# Tagged Questions

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### Prove that f has at least one global minimizer

$f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function such that $\displaystyle\lim_{\|x\| \to \infty} f(x) = \infty$ On a side note: how can a function have more than one global minimizer? Is a ...
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### Feasible Condition with a single constraint

A linear program with a single constraint minimize $z = c_{1}x_{1} + c_{2}x_{2} +· · ·+c_{n}x_{n}$ subject to $a_{1}x_{1} + a_{2}x_{2} +· · ·+a_{n}x_{n} ≤ b$, $x_{1}, x_{2}, . . . , x_{n} ≥ 0.$ (a) ...
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### Linear Programming - Tableau Condition

The following tableau corresponds to an iteration of the simplex method: ...
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### Relaxed optimization [duplicate]

I am facing an optimization problem $\max f(X)$ Can I solve a relaxed optimization problem $g(X)$ $\max g(X)$ if I can prove that $g(X)> f(X)$.
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### Relaxation of optimization problem [duplicate]

Can I solve the following optimization problem, $$f= \max \{h(Y) - h(Y|U)\}$$ by solving an easier upperbound on $f$ for example $g > f$ where $g= \max\{h(Y)-h(Z)\}$. My aim is to prove that ...
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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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### Why Am i standing in a global minimum?

I`been asked the following in optimization If I am located in a point where all the possible factible directions turn out to be worse for the function, Am I located in a global minimum? The answer is ...
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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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### how to differentiate to optimize this function?

I have an optimization function in the following form: $E = \operatorname*{argmin}_{A} \sum_j \| A\bf{x}_j - B \|_2^2 + \mu\sum_i a_{ii}^2$ Where, A is an unknown diagonal matrix with elements ...
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### How to derive dual of this L1 norm approximation problem?

I am working through a question in Convex Optimization by Boyd and Vandenberghe. I've made an image with the original question, and the part of the solution I don't understand: how the dual is ...
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### Primal-dual subgradient method

In these notes, an extension of the subgradient method is presented in Section 8 (page 30). The method is described so quickly and neither convergence analysis (compared to classical subgradient for ...
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### Formulation and computation of “the” unique median of an even-sized list

Consider an even-sized set of numbers $X = \{x_k\}$, such as $X = \{1, 2, 7, 10\}$. The median $m$ is defined as: $$m = \mathrm{arg \min_x} \sum_k \lvert x_k - x\rvert^1$$ Any $m \in [2, 7]$ is a ...
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### Proof of sufficient condition of existence of Lagrange multipliers

Consider the optimization problem $$(P) \quad \inf\{ f(x) : g_i(x) \le 0, i = 1,\ldots, m, x \in \mathbb R^m \}$$ where $f, g_i : \mathbb R^n \to (-\infty, \infty]$ are convex and \$0 \ne ...