Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

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Find Maximum of Lower Envelope

Okay, I'm not really sure whether the title is good. Consider \begin{align*} \min\{ 5x_1 + \frac{5}{2}x_2 + \frac{5}{3}x_3 + \frac{5}{4}x_4, \\ x_1 + \frac{6}{2}x_2 + \frac{6}{3}x_3 + ...
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1answer
21 views

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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35 views

Relation between Riccati Algebraic Equation and optimization problem

Reading this page: http://www.mathworks.com/help/robust/ug/minimizing-linear-objectives-under-lmi-constraints.html I got stuck in the result that says it can be show that minimizing Trace of X (a ...
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8 views

online algorithm for nested optimization

How to construct a sequence {x_t;\theta_t}, which is online algorithm for following optimization problem: $\arg\min_\theta \sum_t \min_{x_t} \ell_t(x_t;\theta)$ For simply, we can assume ...
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1answer
53 views

Why geometric median cannot be solved analytically

$\DeclareMathOperator*{\argmin}{argmin}$ For a given set of $m$ points $x_1,...,x_m$ with each $x_i\in \mathbb{R}^n$, the geometric median (or the weber point) is defined as $$\argmin\limits_{y \in ...
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1answer
41 views

Dual of concave function is convex

If $U(x)$ is strictly increasing and strictly concave and $lim_{x \rightarrow \infty}$ U'(x) = 0, prove that its dual: $$U^{*}(y) = max_x \{U(x) - xy\}$$ is convex. Does anyone know how to prove ...
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1answer
24 views

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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30 views

Linear Programming - Tableau Condition

The following tableau corresponds to an iteration of the simplex method: ...
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29 views

Easy elementary proof of Farkas Lemma?

Is there any elementary proof of Farkas lemma which does not use convex analysis and hyperplane separation theorem? What about special case below: If the Matrix $A$ is invertible, then there is ...
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1answer
312 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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21 views

Convex sets and minimum points

Let $X$ be the convex set formed by the convex combination of the $n$ points $\{x_1, x_2, ... x_n\}$ in $\mathbb{R}^n$. Let $X^* \subseteq X$ be the convex set of minimal points w.r.t to the convex ...
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19 views

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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32 views

weights go to infinity in logistic regression with linearly separable data

I have the loss function of logistic regression $L(W)$ = - $\sum_{i=1}^n {y_i}.log[\sigma(w^Tx)] + {(1-y_i)}.log[1- \sigma(w^Tx)]$ I have derived the Hessian and proven it's positive semi-definite ...
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14 views

Eliminating variables in convex program

This is a basic convex optimisation question. I have the following problem: $$\max_{\substack{t\le e\\ At\le b}} e^\top t$$ How do I find the optimum $t^*$? I write the KKT conditions, get ...
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1answer
16 views

Showing the multivariate normal is log-concave?

I'm trying to show that $\log p(x) = -\frac{1}{2}(x-\mu)^{T} \Sigma^{-1}(x-\mu)$ is concave. How would I go about this in $\mathbb{R}^n$? I've tried taking derivatives but I'm getting stuck once I get ...
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50 views

Show that this function is convex?

So I'm supposed to show that this function is convex, but I have no idea how to go about it...I've been told to use Cauchy Schwarz in order to show that the Hessian is non-negative definite, but I'm ...
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Want to show that Farkas lemma is equivallent to corolary 2.7. [closed]

I want to show that Farkas lemma (lemma 2.6) is equivalent to corollary 2.7. Any help?
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10 views

A strange sufficient condition for quasiconcavity

I think I learned this from a lecture today If a multivariate C2 function is increasing and its bordered Hessian has positive determinant, then f is quasiconcave. ...
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6 views

Verification of the Approach to a given non-convex integer programming problem

I need to verify my approach to a non-convex integer programming problem. It would be interesting to see other approaches as well. Let $\mathbf{C}_1,\dots,\mathbf{C}_R$ be $N\times N$ hermitian ...
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10 views

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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362 views

Derivative of nuclear norm

I'm trying to take the derivative of nuclear norm with respect to its argument. nuclear norm is defined in the following way: $$\|x\|_*=\mathrm{tr}(\sqrt{x^Tx})$$ I'm trying to calculate: ...
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27 views

Least squares and simplex

I am interested in the linear least square problem with the solution with the following constraints : $$ \min_x \|Ax-b\|^2$$ subject to $0 \le x_i \le 1$ and $\Sigma_{i=1}^n x_i= 1$. Because of the ...
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optimization problem gaussian maximizes entropy

Let $X_1, X_2, Z_1$ be random variables and define $$Y=aX_1+bX_2+Z_1$$ I have the following optimization problem of difference of entropies, $$f=\max_{p(x_1x_2)} h(Y) - h(Y|X_2)= \max_{p(x_1,x_2)} ...
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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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1answer
24 views

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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1answer
38 views

normal cone to sublevel set

I came across the following interesting and important result: Let $f$ be a proper convex function and $\bar{x}$ be an interior point of ${\rm dom} f$. Denote the sublevel set $\{x:f(x)\leq ...
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1answer
42 views

min : sum of L2 norm and squared-L2 norm.

Is there a closed-form solution of the following convex problem: $$\min_x \| x - u \| + C \| x - v \|^2$$ where $\| \cdot \|$ is the L2 norm.
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302 views

Solving Linearly Constrained Quadratic Programming with Coordinate Descent

Does anybody have any idea about how to solve the following problem with Coordinate Descent? \begin{align} \min &\quad \mathbf{x}^{\top}P\mathbf{x} + b^{\top}\mathbf{x}\\ \text{Subject to}& ...
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60 views

$x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$, for $xyz=1$ [closed]

$x,y,z > 0$ such that $xyz=1$ can you prove that $x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$ without lagrange multiplier Thanks
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1answer
21 views

A question Kolmogrov's generalized inequality for projection onto convex sets

Kolmogrov's inequality says that, if $C$ is a convex set, and $P_C(x)$ is an operator for projecting point $x$ into the convex set $C$, if $z = P_C(x)$, then for any $y \in C$ we have $$ (z - y).(x - ...
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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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34 views

Projection onto hypercube [0,1]^n

Given a positive n-dimensional vector $\mathbf{z}$ (all its elements are positive), my goal is to project it to a unit hyperplane $[0,1]^n$. However my projection is defined with respect to ...
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248 views

Karush-Kuhn-Tucker (KKT) conditions

I am having difficulties understanding the graphical interpretation as well as why the two following KKT conditions is necessary for a point x* being a minimum. It is my understanding that the (d) ...
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48 views

Solving LP with two $L_1$ inequality constraints

Is there a "fast" way to solve the following LP formulation with the following constraints: $$ \max_{\mathbf{f}} \mathbf{f}'.\mathbf{g} \\ \mathbf{1}'\mathbf{f}=1\\ \|\mathbf{f}-\mathbf{h}\|_1\le ...
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1answer
26 views

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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2answers
311 views

Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...
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Lagrange multipliers for minimax optimization

Does the equally constrained minimax optimization can be solved by using Lagrange multiplier method? Thanks in advance.
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Let $P \subseteq R^n$ be a polyhedron. Why does $\{ x + \alpha d \mid \alpha > 0\} \subseteq P$ for some $x \in P$ imply $d$ is a recession direction?

Suppose we have a polyhedron $P \subseteq R^n$ and let $d \in P$ be a recession direction, that is $\{ x + \alpha d \mid \alpha > 0\} \subseteq P$ for all $x \in P$. Why does $\{ x + \alpha d \mid ...
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1answer
264 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
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313 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
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159 views

Is this function concave or can it be made concave?

I am working with a point process with an event arrival rate of: $$ \lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$$ where $ t_1,..t_n $ are the event arrival times. The log ...
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133 views

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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Complexity analysis of convex optimization problem

I am studying an optimization problem \begin{equation} \mathbf{x}^*=\text{argmax}\quad\sum_{d=1}^{D}\log(\mathbf{a}_d^T\mathbf{x}+b)+\mathbf{c}_d^T\mathbf{x}+f_d\\ \text{subject to}\quad ...
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Frobenius norm and Gaussian noise

Why Frobenius norm is considered to a good tool for dealing with Gaussian noise?
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1answer
27 views

closed form vs gradient descent baseed methods

I am a beginner to optimization. Could anybody give me a simple example to illustrate when I should use closed form and when I should use iterative methods like gradient descent? Thanks in advance.
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103 views

Closed form solution

I have the following optimization problem: $$\min_{\mathbf{G}} \|\mathbf{B(A+G)\|_F^2} \quad{} \\\text{subject to} \quad{} \mathbf{\|C^T(A+G)\|_F^2\leq \gamma \|A^T(A+G)\|_F^2 } \quad{}, \\ ...
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Are iterations involving quantization going to converge?

For $i = 1,2,3$, let $~f_i(y_i)~$ be a convex and differentiable function and $y_i$ a scalar variable. Consider the following iteration $$\left[ \begin{array}{c} \nabla f_1(y_1^{k+1}) \\ \nabla ...
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257 views

convex hull function in matlab

Is there anyway to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
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1answer
21 views

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 ...