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

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Describing a Dual Cone

1)Does dual cone define just for proper cone or all kinds of cone ? 2)Can someone show me a figure that shows a dual cone of a cone ? In Convex Optimization by Boyd (P.51) said that " $y\in k^*$ iff ...
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Normal vector of a Hyperplane

I'm reading convex optimization by Boyd and I have a problem with normal of a hyperplane how many normal can we assume for a hyperplane at just one point? is it true that we can assume many vectors ...
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How can I show that two objective functions are the same?

I am trying to understand the relationship between the constrained and unconstrained versions of a convex optimization problem. The unconstrained problem is as follows: $$\min_{X}||X-Y||_2^2 + \lambda ...
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96 views

Maximizing minimum distance between points placed in a polygon

I would like to maximize the minimum spacing between a fixed number of points ($x_i \in \mathbb{R}^2$) placed inside a polygon in the plane. The minimum spacing includes distance to the polygon. ...
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Online convex programming: Projection followed by normalization

I have the following projected gradient descent online linear programming problem which has been well studied in www.cs.cmu.edu/~maz/publications/techconvex.pdf‎ $\mathbf{y}_{t+1}=\mathbf{w}_t - ...
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How can I prove this problem is quasiconvex?

I'm doing a convex optimization problem. It requires me to fit a rational function to an exponential function. I assumed the original problem would be a quasiconvex optimization problem and based on ...
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55 views

Can SVD help to solve (inequality) constrained least squares problem?

Consider the following minimization problem: $$ ||Q u - h^{o} ||^{2} \to min \;\;\; s.t. \; u \geq 0 $$ where $Q$ is $m \times n$ matrix and $u$ is $n$-dimensional vector and $h^{0}$ is ...
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Nonempty interior feature of a proper cone

one of feature of proper cone is solid which means a proper cone has nonempty interior what dose nonempty interior mean ? I was reading Boyd convex optimization and I saw this term "Nonempty ...
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68 views

Solution in general for a seemingly simple problem

Let $\mathbb{S}$ be a closed, bounded, convex set in $\mathbb{R}^N$. Let $\mathbb{x}=[x_1,\dots,x_N]$ be any arbitrary vector in $\mathbb{S}$. Then what can we comment on the problem \begin{align} ...
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29 views

Prove that a point is optimal in LP-problem

I have the following LP-problem: Minimize $B_1^t Y_1 + B_2^t Y_2 + B_3^t Y_3$ subject to $$ (C_1,C_2,I) \begin{pmatrix} Y_1 \\ Y_2 \\ Y_3 \end{pmatrix}\geq 2 \text{ and } Y\geq 0 $$ where $B_1$ is ...
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50 views

How can the sum of two closed cones be not closed?

Can there be two closed cones $K_1$ and $K_2$ in $\mathbb{R}^3$ such that $K_1+K_2$ need not be closed?
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11 views

Conic hull of a proper function

Suppose $f$ is a proper function pn $\mathbb{R}^{n}$with $f(0)>0$.Now consider $$ g(x) = \text{inf}\{t: (t,x) \in \text{cl(cone(epi(}f)))\} $$ Can I always say that $\exists y \in \mathbb{R}^{n} : ...
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62 views

Difference between maximize $\sum\limits_{k=1}^Kg_k(\mathbf{x})$ and $\sum\limits_{k=1}^{K}\log(1+g_k(\mathbf{x}))$ in convex optimization

I have a problem of the following form: maximize $\;\;\;\,\sum\limits_{k=1}^Kg_k(\mathbf{x})$ subject to: $\;\,\,f_i(\mathbf{x})\leq\,1\,\forall\,i\in\{1, 2, \dotsc, m\}$ ...
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52 views

Expressing rank condition of a matrix in terms of its elements

Let $x \in \mathbb{R}^{n}$, define $X = xx^{T}$. I have an optimization problem with some linear constraints and few quadratic constraints, and I have to solve for $x$. Using $X$ as the unknown ...
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269 views

Semi-positive definite Hessian matrix and local minimum

Suppose we have a function $F(x)$ defined as \begin{equation} F(x) = \frac{1}{2}x^TAx + b^Tx +c, \end{equation} where \begin{equation} A = \begin{bmatrix} 4 & 2 \\ 2 & 1 \\ ...
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24 views

Sparse coding with local sparseness of dictionary

The title is probably pretty unclear, I hope I am able to explain it better here. I am currently working on a problem in the field of sparse coding, that is Principal Component Analysis, Non-negative ...
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30 views

Is this a polyhedron?

Is $S$ a polyhedron? $$S=\{x\in\mathbb{R}^n|\|x-x_0\|\le\|x-x_1\|\}$$ where $x_0, x_1$ are given. $S$ is the set of points that are closer to $x_0$ than to $x_1$. I was thinking the ...
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21 views

Quadratic Program over Box Constraints

Consider $f:\mathbb{R}^n \rightarrow \mathbb{R}$ defined as $$ f(x) := x^\top x + c^\top x $$ for some $c \in \mathbb{R}^n$. Define the (compact) "Box" $$X := \{ x \in \mathbb{R}^n \mid x_i \in [ ...
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79 views

Is the following objective function jointly convex?

I have the following optimization problem: $$ \begin{aligned} & \underset{\alpha, \gamma}{\text{minimize}} & & \end{aligned} \frac{1}{2} \|y - \sum\limits_{i=1}^{S}\gamma_{i}\cdot ...
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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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59 views

KKT conditions for nonsmooth convex problems

What are the KKT conditions for a non-smooth convex function? Is the vanishing gradient of Lagrangian, replaced by $0$ in sub-differential of the Lagrangian, and all other things remain the same? I ...
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41 views

Reference table of “tricks” for converting problems to standard LP, QP, SOCP, etc. form?

Where can I find a decent source/reference that which I can use to look up the various standard "tricks" for converting typical problems to standard form in LP, QP, SOCP, etc.? The Charnes-Cooper ...
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76 views

Explain the convexity by looking the hessian matrix of a function

The hessian matrix of a function is given by, $$ H = \begin{bmatrix} a & b & c \\[0.3em] b & b & 0\\[0.3em] c & 0 & c \end{bmatrix} $$ where, ...
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As a beginner, I would like to solve convex quadratic maximization problem with a gradient descent variant in probability simplex?

I know the basics of gradient approaches to optimize the function iteratively, but for this case have have a equality constraint as $\sum_{i=1}^Nx_i = 1$ where each $x_i \geq 0$ with the objective ...
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61 views

How to find a positive semi-definite linear combination?

Suppose we are given two explicit symmetric matrices $X$ and $Y$ and we'd like to find a non-zero real linear combination $aX+bY$ that is positive semi-definite (if possible). Is there a way to go ...
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13 views

NNLS for under determined system

I have system of equations to solve Ax=B under x>=0. I read that Non Negative least Square(NNlS) algorithm proposed by Lawson and Hanson could solve this system for over determined case( number of ...
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55 views

How to solve this optimization problem?

Suppose I have the following problem: Maximize: $\quad\quad x_1+x_2+x_3+x_4$ Subject to: $\quad\quad \dfrac{\gamma\;a_1\;x_1}{\gamma\;a_2\;x_4+1}\geq1$, $\quad\quad\quad\quad\;\;\quad\quad ...
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54 views

Partial derivative on convex set

If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$. How can we prove that $f$ ...
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Proof of the Moreau decomposition property of proximal operators?

Given the prox operator i.e. $ prox_h (x) = arg min_u (h(u) + 1/2 ||u-x||^2_2) $ the Moreau decomposition property says that $ x = prox_h (x) + prox_{h^*} (x) $ where $h^*$ is the conjugate of ...
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Lagrangian with inequalities

I have a toy question on SVM , where i have to find the weight $w$ by solving the Lagrangian multiplier method by hand . I know Lagrangain with equalities only . Here I have to deal with inequalities ...
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36 views

Express a function as difference of convex functions (DC)

is there a way to express the function $$1-\exp \Big( \frac{-\max(0,x)^2}{\alpha} \Big)$$ as the difference of two convex functions (DC)? Thanks
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47 views

Minimize $L_2$-norm of $x1-b$ where $x \in R, b \in R^n$

Minimize $L_2$-norm of $x1-b$ where $x \in R, b \in R^n$ $||x1-b||_2 \rightarrow ||x1-b||^2_2=||x1-b||^T||x1-b||$ (squaring $L_2$-norm doesn't change outcome and yields quadratic) ...
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19 views

Null space and minimization

Let $x^*\in\text{argmin }f(x)=\text{argmin }\frac{1}{2}\|Ax-b\|^2$ where $A$ is a linear operator. Show that $\text{argmin }f=x^*+\text{Null}(A)$. For $x\in x^*+\text{Null}(A)$ we have ...
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51 views

Relax equality into inequality in convex problem

Let $\mathbf{x}, \mathbf{z}, \underline{\mathbf{x}}, \overline{\mathbf{x}} \in \mathbb{R}^{I}$, where the first two are variables and the last two are given data. I have the following problem: ...
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40 views

Minimizing difference and individual variables in convex problem

Let's say I have the following optimization problem: $$ \begin{align*} \min_{\mathbf{x},\mathbf{y}} & \sum_i x_i-y_i \\ \mathrm{s.t.} & \{\mathbf{x},\mathbf{y}\} \in ...
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30 views

Prove that the intersection of convex sets is convex using the following three points…

I want to prove each point, then, use points (1) and (2) to prove (3). $C_{1} = \lbrace x \in \mathbb{R}^{n} \mid h(x) = 0 \rbrace $ is convex iff $h(x)$ is affine in $C_{1}$ $C_{2} = \lbrace x ...
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Prove that $e^{tx} \le xe^t + 1-x$

Prove that $$e^{tx} \le xe^t + 1-x$$ for $t \ge 1$ and $0 \le x \le 1$ I think I need to use the fact that e is convex? But I can't quite see it. Any help appreciated Thanks.
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Solution to a Quadratic Minimization with Norm Constraint

How do I solve the optimization problem \begin{align} &\min_{\mathbf{x}\in\mathbb{C}^N}\mathbf{x}^H\mathbf{A}\mathbf{x}+2\Re\{\mathbf{b}^H\mathbf{x}\} \\ \mbox{subject to }\\ ...
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Logistic function as “difference of convex functions” (DC)

is there a way to express the logistic function $$\frac{1}{1+\exp(-x)}$$ as the difference of two convex functions? Thanks
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140 views

Why is this weighted least squares cost function a function of weights?

Here is a picture from my book regarding weighted least squares: Totally lost here, so I extracted the main nested issues confusing me: First Question: I know that in any LSE we want to minimize ...
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How to obtain primal problem from Lagrangian?

If you're trying to optimize $\min_x f_0(x)$ subject to $f_i(x) \leq 0$ then the Lagrangian would be $$L(x, \lambda) = f_0(x) + \sum_i \lambda_i f_i(x)$$ The dual problem is $\max_\lambda g(y)$ ...
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“Support function of a set” and supremum question.

I have already learned about what a supremum means from wikipedia and from another answer here. However I am not quite sure what 'supremum over a set of functions' means exactly. As an example, my ...
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Inequality involving max is confusing me.

I am trying to understand one line in a derivation here. Simply put, the statement is that: $$ max\{\theta \ f_1(x) + (1-\theta) \ f_1(y) \ , \ \theta \ f_2(x) + (1-\theta) \ f_2(y)\} \leq \theta \ ...
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Examples of affine functions and convex sets

I'm just learning about convexity and affineness, and I've read over some similar questions asked here, but those were more about general properties. I need some help applying those properties to a ...
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19 views

Conic Farkas lemma for dual programm

Suppose $C\subset \mathbb{R}^n$ is a regular cone (convex, closed) and $C^*=\{y\in\mathbb{R}^n \mid y^Tx\geq 0\}$ the dual cone and $A\in\mathbb{R}^{m \times n}$. I want to show that 1) There exist ...
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32 views

Could anyone give me an example of non-smooth strong convex function? [closed]

Could anyone give me an example of non-smooth strong convex function? I cannot figure out one.
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80 views

Confusion related to augmented lagrangian multiplier method

I have this confusion related to the augmented lagrangian multiplier method from this tutorial How come the gradient wrt y is equal to $\rho(Ax^{k+1}-b)$
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Singular Values of Matrix as Optimization Problem

Assume that $A$ is a positive semidefinite symmetric matrix. It is known that $$\max_{||y||\leq1} \quad y^TAy$$ Has an analytical solution which is the maximum eigenvalue of $A$. This isn't hard ...
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89 views

SDP formulation of noisy low rank matrix completion (2)

Thank you Michael for the answering my previous question, SDP formulation of noisy low rank matrix completion. It seems that I overlooked the problems in my initial question. I didn't recognize the ...
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Linear least squares with sparse inequality constraints for support function estimation

The initial problem is the following: $$ ||h - h^{0}|| \to min \; \; s.t. Qh \leq 0 $$ where $h^{0} \in \mathbb{R}^{n}$ is known vector and $Q$ is a $m \times n$ matrix. The problem arises in specific ...