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

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Quasi convexity and strong duality

Is there a possibility to prove the strong duality (Lagrangian duality) if the primal problem is quasiconvex? Or is there a technique that proves that the primal problem is convex instead of ...
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establish that a minimization problem is indeed a convex minimization problem and then solve it

I have a minimization problem of the following form : $$\underset{\lambda (x) \in [0,1]}{min} \int_\Omega (1- \lambda(x))C_s(x)dx + \int_\Omega \lambda(x)C_t(x)dx + \alpha\int_\Omega ...
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730 views

Trace Eigenvalue Relation

I am reading Boyd's Convex Optimization text, and I am looking at a relation between the trace and the eigenvalue of a matrix. It is on page 92, example 3.23, line 7. The matrix $Y$ (apparently) has ...
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648 views

Converse supporting hyperplane theorem

Exercise 2.27 in Boyd and Vanderberghe: Suppose the set C is closed, has nonempty interior, and has a supporting hyperplane at every point in its boundary. Show that C is convex. Seems to me one ...
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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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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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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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Prove this function is convex

we have $ g: R^n \rightarrow R$ is a concave function and $S$={$x :g(x)> 0$} and $f:S \rightarrow R$ and $f(x)$=$1/g(x)$ so we must show that $f$ is a convex function
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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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Can we ever have E(argmin(f)) = argmin(E(f))?

Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}:\ \mathbb D^N \rightarrow\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ ...
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Show that scalar-valued function of a matrix is convex

Consider the mapping $$f(X) = g\left(\frac{b}{a^TXa}\right),$$ where $g$ is a convex function, $b$ is a strictly positive scalar, $a$ is a real vector, and $X$ is restricted to be symmetric and ...
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What is the definition of “convex” and “relaxation” concepts in clustering?

I have following text from a paper i am trying to understand: I don't understand what does below sentence refers to as being convex/non-convex The problem is that even though the objectives ...
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Express this linear optimization problem subject to a circular disk as a semidefinite problem.

I have to express following problem as a semidefinite problem: $ min \, F(x,y) = x + y +1$ subject to (1) $(x,y) \in \mathbb{R}^2 : (x-1)^2+y^2\leq 1$ Only affin equality conditions should be used. ...
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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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Optimization of competitive scenario

Suppose we have a function $f(x_1,x_2)$ with the following properties: Let $x^*=\arg \max_{x_1} f(x_1,x_2=x^*)$ and $x^*=\arg \min_{x_2}f(x_1=x^*,x_2)$. $f(x_1,x_2)$ is concave in $x_1$. ...
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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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Why test problems in convex optimization are mostly random?

Very often people who compare performance of different algorithms in convex optimization use randomly generated data. For instance, this often happens in compressed sensing and signal processing. Is ...
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34 views

Toeplitz equality constrained least-square optimization

What is the fastest known algorithm for least-square optimization problem with a linear equality constrain \begin{align*} &\min \|K x - y\|^2 + \mu \|x\|^2\\ \text{s. t. }& Q x = v ...
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45 views

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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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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Difference between tangent cone and this constraint cone

Define the cone $G(\textbf{x} = \{\textbf{p} \in \mathbb{R}^n | \nabla g_i(\textbf{x}^T \textbf{p} \leq 0, i \in I(\textbf{x})\}$ So this is a cone associated with the inequality constraints that is ...
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28 views

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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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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30 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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39 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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77 views

Argument to “linearize” an objective function

I have this optimization problem on the variables $\lambda_\ell^+, \lambda_\ell^-$ such that $ \lambda_\ell^+ \geq \lambda_\ell^-$ with $\ell=1,\ldots,n$ , and fixed $P\in [1/(n+1),1]$ \begin{align} ...
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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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108 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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Relation between Riccati Algebraic Equation and optimization problem

Reading this page: http://www.mathworks.com/help/robust/ug/minimizing-linear-objectives-under-lmi-constraints.html I got stuck in the result that says it can be show that minimizing Trace of X (a ...
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Why geometric median cannot be solved analytically

$\DeclareMathOperator*{\argmin}{argmin}$ For a given set of $m$ points $x_1,...,x_m$ with each $x_i\in \mathbb{R}^n$, the geometric median (or the weber point) is defined as $$\argmin\limits_{y \in ...
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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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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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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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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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Knuth's Sandwich Theorem: requesting proof clarification

The question is about F6 of Section 8 ("Elementary facts about cones") in Donald Knuth's Sandwich Theorem (http://arxiv.org/pdf/math/9312214.pdf). He claims to prove $(A \cap B)^* = A^* + B^*$ when ...
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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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Finding analytically proximal operator of $g(P)=\|PX\|_1$ using Moreau decomposition

$\newcommand{\prox}{\operatorname{prox}}$ Consider the function $g(P) = \|PX\|_1$, where $P \in R^m \times R^m$ and $X \in R^m \times R^n$ is a known constant matrix. In the paper by Parikh and Boyd ...
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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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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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Stochastic convex (conave) functions vs. convex (concave) function

Can someone help me understand the difference beween stochastic convex (conave) functions and convex (concave) function
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89 views

Proximal Mapping for maximum of linear and quadratic function

I was wondering if there is an efficient way of calculating the proximal mapping of the following function $f : \mathbb{R}^3 \rightarrow \mathbb{R}$, $b_i \in \mathbb{R}^3$, $c_i \in \mathbb{R}$ : $$ ...
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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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Could I get the explicit solution to the following problem relate to generalized rayleigh quotient?

$\bf x$ and $\bf a$ are complex vectors, $\bf C$ is positive definite complex matrix, $\bf B$ is positive-semidefinite complex matrix. What's the objective value? Thanks! $$\max_{\bf x} ...
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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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Convexity of a rational function

I am attempting to (dis)prove that the function $$\frac{4x+3y+2}{x^2+xy+2x+y}$$ is convex for $x,y>0$. Attempting to differentiate the function does not seem like a good idea (or am I making a ...
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accomodating non-negativity constraint in the dual

Suppose the objective implicitly imposes non-negativity constraint, say, the objective is sum of square roots of the decision variables. Is it necessary to consider the inequality constraints imposing ...