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

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Compressed sensing: converse of a theorem

Theorem: Suppose that the signal length $N$ is a prime integer. Let $\Omega$ be a subset of $\{ 0, \ldots , N-1 \}$ and let $f$ be a vector supported on $T$ such that $|T| < \dfrac{1}{2}|\Omega|$ ...
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116 views

Dynamic programming problem for discrete linear time varying system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot ...
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38 views

How nuclear norm is convex whereas weighted nuclaer norm is not?

In (http://nuit-blanche.blogspot.in/2014/05/wnnm-weighted-nuclear-norm-minimization.html), it is stated that nuclear norm of a matrix $\mathbf{X}$, given as $||\mathbf{X}||_{*}=\sum_{i} ...
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Entropy proerty

Let $a,b,c>0$ be distinct postive reals. Define four different probability distributions: $$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ ...
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29 views

The optimization problem of soft margin Support Vector Machine: How to interpret?

I try to understand what exactly we are trying to optimize in the case of Support Vector Machine problem, which supports soft margins. The original problem is posed first as, without soft margins ...
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33 views

Minimization using logarithmic barrier function

I'm thinking of the quadratic problem(QP) \begin{align} &\underset{x\in \mathrm{R}^n}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x\\ &\mathrm{subject\ to}\ \ \ \ a_ix \leq b_i\ ...
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Conic hull of outer products

Consider the set of rank-k outer products, defined as $\{XX^T | X \in R^{n\times k}, rankX = k \}$. Describe its connic hull in simple terms. I have found the solution of this exercise but I have ...
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60 views

Using Taylor's theorem and Lagrange form of the reminder to prove the second order condition for convexity

I try to prove the second order condition for convexity. So far' I've done the following: First, I prove second order => convexity: Let $f$ be a function with positive semi-definite Hessian. Using ...
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34 views

Property of Newton step for self-concordant convex functions

Suppose $f(x)$ is a convex and self-concordant function minimized at $x^*$. I have two starting points $\tilde{x}_0$ and $\hat{x}_0$ such that $|\hat{x}_0-x^*| \le |\tilde{x}_0 - x^*|$. We also know ...
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Distributed convex optimization problem

Consider the optimization problem $$ \min_{ x_1, \ldots, x_N } \sum_{i=1}^{N} f_i( x_i ) \\ \text{s.t.: } \sum_{i=1}^{N} x_i \in X, \ x_i \in X_i \ \forall i \in \{1, \ldots, N\} $$ where $f_1, ...
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About Intersection of two convex polytope?

the intersection of two convex hull of two polytope P and Q , is it the convex hull of the intersection of P&Q ? Conv(P) ∩ Conv(Q) = conv(P∩Q) ???.
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Is there any way to transform a non-convex optimization problem into a convex one?

I have an optimization problem which is described as $$\begin{array}{ll} \text{minimize}_x & c^{T}x\\ \text{subject to} & Gx \preceq h\\ & -x^{T}Px - qx - r \leq 0 \end{array} $$ where ...
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27 views

Is there a good textbook/book out there that explains sub gradients thoroughly?

I was interested in learning and understanding sub gradients as much as I could from some good resource. I know what the definition is, but I seem unable to apply the definition to prove basic facts ...
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Partial concave maximization of subset of variables

Let $f(x_1, \dots, x_N)$ be a concave function in $x_1, \dots, x_N$. For arbitray $n>1$, prove that the (constrained) truncated function defined by $$g(x_1, \dots, x_{n-1}) = \max_{x_n, \dots, x_N ...
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61 views

What is the left derivative of the hinge loss function in the context of subgradients?

Let: $$|a|_+ = max\{0,a\}$$ Then the Hinge loss function (in the context of classification in Machine Learning) is: $$V(-yf(x)) = |1 - yf(x)|_+$$ Note that $y \in \{-1,1\}$ Let $f(x) = \langle w, ...
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Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
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38 views

When result of max of min problem is equal to min of max problem

Let's assume there are two functions $f(x)$ and $g(x)$. I want to know when the optimal $x$ of max of min of $f(x)$ and $g(x)$ is not equal to optimal $x$ of min of max of $\frac{1}{f(x)}$ and ...
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129 views

Proof of Simplex Method, Adjacent CPF Solutions

I was looking at justification as to why the simplex method runs and the basic arguments seem to rely on the follow: i)The optimal solution occurs at some vertex of the feasible region (CPF points) ...
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Extra information needed to distinguish combinatorially isomorphic polytopes

The title pretty much sums up my question: what extra information do we need (or what is an example of sufficient information) on top of the face lattice in order to completely characterize a convex ...
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120 views

Show that every extreme point in Q is either an extreme point of P or a convex combination of two adjacent extreme points of P

P is a bounded polyhedron in $\mathbb{R}^n$, $a$ a vector in $\mathbb{R}^n$, and $b$ some scalar. Define $$Q = {x \in P | a'x = b}$$. Show that every extreme point in Q is either an extreme point of P ...
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81 views

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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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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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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58 views

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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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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103 views

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 ...
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22 views

Can this type of constraint be recasted to a convex constraint?

I have an optimization problem where all the constraints are linear but some of the type: $$ y_i = \frac{x_i}{\sum_k x_k} $$ It seems that the equality can be relaxed to an inequality adding the ...
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When does l1 regularisation give a sparse solution?

I was maximising a likelihood function, which is convex. I know that the system has a K-sparse solution. I wanted to know the conditions (or some sufficient conditions) on the likelihood function ...
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An Orthogonal Projection with Weighted Norm

In the context of solving a convex program via projected gradient descent i am facing the following problem: $$\min_{x\in\mathbb R^2}\lVert x-y\rVert_M^2,\qquad\lVert x\rVert\le1$$ or written ...
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43 views

find the angles of a given vector sum

Assume you have n vectors in 2D space, with different fixed magnitudes $l_i$. The problem is to find the angle of each vector such that vector sum is a specific vector. That is, $\sum l_i \cos ...
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Primal-dual subgradient method

In these notes, an extension of the subgradient method is presented in Section 8 (page 30). The method is described so quickly and neither convergence analysis (compared to classical subgradient for ...
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Confusion related to proximal newton method

I was reading this method related to proximal newton methods http://machinelearning.wustl.edu/mlpapers/paper_files/NIPS2012_0388.pdf. I came across this page I didn't get what this part means $ ...
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First order necessary conditions for nondifferentiable nonconvex minimization problem

I am interested in first order necessary conditions for the following minimization problem where the function $f$ is continuous, nondecreasing and concave, with $f(0)=0$, but not necessarily ...
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Detecting faces of polytopes

I am working in convex geometry for the summer with little experience beforehand. It's a lot of fun but it does mean I don't know some of the basic things. I'm interested in the orbits of finite ...
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Infinity norm minimization

I am wondering how to minimize an objective function of the following form: $$\min_{\mathbf{x}\in\mathcal{R}^{MN}} \|\mathbf{x}-\mathbf{y}\|_\infty + \lambda\mathrm{TV}(\mathbf{x})$$ Here, ...
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Interpreting constraints in an optimization problem

I am working on an optimization-based image denoising project in which I have three "flavors" of an optimization problem, one constrained and two unconstrained. They are given as follows: ...
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Formal definition of convexity for multivariate function?

Let $M\in R^{M\times N}$, a function $f: M\rightarrow R$ is called convex on $M$ if $f\big((1-\lambda)X1+\lambda X2, (1-\lambda)Y1+\lambda Y2\big) \leq (1-\lambda)f(X1,Y1) + \lambda f(X2,Y2)$ For ...
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restricted set of a convex set

Let $S \subset \mathbb R^n$, $S$ is convex and let $||.||$ be a norm on $\mathbb R^n.$ For $a \ge 0$ we define $S_{-a} =\{ x | B(x,a) \in S\}$, where $B(x,a)$ is the ball (in the norm $||.||)$, ...
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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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186 views

Until now, what is the fastest optimization algorithm of non-smooth convex functions

I am wondering if I minimize a non-smooth convex function, which solver should I choose. I think I should choose a fastest one with a big convergence rate. Subgradient descent is always on the ...
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Dual of the mixed $\ell_1/\ell_2$ norm?

The mixed $\ell_1/\ell_2$ norm $\Omega_{12} $ is defined as $\Omega_{12}(x) = \sum_g ||x_g||_2$ where $x_g$ are disjoint subsets of the elements of the vector $x$. This is used in machine learning ...
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can we prove that a certain supremum of affine functions is frechet differentiable or at least continuous?

Let $X$ be a Hilbert space. Let $A\colon \operatorname{dom} A\to X$ be linear operator with closed graph (not necessarily bounded). Define $$ g\colon \operatorname{dom}A\to \mathbb{R}:x\mapsto ...
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Uniqueness of the solution to a quadratic opt problem

Consider a positive definite matrix $\boldsymbol H$, the known vectors ${\boldsymbol b}$ and ${\boldsymbol a}_i$. Now the minimization problem is casted with respect to the vector ${\boldsymbol x} $ ...
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270 views

Minimum L1 norm may not obtain the sparsest solution?

Here is a paper called For Most Large Underdetermined Systems of Equations, the Minimal L1-norm Near-Solution Approximates the Sparsest Near-Solution. However, I did not quite get its definition of ...
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Integral Farkas Lemma

The context of this question is commutative algebra, however the question itself is more related to convex geometry. All necessary information is given. In the proof of Lemma 3.1.1 in the book ...
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Is $A^TP+PA<0$, $P>0$ and $A^TP+PA\leq-I$, $P\geq I$ equivalent?

Consider the LMI, where $A$ is a Hurwitz matrix: $A^TP+PA<0$, $P>0$, minimize trace(P) According to Stephen Boyd's book, the inequalities are homogeneous in P and hence can by replaced with ...
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Cauchy point,derivation: whe the constrianed optimizitaion is not used

Sorry for the slightly longer question. Consider the following definition of the Cauchy point $h_{i}^{C}=\alpha_{i}^{C}h_{i}.$ It can be found minimizing a quadratic form ...
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Some problems with a proof of the Farkas Lemma

The following is a proof of the Farkas Lemma that is creating me quite some problems. [I presented the all proof simply to point out the notation used by the author.] My problem is with the last part ...
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Differentiability of the Value (Support) Function

Consider the following problem, \begin{align} c(y,\mathbf{w})=\inf_{\substack{\mathbf{x} \in \mathbb{R}^n_{+} \\ \text{s.t. }f(\mathbf{x}) \geq y }} \mathbf{w} \cdot \mathbf{x} \end{align} where ...
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160 views

Whitening matrix for Fast ICA

I have a matrix $X $ with dimension say $ m \times n $ with $ m> n $. I am trying to whiten this matrix in matlab by first taking the $C= \operatorname{covariance}(X)$ followed by eigenvalue ...