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

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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
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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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1answer
57 views

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

$x,y,z > 0$ 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
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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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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1answer
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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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1answer
24 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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1answer
42 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
38 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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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
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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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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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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1answer
34 views

Convex matrix function

Please give me some hints for the following problem: Let $S = \{D \in R^{m \times n}, \|d_i\| \leq 1, i = 1, 2, \dots, n \}$. Find condition of $F \in R^{m \times m}$ such that the function: $ f(D) ...
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1answer
27 views

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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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 ...
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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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28 views

The convexity of the range of concave function

Given a concave function $f: X \rightarrow \mathbb{R}^n $ with a convex domain $X \in \mathbb{R}^n$, is the range of $f$ a convex set also?
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Farthest point on a parallelotope from the origin

I have two related questions. First, consider a maximal independent set of vectors $\{v_1,\cdots,v_k\}$ in $k$ dimensional space. The rows of a square matrix $A$ are from those vectors. The origin is ...
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1answer
102 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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2answers
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Solving for gradient of Frobenius norm term

Let's first define a couple of variables: $A,B,C \in \mathbb{R}^{m \times n}, D \in \mathbb{R}^{n \times n}$, and $\mu$ is a scalar. Say I have an ADMM sub-problem that looks like this: $\arg ...
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1answer
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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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Shrinkage operator for matrices

Here http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf, on page 188, you can see the derivation of the soft thresholding operator or shrinkage operator for the case of vectors using Moreau ...
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2answers
48 views

$\max \{\sum \limits_{i=1}^n z_i w_i \}$ where $\sum \limits_{j=1}^n |w_j| = 1$

How can we find $\max \{\sum \limits_{i=1}^n z_i w_i \}$ where $\sum \limits_{i=1}^n |w_j| = 1$? $z, w \in \mathbb{R}^n$
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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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28 views

what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
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Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
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3answers
131 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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1answer
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Reentrant constraints in active set algorithm?

Problem definition Supposing you're trying to solve a quadratic program: $$ \min_x f(x) = \frac{1}{2}x^T Q x + c^T x \\ \mbox{s.t} \, \; A x \ge b$$ Where Q is square ($n$x$n$), positive semi ...
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Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
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Closed-form solution of the following LP problem

I am considering the following LP problem: $$ \begin{array}{cl} \text{maximize} & c^Tx\\ \text{subject to} & a^Tx\geq0 \\ & 0\leq x\leq x^\max \end{array} $$ where ...
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2answers
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Augmented Lagrangian

Consider the following equality constraint minimization problem: minimize $\text{ }f(x)$ subject to $Ax=b$ Its Lagrangian is then: $L(x,y) = f(x) + y^T(Ax-b)$ We can use then gradient ascent to ...
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3answers
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Optimizing an expression containing sum of square roots of squared terms

For optimization problems involving square root, it is common to optimize the squared expression instead of that containing the square root. What if we have sum of squared expressions ? Consider the ...
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Applications of low-rank matrix approximation

There was a similar question here Use of low rank approximation of a matrix that has unfortunately remained unanswered. Although being along the same lines, my question will be formulated in a little ...
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Intuition behind accelerated first-order methods

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ Suppose that we want to solve the following convex optimization problem: $\min_{x \in \mathbb{R}^n} g(x) + ...
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Convex Inequality describing Functions inside specific area

Let us assume that we have two functions $f_1$, $f_2:[0,1] \rightarrow \mathbb{R}^{2}$, which describe each a point trajectory on the plane. Let us further assume that we parametrize those functions ...
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2answers
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How to check for convexity of function that is not everywhere differentiable?

I have a question. I have just been introduced to the subject of convex sets and convex functions. I read this in wikipedia that a practical test for convexity is - to check whether the 2nd ...
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67 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ $$ \min_{x \in \mathbb{R}^2} \left\| x-p \right\|^2 $$ $$ \text{sub. to: } \ A x \leq b, \ \ x_1^2 + x_2^2 = 1 $$ where $p \in \mathbb{R}^2$ is a ...
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Model $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty $ into standard form

I need to solve the following convex optimization problem: $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty$ s.t $x ...
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3answers
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Is this optimization problem solvable?

I have the following optimization problem: $$ \min_\mathbf{x}\|\mathbf{a+Bx}\|^2 ~~\text{s.t}~~ \|\mathbf{y+Ax}\|_\infty \leq \beta\|\mathbf{y}\|_\infty ~~,~~ \|\mathbf{x}\|^2 \leq \alpha^2$$ where ...
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1answer
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Generalization of log-convexity (log-concavity): log-log-convexity (log-log-concavity)?

$\underline{\mathrm{Background\; on\; function\; Convexity}}$ A function, $f$, is convex if: $$f( x\theta+y(1-\theta) ) \leq \theta f(x) + (1-\theta)f(y).$$ $f$ is concave if $-f$ is convex, [1]. If ...
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1answer
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Complexity of convex optimization problem with interior-point

Assume I have a matrix $Q \in \mathbb{R}^{n \times n}$ which is positiv definite and symmetric, and I want to minimize $$ \frac{1}{2} x^T Q x + x^T f $$ on a non-empty convex set $x\in D=\{ v \in ...
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Linearized ADM and matrix adjoints

$\newcommand{\argmin}{\operatorname{argmin}}$ In this question I am referring to this paper ...
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How to convexify (relax) this L0 eigenvalue optimization problem?

Let $C_1,\dots,C_L$ be $N\times N$ hermitian matrices. Let $d<0$ be a given negative constant. Then consider the optimization problem \begin{align} \max_{r\in \mathcal{R}^{L\times 1}} &\mid\mid ...
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How to solve this optimization problem? (may be gradient descent?)

I have the following optimization problem. $$\operatorname*{argmax}_{w} \|(1-w)\boldsymbol{X} -w\boldsymbol{Y}\|^2 \\ s.t. \quad 0<w<1 $$ How can I find the solution of this problem? May be ...
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Solve the linear program

Please help to solve this problem. I am new to this type of problems and any help will be greatly appreciated $$\text{ Minimize } 7x-5y+3z$$ $$\text{ Such that } \ \ \ 0 ≤ x ≤ 6 , -2 ≤ y ≤ 7 , -4 ...
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Alternating Direction Method of Multipliers (ADMM) application

$\newcommand{\argmin}{\operatorname{argmin}}$ Recall, that ADMM algorithm solves the problem of the form: $\min \text{ } f(X) + g(Z)$ $\text{s.t. } AX + BZ = C$ where $X$, $Z$ and $C$ are real ...