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.

5,836 questions
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Problem calculating the Dual of a convex problem

The problem is $$(P) \hspace{1cm} \begin{array}{ll}\min & e^{-x} \\ \text{s.t.} & \frac{x^2}{y} \leq 0 \end{array}$$ over the domain $\mathcal{D}= \{(x,y)| y>0\}$, which I tried to ...
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Why does strong convexity imply this inequality?

From Convex Optimization by Boyd and Vandenberghe: Let $f: \Bbb R^n \rightarrow \Bbb R$ be continuous and twice differentiable. Assume $$\|\nabla f(x)\|_2 \le \eta \le 3(1-2 \alpha)m^2/L$$ ...
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Star-Convex Set Centers Form Convex Set

I have the following proof that all centers of a star-convex set $U\subset\mathbb{R}^n$, $Z(U)$ form a convex set: Suppose $Z(U)$ is not convex, then there are elements $a,b\in Z(U)$ such that the ...
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Gradient always points away from the minima of convex functions?

I'm reading a paper by Léon Bottou, "Online Learning and Stochastic Approximations". He studies online learning with cost functions $C(w)$ satisfying two conditions: $C(w)$ has a single minimum $w^*$....
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Prove the convexity of this function [on hold]

Prove the convexity of this function without verifying the definiteness of its Hessian matrix $$f(x_1,x_2) = x_1 *\log_2(1+x_2/x_1)$$ with $$x_1,x_2 \geq 0 ;\: x_1,x_2 \in \Bbb R$$
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If $\int_0^1 f(x+\theta(y-x))d\theta \leq \frac{f(x)+f(y)}{2}$ then f is convex

Suppose that we have $f: \mathbb{R}\rightarrow\mathbb{R}$ is Riemann integrable and $$\int_0^1 f(x+\theta(y-x))d\theta \leq \frac{f(x)+f(y)}{2} \ \ \ (\ast)$$ for all $x,y\in\mathbb{R}$. Is this ...
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convex hull for tangent function

I would like to ask if I can obtain a convex hull for the function tan(S2-S1) where S1 and S2 are unknown angles while knowing the maximum/minimum limits on (S2-S1).
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convexity of negative log likelihood under constraints

The negative log-likelihood function \begin{align} \text{min}_B -log|\Sigma^{-1}| + \text{tr}[(Y-XB)\Sigma^{-1}(Y-XB)] \end{align} is a convex optimization with respect to $\Sigma^{-1}$. However, ...
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Why isn't this function affine?

I'm studying convex optimization using Convex Optimization (Boyd & Vandenberghe) and had a question from an example used in Chapter 4.2: Convex Optimization. The specific example is as follows: ...
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Strict inequality of the H1 seminorm of a curve emerging from a convex set and its convex projection

Let $\gamma(t) \in W^{1,2}([0,1]; \mathbb{R}^d)$ such that $\gamma(0) \in C$, with $C \subset \mathbb{R}^d$ some closed convex set, and $\gamma(t) \not \in C, \ \forall t\in (0,1]$. Furthermore we ...
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Finite set of linear functional attaining certain values (Brezis functional analysis exercise 1.12)

Problem: Let $E$ be a vector space. Fix $n$ linear functionals $(f_{i})_{1\leq i\leq n}$ on $E$ and $n$ real numbers $(\alpha_{i})_{1\leq i\leq n}.$ Prove that the following properties are equivalent....
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Using notation of conjunctive normal forms for multi-objective optimization.

I need to maximize several objective functions $f_i(x)$, that I have arranged in a vector. $f(x) = [ f_1(x) , f_2(x) \cdots, f_K(x) ]^T$. Essentially, my question is whether I can represent the ...
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Is $K$ is convex?

Consider the sup-normed Banach space $C(S)$, which designates the space of continuous functions over a compact Hausdorff Space $S$. Let $K=\{\mu\in A^{\perp}:||\mu||\leqslant 1\}$. I tried ...
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If $f:(a,b)\to\mathbb R$ with $f''(t)\geq0\,\forall t\in(a,b)$, then $f$ is convex
$\textbf{Definition we use:}$ We call a function $f:(a,b)\to\mathbb R$ convex if, whenever $x,y\in(a,b)$ and $0\leq\lambda\leq1$, then $$f(\lambda x+(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y).$$ ...