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

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Optimization problem: $\min \limits_{\mathbf{q}} \sum_{n=1}^N q_n$, s.t. $\frac{c_{nn} q_n }{\sum_{m \ne n} c_{nm} q_m } \ge a$

\begin{array}{rl} \min \limits_{\mathbf{q}} & \sum_{n=1}^N q_n \\ \mbox{s.t.} & \frac{c_{nn} q_n }{\sum_{m \ne n} c_{nm} q_m } \ge a, \forall n \in \{1,\ldots,N\} \end{array} For this ...
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15 views

Closure of intersection of convex sets

Let $C_i$ be a convex set in $R^n$ for $i\in I$, suppose sets $ri \, C_i$ have at least one point in common, then how to prove this: $cl\bigcap\{C_i\mid i\in I\} = \bigcap\{cl \, C_i \mid i\in I\}$
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33 views

Conditions for all positive $x$ solved by $\min \bf{x}^T\bf{x}$ $s.t. Ax=b$

I want to find a condition for having all nonnegative $x_i$ in $\min \bf{x}^T\bf{x}$ $s.t. \bf{Ax}=\bf{b}$ where $\bf{x}\in \mathbb{R}^{n\times 1}$, $\bf{A}\in \mathbb{R}^{m\times n}$, $\bf{b}\in ...
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22 views

Stability of a semidefinite programming problem

For the minimum trace factor analysis problem, I want to prove that if I change a parameter in the optimization problem, the solution will be stable. Let $\mathbf{D}^p$ denote the set of $p \times ...
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22 views

how to minimize huber loss? (SVM related) [on hold]

I don't know how to minimize huber loss which is convex and continuous, but how can I minimize that? Can I express it in only one ojective function and obtain its dual problem? My object is to ...
2
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1answer
33 views

Non-convex QCQP

Consider the following optimization problem: $$\begin{array}{ll} \text{minimize} & \mathbf{x}^{T} \mathbf{A} \mathbf{x}\\ \text{subject to } & \mathbf{x}^{T} \mathbf{P}_i \mathbf{x} > 0, ...
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9 views

Difference between mirror descent and dual averaging [on hold]

What is the main differences between mirror descent and dual averaging methods? When the number of steps or accuracy are fixed, they are equivalent. But what can be said when these parameters are not ...
2
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1answer
25 views

Existence of direction for polyhedral set

I refer to Lemma 4.42 of this lecture notes on linear programming, about the relationship between the boundedness of a polyhedral set and its directions. Let $P$ be a non-empty polyhedral set. ...
2
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1answer
28 views

Solving a quadratic convex optimization problem

There's this convex optimization problem which I got stuck after writing the Lagrange equation. I simply couldn't find a way to eliminate the Lagrange multiplier. $$\begin{array}{ll} \text{minimize} ...
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13 views

References to: If $C\subset\mathbb{R}^n$ is convex and $0\notin C$ then there exists $v\in C$ such that $C$ is in the closed halfspace $H_v$.

For each $v\in\mathbb{R}^n$, we define the notation $H_v=\{u\in\mathbb{R}^n:\langle u,v\rangle\geq0\}$, where $\langle\cdot,\cdot\rangle$ denotes the usual inner product in $\mathbb{R}^n$. Recently, ...
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36 views

Good text book recomendation

I would like to do some reading about a technique called sequential convex programming. There is a lot of material about sequential quadratic programming out there, including books (Nocedal & ...
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24 views

A question on Edgeworth Expansion

I'm working Edgeworth Expansion. I couldn't understand one thing . Can you help me about that please. $$Z= \frac{\sqrt {n} (\bar {x} -\mu)}{\sigma}$$ converges in distribution to N(0,1) I have ...
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82 views

integral vertex of the polyhedron

I am trying to prove the following : If $A$ is a $\{0, 1\}$-matrix, then any integral vertex of the polyhedron $P = \{x \mid x \geq 0 ; Ax \geq 1\}$ is a $\{0, 1\}$-vector. But I cannot do it. ...
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1answer
55 views

Product of the differences of two pd matrices and their respective inverses is pd

Given two $\textbf{positive definite (pd), Hermitian}$ matrices X and Y, I am trying to determine whether $(X-Y)(Y^{-1}-X^{-1})$ will always be pd as well, and how to prove this. This formulation ...
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27 views

What's best way to determine optimal solution given 2 objectives to maximize [closed]

I have 3 machines A, B and C. I would like to rank the machines based on which machine maximizes Score1 and Score2. Score1 and Score2 are performance measures that rang from 0-100%. Below is some ...
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1answer
20 views

To what extent can the column generation method for solving linear programs be extended to solving more general convex optimization problems?

Are there column generation approaches to solving classes of convex optimization problems other than LPs, and are they guaranteed to find a global minimizer?
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3answers
461 views

Shortest distance between two lines in 3-dimensional space [closed]

Can someone explain to me how to solve this question? Find the shortest distance between the lines $L_1 = \left\{t \begin{bmatrix} 1\\ 1\\ 1\end{bmatrix} : t \in \mathbb{R}\right\}$ and $L_2 = ...
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36 views

maximum of a concave function in a minkowski sum

Let: $f(x,y)$ - a strictly-concave function, monotonically increasing in $x$ and $y$; $A,B$ - two compact and convex sets in the positive quadrant; $C$ - their Minkowski sum, $A+B$; $(x_A,y_A)$ - ...
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16 views

Diagonal Newton's method for unconstrained optimisation.

Assume you are minimising a convex function $f$. The function is twice-differentiable. The well-known Newton's method consists in starting form some point $x_0$ and then using the iteration below. $$ ...
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1answer
22 views

Property of Conical Hull

Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$ \operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}. $$ (it is a cone in ...
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18 views

Solving optimization problem where objective function is of type (affine+(affine/affine))

I need to solve a non linear optimization of the form minimize $f(x) +\frac{g(x)}{h(x)}$ subject to $p(x)\leq0$ $q(x)=0$ Here $f,g,h,p,q$ are affine functions of $x$ and they are convex in the ...
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7 views

Deriving a minimizer involving proximal operator

The following are from this thesis http://gpu4vision.icg.tugraz.at/papers/2012/werlberger_phd.pdf I have a difficulty understanding the lines: For example, (4.38) seems very weird to me. I think ...
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23 views

Converges extremely slowly, using Douglas-Rachford splitting, how to improve?

my problem looks like this: $\min _{ E,A }{ { \lambda }_{ 1 }{ \left\| E \right\| }_{ 1 }+{ { \lambda }_{ 2 }\left\| A \right\| }_{ * }+{ \left\| D-ME-A \right\| }_{ 2 }^{ 2 } } $ the M is a ...
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35 views

Proving function is convex argmin

How can I show that the following function is convex in which Z is a random variable? $$\rho(Z)=\frac{2}{3}argmin_{t}\{t+10\mathbf{E}[Z-t]_{+}\}+\frac{1}{3}argmin_{t}\{t+5\mathbf{E}[Z-t]_{+}\}$$ I ...
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12 views

Convex solver returns disordered dual variables, how to re-order?

I have the following convex optimization problem: $$ \begin{align*} \min_{x} &f(x) \\ \text{subject to} \\ x_{k+1} &= x_k+\cdots\qquad k=0,1,2,\ldots,N-1 \end{align*} $$ I managed to solve ...
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2answers
98 views

Equivalent characterizations of the dual norm on finite dimensional vector spaces

In their book on Convex Optimization, Boyd and Vandenberghe state that given a norm, $||\cdot||$, defined on $\mathbb{R}^n$, the dual norm is defined as $$||z||_*= \sup \{ z^Tx : ||x|| \leq 1 \}$$ ...
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17 views

How you can model the other players replies in a game theoretic model?

In a game theory field, the payoff function of a player n is basically a function of the other players responses which are considered as constants. I'm trying to solve the maximization of the payoff ...
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38 views

How to get explicit solution of the fractional minimization problem?

I need to minimize $f(x) = \frac{x^tQx}{a^tx-b}$, $Q$ is positive definite, $a,b$ are constant vector. But when I take gradient and set it equal to $0$, it's hard to get an explicit expression for ...
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1answer
17 views

Why the dual and the canonical dual may have the same optimal solution?

For instance let be \begin{cases} \max & 3x_1 &+2x_2 &+4x_3 & -x_4\\ &x_1 &+x_2 &-2x_3&&\le4\\ &2x_1&+3x_3&&-4x_4&\ge 5\\ ...
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1answer
20 views

Has $\max\{cx:Ax\le b\}$ an optimal solution $x_1=\sqrt 2$ with $A\in \{-1,0,1\}^{m*n}$ with exactly one $1$ and one $-1$?

Let be $A\in \{-1,0,1\}^{m*n}$ with exactly one $1$ and one $-1$ and zeroes at each line. $c\in\mathbb{Z}^n$ such that $\sum\limits_{j=1}^{j=n}c_j=0$. $b\in\mathbb{Z}^m$ positive. How to start to ...
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1answer
36 views

A Question On Triple Integration

Can anyone construct a nonzero continuous function $f:[0, 1]\times[0, 1]\times [0, 1]\rightarrow [0, \infty)$ such that \begin{equation*} \int_{t_1=0}^1 \int_{t_2=0}^1 \int_{t_3=0}^1 f(t_1, t_2, ...
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14 views

subdifferential of $\max_{i=1,\cdots,k} x_i+\frac{1}{2}\|x\|_2^2,\ \ \ x\in \mathbb{R}^n$

How to find the subdifferential of $$f(x) = \max_{i=1,\cdots,k} x_i+\frac{1}{2}\|x\|_2^2,\ \ \ x\in \mathbb{R}^n$$ My derivation is: $\nabla \frac{1}{2}\|x\|_2^2=\nabla \frac{1}{2}x^Tx=x$ ...
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54 views

Yale College's Housing Draw Problem — Convex optimization techniques on a modified stable marriage problem?

I'd like to run various optimization techniques on this variation of the stable marriage problem I formulated. Ideally, I'd be able to convert the problem I constructed into one that is more ...
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8 views

Why $x_B=\tilde b +\tilde A x_{\bar B};c^Tx=\psi+\tilde c^Tx_{\bar B}$ doesn't describe an optimal solution iif $\tilde c_i\le 0,\forall i$

How to counterprove the assertion that if a feasible dictionnary in the type \begin{cases} x_B=\tilde b +\tilde A x_{\bar B}\\c^Tx=\psi+\tilde c^Tx_{\bar B} \end{cases} describe an ...
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1answer
20 views

Can geometric programs be solved more efficiently than general convex optimization problems?

I want to solve an optimization problem for which I have already proven that it is feasible and convex. Introducing further variables and considering a special case of the problem, I can formulate it ...
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1answer
37 views

piecewise linear minimization equivalent to linear programming

Why is \begin{equation} \begin{aligned} & \min\max_{i=1,\ldots,n} & &a_i^Tx+b_i\\ \end{aligned} \end{equation} equivalent to an LP \begin{equation} \begin{aligned} & \min & ...
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2answers
34 views

Optimization of a function over probability distributions

I'm trying to solve certain optimization problems dealing with probability distributions. Consider the space of probability distributions $\{ 1, ..., N\} \to [0, 1]$ I have a function $f : (\{ 1, ...
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Finite Subdifferential

Can the set of subgradient vectors (Subdifferential) have finite number of vectors? I know it can be either empty, have one vector, or have infinite number of vectors
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28 views

Minima of non-strictly convex function

I wonder if there is anything that can be said about minimizers of convex, but not strictly convex problems, with regards to initial points provided to an optimization algorithm? Not sure if this is ...
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L1 regularized unconstrained optimization problem

I am encountering an unconstrained minimization problem. The problem is of the form $$\min_x \frac{\|x-a\|_2^2}{2}+\lambda\|x\|_1$$ where $x,a \in R^n$ and $x$ is the optimization variable. ...
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2answers
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References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
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Why this problem is unbounded, even though it's dual is feasible?

In this paper, An Efficient Inexact ABCD Method for Least Squares Semidefinite Programming,page 2, there is a problem called, (P): \begin{align} P: &\min_{X,s} && ‎ \frac{1}{2} ‎\Vert ...
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15 views

Convert this problem with 2-norm cost to an SOCP?

I am solving a non-convex problem via sequential convex optimization. Here is a minimal example of my problem at iteration $i$: $$ \begin{align*} \min_{\Delta t,F_i}&\left( \Delta ...
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27 views

Variational inequality and saddle point problem

I refer to the following paper: (fix the wrong link) https://papers.nips.cc/paper/5723-adaptive-primal-dual-splitting-methods-for-statistical-learning-and-image-processing.pdf In that paper, we ...
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67 views

Maximum concyclic points

Given n points, find an algorithm to get a circle having maximum points.
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19 views

$\max\{c^Tx:Ax\le b,x\ge 0\}=+\infty$ iif it exists $j\in\{1,…,n\}$ such that $\max \{x_j:Ax\le b, x\ge 0\}=+\infty$

Show a vector $\vec c$ exists such that $\max\{c^Tx:Ax\le b,x\ge 0\}=+\infty$ if and only if it exists $j\in\{1,...,n\}$ such that $\max \{x_j:Ax\le b, x\ge 0\}=+\infty$ I'm only asking for a ...
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34 views

Positivity of the last component of non negative least squares based on active set method

I have followed the instructions given in Lawson and Hanson book for non-negative least squares using active set method. I am having a trouble in justifying one of the statements they have made about ...
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22 views

Can a linear program be optimal if its basis is infeasible?

I want to know thanks to the dual theorem wether the following basis is or isn't optimal. That is to say looking for the slack variables. As far as the third line doesn't respect the constraints: ...
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12 views

Which coefficient to start with in the dictionary method?

I used to start with the variable with the biggest coefficient in the goal function (in the case of max). yet I read an article that behaving like this may lead to loop. It is rather preferred to do ...
5
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1answer
82 views

Why is the Barycenter operation in Hadamard spaces Lipschitz continuous?

I am looking into exercise 9.2.22 of "A course in metric geometry" by Burago-Burago-Ivanov. For a Hadamard space $H$ (a complete simply connected metric space of nonpositive curvature in the ...