# Tagged Questions

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### A Difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
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### Quasiconcave condition for a power function

Let $f(x, y)= (ax^2+by^2)^n$ where $a, b, n$ are positive, $x, y\in \mathbb{R}$. What is the condition of $n$ so that $f(x, y)$ is a quasiconcave, and concave function? My idea is only calculate ...
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### LP with a linear cost function $c^Tx$: Prove optimal value is $-\infty$ or there exist some $v \in P$ such that $c^Tv \le c^Tx$ for all $x \in P$

Suppose I have a LP with a linear cost function $c^Tx$, where $P=\{x \in \mathbb R^n : Ax \ge b\}$ is the polyhedron I want to minimize over. How do I see that either the problem is unbounded, that ...
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### Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma.

Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma. I've a Lemma saying: Suppose $P=P(V,E)$ where $V,E \in \mathbb R^n$ are ...
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### Minimizing a non-convex rational function of two variables

I need to minimize the following function $$f(x,y)= \frac{a}{x}+\frac{bx}{y}+\frac{cy}{x}+dy+\frac{e}{y}$$ where $a,b,c,d$, and $e$ are positive constants, and $x$ and $y$ are both strictly positive. ...
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### how to check an optimization function is convex or not

This is the sparse coding optimization function: $\operatorname*{argmin}_{B, \alpha} \sum_j \| \bf{x}_j - B\bf{\alpha}_j \|_2^2 + \lambda\sum_j |\bf{\alpha}_j|_1$ I read in the literature that this ...
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### Distributed Convex Optimization Algorithm

Consider the convex optimization problem $$\min_{x_1, \cdots, x_N, y} \sum_{i=1}^{N} f_i(x_i,y)$$ $$\text{subject to: } x_i \in X_i \ \ \forall i, \ \ y \in Y, \ \ y = \sum_{i=1}^{N} x_i$$ ...
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### Knuth's Sandwich Theorem: requesting proof clarification

The question is about F6 of Section 8 ("Elementary facts about cones") in Donald Knuth's Sandwich Theorem (http://arxiv.org/pdf/math/9312214.pdf). He claims to prove $(A \cap B)^* = A^* + B^*$ when ...
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### Strong convexity of a function with cases

Given a set $S = \{x_1,\dotsc,x_n\} \subset \mathbb{R}$, is the function \begin{align} f&: (0,\infty) \to \mathbb{R} \\ f&(p) = 2p^2 + \frac{1}{n}\sum_{i=1}^n \max(0, -p^2-x_i) \end{align} ...
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### Sufficient condition on convex function such that $f(x) > -\infty$ for all $x$.

Let $f : \mathbb R^n \to [-\infty, \infty]$ convex and let $f(\overline x) > -\infty$ for $\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that $f(x) > -\infty$ for all $x \in \mathbb R$. ...
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### The convexity of convex function's range

Given a convex function $f\colon X \to \mathbb R$ with convex domain $X \subseteq \mathbb R^n$, is the range of $f$ a convex set also?
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### regarding the concept of dual cone

When studying the covex analysis, I am not clear about the concept of dual cone. In the following graph, $\mathcal{K}*$ was the dual cone. I marked two points, the ...
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### Convex optimization: affine equality constraints into inequality constraints

I have the following problem: \begin{array}{cll} \displaystyle \min_{ \mathbf{x} } & & \displaystyle f(\mathbf{x}) \\ \mathrm{s.t.} & & \mathbf{x} \in \mathcal{C} \\ ...
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### Explain the convexity by looking the hessian matrix of a function

The hessian matrix of a function is given by, $$H = \begin{bmatrix} a & b & c \\[0.3em] b & b & 0\\[0.3em] c & 0 & c \end{bmatrix}$$ where, ...
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### Relax equality into inequality in convex problem

Let $\mathbf{x}, \mathbf{z}, \underline{\mathbf{x}}, \overline{\mathbf{x}} \in \mathbb{R}^{I}$, where the first two are variables and the last two are given data. I have the following problem: ...
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### Examples of affine functions and convex sets

I'm just learning about convexity and affineness, and I've read over some similar questions asked here, but those were more about general properties. I need some help applying those properties to a ...
Consider a vector $\mathbf{x} \in \mathbb{R}_{++}^N$. Also consider two functions, $g(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, and $a(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, ...
Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex. If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, ...