# 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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### Convex optimization - comparing optimal values (any counter example?)

Consider the following maximization problems: $\max_{x} x -\gamma p(x)$ subject to $x \in \Omega_1$ $\max_{x} x-\gamma (p(x) + q(x) )+K$ subject to $x \in \Omega_2$ where $\Omega_1$ and $\Omega_2$...
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### If a product of a Lipschitz function and another bounded function is Lipschitz, is the other function Lipschitz?

Let $x=[0,1]^n,\;p: [0,1]^n \rightarrow [0,1]^n$. Define $V(x) = p(x).x$, i.e. the inner product of $p(x)$ and $x$. We are given $V(.)$ is convex. Can we say $p(x)$ is Lipschitz? My approach: $V(.)$ ...
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### How to solve the optimization problem? (any counter example?)

Consider the following maximization problems: $\max_{x} x -\gamma p(x)$ subject to $x \in \Omega_1$ $\max_{x} x-\gamma (p(x) + q(x) )+K$ subject to $x \in \Omega_2$ where $\Omega_1$ and $\Omega_2$...
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Consider the following problem: Suppose that $f$ is a twice differentiable real function such that $f''(x)>0$ for all $x\in[a,b]$. Find all numbers $c\in[a,b]$ at which the area between the graph $... 2answers 57 views ### What are the subdifferentials$\partial f(0)$and$\partial f(1)$? Let$ f: \mathbb{R} \to \mathbb{R} $given by \begin{equation*} f(x) = \left\{ \begin{array}{rl} x \log x -x & \text{if } x \geq 0\\ \infty & \text{if else}\\ \end{array} \right. \end{equation*... 1answer 23 views ### Volume of convex body as an integral of its radial function Let$C$be a compact convex set in$\mathbb{R}^d$. Let the origin$O$by in the internal of$C$. The gauge function$\gamma_C(.) : \mathbb{R}^d \to [0, \infty]$of$Cis defined as $$\gamma_C(x) = \... 1answer 47 views ### convex sets in convex optimization How to prove that the following set is not convex?$$M = \left\{ \mathbb{R}^{3}: x_{1}x_{2}x_{3}\le 1,x_{1}+x_{3}\ge 2,x_{1} \ge 0 \right\}$$Thanks for any help. I tried to write it down as ... 1answer 44 views ### The function f(x)=|x|^p, x\in \mathbb{R}^{n} is strictly convex for p>1? Let p>1. In the paper  below, it says that The function f(x)=|x|^p, x\in \mathbb{R}^{n} is strictly convex. I would like to prove that. By definition, We need to show that the Hessian ... 1answer 24 views ### Infinite-dimensional Bregman divergence Let C be a convex subset of \mathbb R^n with nonempty interior. Let f: C \to \mathbb R be a strictly convex function, differentiable in the interior of C, whose gradient \nabla f extends to ... 1answer 37 views ### Simple Concave Function Question Suppose we have a function f:[0,\infty)\longrightarrow [0,\infty) that is concave and a > b>0. Then given a constant c>0 I claim that f(a+c) - f(a) \le f(b+c) - f(b). If I draw a ... 1answer 23 views ### Clarification about the definition of subgradient I have a basic clarificatory question about the definition of a subgradient from the book Infinite Dimensional Analysis by Aliprantis and Border. Here is the definition. Given a dual pair \langle X,... 1answer 48 views ### Proving an inequality involving a concave function I have encountered the following inequality, but I'm struggling to prove it: Suppose f:\mathbb{R} \to \mathbb{R} is a concave function and \theta > 1. Then$$ \theta^k f(x) \geq f(\theta^k x)... 1answer 14 views ### Prove that the supremum of an affine function is concave Suppose I have a function such that F(\theta x+(1-\theta)y,z)=\theta F(x,z)+(1-\theta)F(y,z) for \theta\in(0,1). I want to show that F(x,z) is concave in the first argument when taking a ... 2answers 53 views ### Convex hulls of a set and its subsets Suppose that P is a set of k>3 points in \mathbb{R}^2. Let \mathrm{Conv}(P) be the convex hull of P. I think that the following claim is true (I know how to prove it geometrically) : \... 1answer 23 views ### Prove this infimum function is convex Let X \subset \mathbb{R}^n be convex, as well as the functions f:X \to \mathbb{R} and g:X \to \mathbb{R}. Prove h(b)=\inf \{f(x): g(x) \le b, x \in X \} defined on \mathbb{R} is convex also. ... 1answer 19 views ### Example of a monotone Lipschitz operator that is not cocoercive? Let \mathcal{X} be a real Hilbert space, let x,y\in\mathcal{X}, and let L\in\left]0,+\infty\right[. I am looking for an operator T\colon\mathcal{X}\to\mathcal{X} which is \begin{align} \text{... 1answer 36 views ### Convexity of constrained profit function [closed] I have to show that the following constrained profit function, \pi, is convex. A firm choses non-negative quantity, x, of inputs with price p and non-negative quantity, y, of outputs with ... 0answers 46 views ### How do I prove that a theoretical set is Convex? I am trying to get my head around convex analysis proofs and I am not sure how to begin. I think the definition I should use for proofs is that a set C is convex if for any u,v\in C, the point tu+(... 1answer 36 views ### Topological properties of convex sets I was trying to prove some 'facts' regarding convex sets which seem natural, but I seem to stuck, so I was wondering whether tbey are indeed true, which I'll describe below. Let X be a topological ... 1answer 29 views ### What are some conditions that are sufficient for the following projection property? Let X be a locally convex topological vector space, and let f: X \times X \to [0,\infty]. Say that f has the projection property if the following holds: For all compact, convex C \subseteq X ... 1answer 39 views ### Example of a strongly convex function where the Lipschitz constant L is equal to the strong convexity parameter u I am trying to come up with like three strongly convex function f\colon\mathbb{R}\to\mathbb{R} where the Lipschitz constant L is equal to the strong convexity parameter u, i.e. for every x,y\in\... 2answers 44 views ### If every x with f(x) = f(\bar x) is local minimum, then \bar x is global minimum. I am writing to ask for your help guys. The question I am having trouble with is: Let f: \mathbb{R}^n \to \mathbb{R} be a contionuous function and \bar x \in \mathbb{R}. Show that the following ... 1answer 31 views ### Convex hull and supporting hyperplanes Let S \subseteq \mathbb{R}^n be a nonempty set and p \in S be a point. A (closed) halfspace H\subseteq \mathbb{R}^n is said to support S at p if S \subseteq H and p \in \partial H. For ... 1answer 76 views ### How to prove B-A \succeq 0 \Leftrightarrow ellipsoid x^TBx \leq 1 contains x^TAx \leq 1? Assume A \in \textbf{S}^n_{++}, an ellipsoid centered at the origin given by\mathcal{E}_A = \{x\mid x^TA^{-1}x \leq 1\}$$Then we have \mathcal{E}_A \subseteq \mathcal{E}_B if and only if B-A ... 1answer 117 views ### Convex quadratic program plus one non-convex term How would you approach a standard convex quadratic program with convex constraints but one non-convex term? Say |x|^{0.4}.$$\begin{array}{ll} \underset{x}{\text{minimize}} & \frac12 x^{T} Q x + ... 0answers 32 views ### Are these two separation results equivalent? LetX$be a real topological vector space, and let$A$and$B$be nonempty, disjoint, convex subsets of$X$. S1. If$A$is open, then there exists$c \in \mathbb R$and a nonzero, linear, continuous, ... 2answers 70 views ### Normal cone of the complement of an ellipsoid On page 68, exercise 2.38 in Boyd & Vanderberghe's book, Convex Optimization, a normal cone of a boundary point$x$of a set$C$is defined as$$\mathcal N_C(x) := \left\{ g \mid g^T x \ge g^T y, \... 1answer 38 views ### Expression for the Clarke subdifferential of a weakly convex function Let$\gamma\in\left]0,+\infty\right[$, let$f$be a proper, convex, lower semicontinuous function from a real Hilbert space$\mathcal{X}$to$\left]-\infty,+\infty\right]$, and set$g=f-\frac{\gamma}{...
Consider a twice-differentiable PDF $f(x)$. If $f(x)$ is log-concave, then so is its CDF, $F(x)$. As a consequence, the Mills' ratio or inverse hazard function $[1-F(x)]/f(x)$ is monotonic ...