# Questions tagged [convex-optimization]

A convex optimization problem consists of either minimizing a convex objective or maximizing a concave objective over a convex feasible region.

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### Equivalent characterization of L-smooth function

If a function $f(x)$ is $L$-smooth, then is it equivalent to say that $$f(x) - \dfrac{1}{2L} || \nabla f(x)||^2 \geq 0 ?$$ Can someone help me prove this? I have the definition that a function $f(x)$ ...
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### Optimizing a linear functional over nested sets

Let $F_0\supseteq F_1\supseteq F_2\supseteq\cdots$ be a sequence of nested closed convex sets in $\mathbb{R}^d$ with some $d\ge 1$. Assume that $0\in F_n$ for all $n\ge 0$. Let $a\in\mathbb{R}^d$ be a ...
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### Two questions about Convex Sets [closed]

There are 2 questions that I can't make any idea while solving. Can you give an idea how I can solve it? For C convex, show that C is closed if and only if C $\cap$ L is closed, for any affine line L?...
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### Why $[(\mathbf{I}_N-\mathbf{A}^\top \mathbf{A})\mathbf{x}]$ is Gaussian with i.i.d. Gaussian $\mathbf{A}$?

1. Background: It is presented in the paper of approximate message passing (AMP) algorithm [Paper Link] that (the conclusion below is slightly modified without changing its original meaning): Given a ...
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### Convex and Lipchitz continuity of functions [closed]

If a function is convex, is it true its gradient satisfies Lipchitz continuity?
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### $\min\{200x+100y\}$ such that $x+3y\geq12$, $3x+2y\geq12$, $x\geq0,y\geq0$

$\min\{200x+100y\}$ such that $2x+3y\geq12$, $3x+2y\geq12$, $x\geq0,y\geq0$ Attempt. I am using the lagrangian approach to attack this problem and I know the optimal solution is $(0,6)$, but I cannot ...
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### Duality optimisation

enter image description here Questions. what does that symbol mean between Ax and b? he has moved the b to Ax-b in the subject too section is this because all constraints have to be on one side ? if ...
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### Optimality condition for a convex function subject to a unit probability simplex

For the optimization problem: $$\text{minimize}~~f(x)$$ $$\text{subject to}~~x\in{\Delta_n}$$ where ${\Delta_n}=\{x\in\mathbb{R}^n|\sum_{i=1}^nx_i=1,x\geq0\}$, and $f$ is a proper convex function such ...
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### Minimize a function regarding two coupling variables

Given known matrics $A\in \mathbb R^{2\times 2}$ and known vectors $b\in \mathbb R^2, c\in \mathbb R^2$, for the two optimization variables $x\in \mathbb R^2$ and $y\in \mathbb R$, how to obtain the ...
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### Strongly convexity of a loss function

I want to calculate the strongly convex parameter $\sigma$ for this loss function: $$l_Z(Z)=||Z-A||^2_F+\lambda tr[Z^TBZ]$$ where $Z\in \mathcal{R}^{n\times m}$, the value of $A,B$ and $\lambda$ are ...
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### How do you represent the number of iterations in the formulation of an optimization problem?

Let's say I want to minimize some function for f(x), with respect to x, in the minimum number of iterations. How would I represent the number of iterations in the formulation of this optimization ...
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### Projecting a point onto a convex set given by Log-Sum-Exp

Motivated by a wish to encode signal temporal logic specs (with linear predicates) as optimization problems w/o mixed integer approaches, I've been attempting to find a way to define the projection ...
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### Modify maximum entropy problem to fix specific Lagrange multiplier

I have an optimization problem of the form $\min_p -H(p) \text{ s.t. } \sum_{x \in \mathcal{X}} p(x)=1 \text{ and } g_i(p)=c_i \text{ for } i=1, \dots, m$ where $p \in [0,1]^{|\mathcal{X}|}$ is a ...
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### How to prove the following Interior point equivalence ?(COROLLARY 6.4.1 in Convex Analysis, Rockafellar) [duplicate]

COROLLARY 6.4.1. Let $C$ be a convex set in $R^{n}$. Then $z \in$ int $C$ if and only if, for every $y \in R^{n}$, there exists some $\varepsilon>0$ such that $z+\varepsilon y \in C$. How to prove:...
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### Why dim$(C_1)$ $<$ dim$(C_2)$? Where $C_1$ is a convex subset of the relative boundary of convex set $C_2$.

How to understand "it would have interior points relative to $\text{aff}(C_2)$"? I only find that $\text{aff}(C_1)=\text{aff}(C_2)$ by assume dim$C_1$=dim$C_2$.
Let $C$ be any convex set in $\mathbb{R}^n$, then $\mathrm{cl}(C)$ and $\mathrm{relint}(C)$ having the same affine hull. I know $\mathrm{aff}(\mathrm{cl}(C))=\mathrm{aff}(C)$ . But how to prove that \...