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

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How to compute norm bound error in robust approximation

I am reading convex optimization, and I am little confused about the following two prolems in norm bound error of robust approximation. How to compute $\{\|\bar{A}X-b+Ux\| | \|U\|\le a\}$ ? For the ...
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reference for convex function results

Here is a simple property of a concave function from $\mathbb{R}$ to $\mathbb{R}$, Given $x,x'\in \mathbb{R}$ with $x'>x$, if $\exists \kappa'\in (0,1)$ such that $$c(\kappa'x+(1-\kappa')x')=\...
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2answers
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$\max 2x_1 +x_2$ unbounded or unfeasible with the constraint $sx_1 +tx_2\le-1$

\begin{cases} \max & 2x_1 &{}+x_2\\ & sx_1 &{}+tx_2&\le-1\\ & x_1,x_2&&\ge 0 \end{cases} Find out when this program is not feasible, bounded Feasibility It is ...
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Use of binary variables in LP problems

I can't figure out how to write the following condition to an LP. I have four nonnegative variables: $X_A$, $X_B$, $X_C$, and $X_D$. The condition which should be satisfied is this: If $X_A$ and $...
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When does a variable goes out with the revised Simplex method?

Let be the following linear program. \begin{cases} \max & 3x_1& +x_2\\ &x_1&-x_2 &\le -1\\ &-x_1 &-x_2&\le -3\\ &2x_1 &+x_2 &\le4\\ x_1,x_2\ge 0 \end{...
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Impact of removing active constraints in convex optimization

In active set methods for non negative least squares, we remove variables from the passive set to active set if the least squares solution gives negative values on those variables. What's the impact ...
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1answer
17 views

Chebyshev's approximation understanding

I am reading Boyd's book on convex optimization. Could you assisst me in understanding what this expression means: $$\text{minimize} \ \ \text{max}_{i=1,...,k}|a_i^Tx-b_i|$$ This is what I think ...
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Some intuition on the support function of a convex set

I have some doubts on the interpretation and properties of the support function of a convex subset of $\mathbb{R}^d$. (1)Let $K$ be a convex set in $\mathbb{R}^d$. (2) The support function of $K$ is ...
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34 views

Residual norm of active set method in non negative least squares

I am having trouble with understanding one of the statements in active set methods done for non negative least squares given in Book written by Lawson. The quadratic problem can be written as \begin{...
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2answers
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Can you prove that this function is convex? $\sqrt{2x_1^2+3x_2^2+x_3^2+4x_1x_2+7} + (x_1^2+x_2^2+x_3^2+1)^2$.

My analysis: The second term can be proven to be convex as follows. It is basically a composition of norm with an affine transformation to the power of four: $(x_1^2+x_2^2+x_3^2+1)^2 = \|(x_1^2, x_2^2,...
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Is the minimum point of a strictly convex function stable?

This is a problem I figured out after seeing the definition of minimum stable point and I think the following tense is true: Let $f(x)$ be a strictly convex function whose minimum values is $f^*=f(...
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12 views

Pseudo-Boolean functions restricted to integers

The Pseudo-Boolean functions are of the following form. $$ f : \mathbb{B}^n \to \mathbb{R} $$ I would like to know if there is a special sub-category of $$ f : \mathbb{B}^n \to \mathbb{Z} $$ with ...
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Variant of conjugate function: $V(s) = \underset{x}\max \{\langle s,x-x_0\rangle-\beta f(x)\}$

Consider one variant of conjugate function: $$V(s) = \underset{x}\max \{\langle s,x-x_0\rangle-\beta f(x)\}$$ You can think $s$ as a linear functional. If I do the following steps: \begin{...
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28 views

How to form a dual problem in convex optimization (in a broad view)

After reading some papers, this problem confuses me. There are different forms of dual problem to the primal problem: $$\underset{x}\min \ \ f(x)$$ where $f(x)$ is a convex function. By ...
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1answer
34 views

characteristic cone of polyhedral

Let $$Q=\{x ∶ Ax ≤ b \}≠∅$$ If $Q = P + C$, where $P$ is a polytope and $C$ is a polyhedral cone, prove that $$\{y|Ay ≤ 0\} = \{y|x + y ∈ Q, ∀ x ∈ Q\}$$ The cone $C = \{y|Ay ≤ 0\}$ is called ...
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$\big\langle\nabla f(x)-\nabla f(y),\,(x-y)\big\rangle\ge m\,\left\| x-y\right\|^2,\;m>0\;$ for strictly and strong convex function [closed]

Prove that $\,\big\langle\nabla f(x) - \nabla f(y),\, (x-y)\big\rangle \geq m\,\left\lVert x-y\right\rVert^2, \;\,m > 0\,$ for strong convex function $f: \mathbb R^n \to\mathbb R$. Is it true for ...
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1answer
42 views

What aspects of convex optimization are used in artificial intelligence, if any?

I work on convex optimization with Stephen Boyd's book. As an example, support vector machines are mentioned as an application of separating hyperplanes theorem. I am wondering if there is any other ...
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25 views

Is there a way to determine if the Convex Hull of two polyhedra is going to be huge?

So in this post: Faster Algorithms for Convex Hulls I was interested in determining if a convex hull of two $n$ dimensional polyhedra can be computed quickly, and the answer was in general: no, ...
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Is a convex cone a convex polyhedron?

Say that I have a convex cone $C=\{t|Ax = t, x\geq 0\}$. where $x\in R^n$, and $t\in R^m$, $A\in R^m\times R^n$. Can I say that this is a convex polyhedron? and why? EDIT: Just in order to avoid ...
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2answers
226 views

Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is non-...
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Can Somebody Help Me Find A Certain Paper about Hybrid Proximal Extragradient method for Bregman Functions?

I have read these two papers by Svaiter and Solodov. The first one, published in 1999 (http://pages.cs.wisc.edu/~solodov/solsva99Teps.pdf) presents an error criterion for the hybrid proximal ...
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Is this the correct proximal operator?

I'm supposed implementing certain optimization algorithms (ISTA, FISTA) to minimize: $$\frac12 ||Ax-(Ax_0+z)||_2^2 + \lambda ||x||_1.$$ $A$ is a matrix, $x$ is a vector, $z$ is some noise filled with ...
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1answer
35 views

Choosing $\lambda$ to yield sparse solution

I'm supposed implementing certain optimization algorithms (ISTA, FISTA) to minimize: $$\frac12 ||Ax-(Ax_0+z)||_2^2 + \lambda ||x||_1.$$ $A$ is a matrix, $x$ is a vector, $z$ is some noise filled with ...
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1answer
43 views

Convex optimization: Piece-wise, quadratic objective

This question is about convex optimization with a convex objective function, which is defined piecewise. We have two functions, a concave function A(x) and a strictly convex, increasing function B(x), ...
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26 views

Show convexity of $f(x,y,z)= x^2+y^2+z^2+xyz$

Let $f(x,y,z)= x^2+y^2+z^2+xyz$. Show that $f$ is convex on $\Omega=${$(x,y,z)\in R^3 : x^2+y^2+z^2<\frac{5}{2}$}. To prove it, I want to show that $\nabla^2f(x,y,z)$ is positive definite. I ...
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Is the sum of a bivariate convex function Schur convex?

A well known fact is: Let $f(x)$ be a convex function. Then $g(\vec{x})=\sum_{i=1}^N f(x_i)$ is Schur convex in $\vec{x}$. Supose $f(\vec{z})$ is a convex function of $\vec{z}=(x,y)$. How to ...
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Solution for $\min_{x^Tx=1} x^TAx-c^Tx$

$\min_{x^Tx=1} x^TAx-c^Tx$ looks like a simple QPQC problem. If $A$ is positive semi-definite, can I get the solution by first getting $x=A^{-1}c$ and then projecting $x:=\dfrac{x}{||x||}$ to make ...
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Finding the lower bound of a linear program with the duality method

The issue I have some difficulties understanding the lower bound of a program when applying the duality method. It seems that it comes from $$c^T\underbrace{\le}_{x\ge 0\\y^TA\ge c^T} y^TAx\...
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3answers
446 views

Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and $g_i(...
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Can the constrained optimization problem (1) be transformed into the unconstrained form (2)

(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{\mathcal{C}_k} & \text{rank}(\mathcal{C}_k)\\ \mathrm{s.t.} & \mathcal{E}(\phi_{j}^{k})\le \epsilon \end{...
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general framework for proof of convexity in least square problem

Let $f(x_i,\theta) : X \times \Theta \rightarrow Z$ be a parametric function with parameter $\theta$ that we wish to fit to set of samples $S =\{(x_i,z_i)\}_{i=1}^N$, where $(x_i, z_i) \in X \times Z$....
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Convex set: extreme points and distance to the origin

I'm fairly sure the following is true, although I wouldn't mind being proven wrong. If true, I would like to see an elegant proof, as my attempts are kind of messy. Let $K\subset\mathbb R^2$ be a ...
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Equality constraints into inequalities constraints through elimination

I read here in Section 10.1.2 of this text that a way to eliminate linear equality constraints of the type $$Ax = b$$ in convex optimization problems is to parameterize the related affine space as a ...
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26 views

Are the constrained optimization problem equal to the unconstrained one?

(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{x} & \|Ax-b\|_2\\ \mathrm{s.t.} & \|x\|_1\le \epsilon \end{array} \end{equation} (2) \begin{equation}\label{...
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Why the trace norm of a tensor is a good approximation to its rank?

In many literature, they use the trace norm of a tensor to approximate its rank, which is defined as follows: $$\|\mathcal{X}\|_{\ast}:=\sum_{i=1}^{n}\alpha_i\|X_{(i)}\|_{\ast}$$ where ${\alpha_i}'s$ ...
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Lower bound of averaging gradient method (Prof. Yurii Nesterov's paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems The last inequality confuses me: (p.231) Note: 1. The ...
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How to efficiently solve a quadratic program repeatedly?

I have a quadratic problem, \begin{align} \min_{x\in [0,1]^n} x^T p+ \frac{1}{2\lambda} x^T Q x \end{align} ($Q$ is semidefinite) which I want to solve repeatedly, with the slight change of p and Q, ...
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A ratio of two convex functions with different minima cannot be monotone. Proof?

Let $\lambda(x)=\frac{f(x)}{g(x)}$ where $f(x)$ is a differentiable function minimized at $x=x_1$ and $g(x)$ is a differentiable function minimized at $x=x_2\neq x_1$. How can I show that $\lambda(x)$ ...
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Who proved that the equilibrium problem is equivalent to a monotone inclusion problem?

I'm looking for the original reference where it was proved that given a subset $X$ of a space $E$ and a function $f:E \times E \mapsto \mathbb{R}$, the equilibrium problem of finding $x \in X$ such ...
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Expressing $\forall$ in linear programing

I'm doing a linear program to a game and I don't know how to express $\forall$ in linear programing (or if I had the right intuition to do it). Here is the problem: I have several vessels that are ...
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36 views

Show non-convexity of a function with vector input

How does one go about proving non-convexity of the function d? $$ d(v) = 1/2*||F(v)- p||^2 $$ $$ F(v)=\sum_{i=1}^n l_i*\begin{pmatrix} cos(\sum_{j=1}^i v_j) \\ sin(\sum_{j=1}^i v_j \end{pmatrix} $$ ...
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1answer
32 views

Gradient and Hessian of a function defined in terms of matrix inverse

Let $\mathbf{I}_m$ be the $m$-dimensional identity matrix and $\mathbf{0}_m$ be the $m$-dimensional zero matrix. The matrix $D(\mathbb{x})$, where $\mathbb{x} = (x_1, \dots, x_n)^T$, is defined as: $$ ...
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370 views

Gradient descent vs ternary search

Consider a strictly convex function $f: [0; 1]^n \rightarrow \mathbb{R}$. The question is why people (especially experts in machine learning) use gradient descent in order to find a global minimum of ...
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Subdifferential optimality conditions

I need help with subdifferential optimality. Let $f(x_1, x_2)=x_1^2 + x_2^2 + |x_1 -x_2 - y|$. Find: \begin{align} \min_{x_1, x_2} f(x_1, x_2) \end{align} This is convex, so must have unique ...
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Be C a matrix n x n positive semi definite. proof x'Cx is convex and sqrtroot(x'Cx) is convex.

Hi I have a homework from optimization and I want to know how to do the following exercise: Be C a matrix n x n positive semi definite. proof that: (1)$x^tCx$ is convex. (2)$\sqrt{x^tCx}$ is convex. ...
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26 views

equality constraints and conic constraints in Sedumi (SOCP)

I´m starting playing with Sedumi. I want to solve a problem in the form $$ \min c_0' x $$ s.t. $$ A_1 x = b_1$$ $$ ||A_2 x + b_2|| <= c_2'x+d_2 $$ where $x \in R^n$, $ A_1 \in R^{m_1,n}$, $ ...
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4answers
135 views

Convex set. Proof.

Prove that if A is convex set and $\alpha, \beta ≥ 0$ then $(\alpha + \beta)A = \alpha A + \beta A$ What came first on my mind is that I have to show that $(\alpha + \beta)A\subset \alpha A + \beta A$...
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14 views

Joint convexity through expected value and max operators

I am trying to minimize the following function by choosing $q$ and $z$, where $X$ and $Y$ are random variables, and $r$, $a$, and $b$ are constants. $C(q,z)=E_{X}[a \cdot max(0,X - q)] + E_{Y}[b \...
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Convert Quadratically constrained basis pursuit to LASSO

The Quadratically constrained basis pursuit is to solve \begin{align} \hat{\boldsymbol{x}} &= \arg\min \|\boldsymbol{x} \|_1 \\ s.t. & \| \boldsymbol{Ax} - \boldsymbol{y} \|_2^2 < \eta \...
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LASSO constrained and penalised forms: first - order conditions?

The LASSO problem can be formulated in two ways: 1) Constrained formulation: $$ \|Xw-y\|^2\to\min_{w}\\ \text{s.t. } \sum_{i}|w_i|\leq{t}. $$ 2)Penalised formulation: $$ \|Xw-y\|^2+\lambda\left(\...