# 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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### Convexity of a complex function

I have a problem in which I need to find the minimum of the function $f:\mathbf{C}\rightarrow\mathbf{R}$ given by $$f(s) = v^*M(s)^*M(s)v$$ with $v \in \mathbf{C}^{n+1}$ and so that $|v_i|$ ...
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### An inequality regarding the derivative of two concave functions

Let $f:\mathbb{R}^+\to \mathbb{R}^+$ and $g:\mathbb{R}^+\to \mathbb{R}^+$ be two strictly increasing, strictly concave and twice differentiable functions with $f(0)=g(0)=0$ and $f'(0)=g'(0)>0$. We ...
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### Log-concave function changes when scalar is added

A function $f$ is log-concave if $\log(f)$ is concave. Intuitively, one might guess that adding a scalar to a function would not affect properties like concavity, log-concavity, quasi-concavity etc., ...
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### Subgradient of function of two variables

i do not have any experience in convex analysis and I would be most grateful if you would help me with the concept of subgradient. I get the concept of subderivative (one dimension) but it is hard ...
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### Subgradient inequality for strongly convex functions

I need some help to follow the argument made here which says that $$f(x_t) - f(x^*) \geq \frac{\alpha}{2}\|x_t-x^*\|^2$$ if $f$ is $\alpha$ strongly convex and $x^*$ the minimizer of $f$. From the ...
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### Prove that $tx+(1-t)x \ge x^ty^{1-t}$

Given conditions are $x>0$ $y>0$ and $0 \le t \le 1$ There is a hint given which says $Log$ is a concave increasing function. How do I apply this here? There is also a generalization of this ...
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### Convexity of n $4u^4x^{2.5}y^{-5}$ over $x>0, y >0$ and $u>0$

I am trying to find if the function $4u^4x^{2.5}y^{-5}$ is convex over $u>0, y>0$ and $x>0$. The thing which comes to my mind immediately is to check the positive semi-definiteness of the ...
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### Explain why for an extreme point some inequalities in the constraints become equalities

Suppose we have a linear program $$\max_x c^Tx\\ \text{subject to } Ax \leq b$$ where $\leq$ denotes pairwise inequality, i.e. $a^T_i x \leq b_i, i = 1,..., n$. If $y$ is an extreme point, why is ...
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### Is the intersection of 2 convex hulls a convex hull?

Given two finite sets of points, $X$ and $Y$, in $\mathbb R^d$ and assuming that $\text{conv}(X)\cap\text{conv}(Y)\neq\varnothing$ I would guess that the intersection is a convex hull of some other ...
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### Convexity of the Pareto front: formal definition

Does anyone have a reference to a formal definition of what convexity of a Pareto front in multiobjective optimisation means? All literature I've seen uses the term without defining it.
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Let $f : \mathbb{R}^N \longrightarrow \mathbb{R} \cup \{+\infty \}$ be a lower semi-continuous convex proper function. Let $dom f$ be the domain of $f$, i.e. $dom f:= \{ x \in \mathbb{R}^N \ | \ f(x) ... 0answers 27 views ### Get the global minimum with functions convex in a subset of the domain; numerical methods. I have a$C^\infty$function$f:\mathbb{R}^n\to \mathbb{R}$that is positive and known to have a zero and a global minimum in an unknown point$x$. Furthermore,$f$is convex in the set $$S = \{x'... 1answer 71 views ### To show a closed convex set S \subseteq R^n is bounded if and only if S contains no rays. I want to show that a closed convex set S \subseteq R^n is bounded if and only if S contains no rays. Where r \in S is a ray of S if x \in S implies that x+\mu r \in S for all \mu \in R_{... 1answer 41 views ### Strong duality of SDPs On pp. 654 of Boyd's book, it is claimed that strong duality holds between the SDPs B.2 and B.3 (at the bottom of this page). Does it require additional assumption that one of them is strictly ... 0answers 39 views ### A question about Hilbert Spaces and convex sets I am struggling with this and could really do with some help: Let H be a Hilbert space over \mathbb{R}, \{v_n\} be a sequence of vectors in H, and C be a convex subset of H containing \{... 2answers 72 views ### Jensen's inequality; what's the need for the probability measure? Jensen's inequality states that if: \mu is a probability measure on \Omega, f is integrable function (on \Omega) and \phi is convex on the range of f then: \phi \left( \int_{\Omega}g(x)\,d\... 1answer 47 views ### Is f(x) convex if \log f(x) is convex? One of the convex composition rules states that h(g(x)) is convex if h(x) is convex and non-decreasing, and g(x) is convex. Now I want to go the other way - I know that \log(f(x)) is convex ... 0answers 90 views ### Convex and continuous function on compact set implies Lipschitz Let the function f: C \rightarrow \mathbb{R} be convex and continuous, where C \subset \mathbb{R}^n is a compact set. Prove or disprove that f is Lipschitz continuous on C. Comments: If f ... 1answer 54 views ### what is the closed form solution for \min_x ||y-x||^2_2+\lambda ||x||_2 y and x are vectors. \|\cdot\|_2 is Euclidean norm. In one paper I read, they say the closed form solution is x=\max\{y-\lambda \frac{y}{\|y\|_2}, 0\}. I don't know why x need to be non-... 1answer 55 views ### f' decreasing everywhere but not defined in one point. Is f concave? Small issue: Suppose that f:[a,b] \rightarrow \mathbb{R} is a continuous function, differentiable except on a finite set of points, let say in one point y. For x<y and x>y we have f''&... 1answer 47 views ### Is the set of probability density functions convex? Given is the set of probability density functions defined as P:=\left \{ p(x)\mid p(x)\, is\ a\ probability \ density \ function \right \} Is P a convex set? I am not sure that here i have to ... 1answer 52 views ### What is the closed form for this norm \|Z\|=\sup_{\|u\|=1,\|v\|=1} u^TZv? I read a paper, which has the equation above \|Z\|=\sup_{\|u\|=1,\|v\|=1} u^TZv. Is this a well known norm and what is the closed form for this norm. Also what does this norm describe? Thanks in ... 0answers 26 views ### Sufficient condition for self-concordance? I've revised a previous question that was ill-formed. Consider the following two definitions. Def'n 1 (Lipschitz continuity of Hessian): A function f:\mathbb{R}^n\to\mathbb{R} is said to have a ... 0answers 46 views ### Log convexity and the gamma function I am writing an essay on the gamma function. I have learnt and understood convex theory and how the log-convex nature of the gamma function makes it a unique extension of the factorials (Bohr-Mollerup)... 2answers 106 views ### Can a “continuous” convex combination not be element of the convex hull? Short version of question: can a "continuous" convex combination not be element of the convex hull? I am not a mathematician, so please excuse me if I am not precise. I consider first, e.g., 4 ... 1answer 34 views ### Boundedness of sublevel sets of an integral function Let f: \mathbb{R} \rightarrow \mathbb{R} be an increasing function, i.e., x<y \Rightarrow f(x) < f(y) for all x,y. Assume that \lim_{|x| \rightarrow \infty} |f(x)| = \infty Define F(x)... 0answers 53 views ### convex polygon triangulation Suppose we have given a convex polygon on n vertices P= \{ a_1, \cdots , a_n \} in the plane (arranged clockwise). How can we prove that there exist atleast two indices i such that circle a_{i-... 1answer 52 views ### Is this \underset{x<1,y<1}{\sup}f(x,y) = \underset{x<1}{\sup} \underset{y<1}{\sup}\ f(x,y) correct? \underset{x<1,y<1}{\sup}f(x,y) = \underset{x<1}{\sup} \underset{y<1}{\sup}\ f(x,y). Intuitively, I think the above equation holds for all f(x,y). Am I right? 0answers 31 views ### A supremum problem Let a=\underset{\|u\|_c\leq 1, \|v\|_r\leq 1}{\sup}v^TY^Tu. If \lambda<a, \underset{u_m, v_m}{\sup}v_m^TY^Tu_m-\lambda\|u_m\|_c\|v_m\|_r=+\infty. While if \lambda > a, then \... 0answers 59 views ### A power inequality with convexity / majorization flavor Let a,b,c,d\in\mathbb R_+ be nonnegative reals. Define the function f:\mathbb N_+\to\mathbb R as$$ f(k):=\left(\frac{(a+b+c+d)^k+(a-b+c-d)^k-(a+b-c-d)^k-(a-b-c+d)^k}{4}\right)^{1/k} $$Is it ... 0answers 20 views ### Duality between Ax<b and another system, but Gordan's just misses I am trying to show that$$Ax < b$$Is feasible iff$$ A^T y =0 , b^Ty + s = 0, (y,s) \ge 0, (y,s) \ne 0$$Is infeasible. Work So Far Now when I try to hit this with Gordan's Lemma I seem ... 0answers 38 views ### How can the ADMM algorithm be distributed? I am reading Boyd's tutorial paper on ADMM. One question I had after I reached the formulation of the algorithm on pp.14 is: how can the alternating or sequential fashion of x-minimization step and z-... 3answers 39 views ### sup_x [ f(x)+sup_x g(x)]= sup_x[f(x)+g(x)]? sup_x [ f(x)+sup_x g(x)]= sup_x[f(x)+g(x)] is the statement correct? Can I prove like this: sup_x [ f(x)+sup_x g(x)] = sup_x[f(x)] + sup_x[g(x)] = sup[f(x)+g(x)]. 1answer 17 views ### Showing that ye^{x/y} = \max_{\alpha>0} \left[ \alpha(x+y) - y\alpha \log(\alpha) \right] In my optimization textbook, the author states without proof that$$ ye^{x/y} = \max_{\alpha>0} \left[ \alpha(x+y) - y\alpha \log(\alpha) \right].$$To be honest, this does not seem very obvious ... 0answers 21 views ### convex envelop on a unit ball What does that exactly mean by "the function$g$(continuous) is the convex envelop of the function$f$(discrete) on a unit$\ell_\infty$ball". I understand the part of "a function being the convex ... 2answers 128 views ### Opening and closing convex sets It seems true that, given$K \subseteq \mathbb{R}^n$a convex set with$K^\circ \neq \emptyset$, then$\overline{K^{\circ}} = \overline{K}$and$\left ( \overline{K} \right )^\circ = K^\circ$. I am ... 2answers 31 views ### Tangents of a Strictly Convex Fuction Let$f:\mathbb R \rightarrow \mathbb R$be differentiable and strictly convex. Is it true that$x,y \in \mathbb R$and$x \neq y$imply $$f'(x)(y-x) <f(y) - f(x)$$ If ... 1answer 36 views ### Smallest convex body inscribed in$n$-cube with all its symmetries? [closed] Consider the cube$[-1,1]^n$and convex bodies inscribed in it, such that all these bodies have the symmetries of the cube. Is there a lower bound on the volume? Which shapes attain it? 0answers 22 views ### Smallest volume of a centrally symmetric convex body inscribed in$n$-cube We consider several centrally symmetric convex bodies inscribed (intersecting all its facets) in an$n$-cube ,$[-1,1]^n$, with volume$2^n$. For instance, such a crosspolytope (its polar dual) has ... 2answers 35 views ### Strictly Convex Implies Invertible Gradient? If$f:\mathbb R^n \rightarrow \mathbb R$is strictly convex and continuously differentiable, does this imply that$\nabla f$is a one-to-one mapping? To be precise, can we say that$x, y \in \mathbb ...
I am hoping to have someone help explain to me the steps of this proof of the triangle inequality for $L^p$ spaces, $p\ge 1$. Specifically the first two applications of the homogeneity property don't ...
Suppose that $k \geq 3$, and let $X_1, \dots , X_k$ be $k$ points in $\mathbb{R}^2$, and let $w_1, \dots , w_k$ be $k$ positive real-valued constant weights. Given this, consider the function \$g(p) = \...