### Is the constraint $A^2 = B^2$ convex

I am trying to use a continuous constraint to replace a discrete equation $A = |B|$ in my model. Since the linear programming method for absolute value is inapplicable in my model, I come up with ...
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### Is minmax equivalent to maxmin?

More precisely, problem $1$ is as follows: \begin{eqnarray} &\max_{1{\le}i{\le}N}\min_{[\gamma_m^i]}\left[\lambda_i - \sum_{m=1}^M\phi_m\gamma_m^iF_m^i\right] \\ &\mbox{subject to} \sum_{i=1}^...
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### geometric representations in convex analysis

Do you have any advices that help having geometric representations in convex analysis ? (for instance examples you always keep in mind when you are working, websites with simulations, graphs , ...) ...