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

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Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
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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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Derivation of soft thresholding operator

I was going through the derivation of soft threholding at http://dl.dropboxusercontent.com/u/22893361/papers/Soft%20Threshold%20Proof.pdf. It says the three unique solutions for $\operatorname{arg ...
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Convexity of the product of two functions in higher dimensions

Exercise 3.32 page 119 of Convex Optimization is concerned with the proof that if $f:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto f(x)$ and $g:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto g(x)$ are both ...
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minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
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Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate

How do I prove an optimal solution to dual is not unique if an optimal solution to the primal is degenerate? I have no idea how to start this. Anyone know any books with these kinds of questions (and ...
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Please explain the intuition behind the dual problem in optimization.

I've studied convex optimization pretty carefully, but don't feel that I have yet "grokked" the dual problem. Here are some questions I would like to understand more deeply/clearly/simply: 1) How ...
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Difference between supremum and maximum

Referring to this lecture , I want to know what is the difference between supremum and maximum. It looks same as far as the lecture is concerned when it explains pointwise supremum and pointwise ...
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5answers
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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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Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernce to this paper [Olivier Chapelle, ...
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Second derivative positive $\implies$ convex

In proof of the following theorem; If $f$ has a second derivative that is non-negative (positive) over an interval then $f$ is convex (strictly convex). $f$ is in real number space., the book I ...
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Prove or disprove that the given expression is “always” positive

I have previously asked a question and I tried to solve it by my own and it led to the question below: Prove or disprove that ...
2
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283 views

Simple resource for Lagrangian constrained optimization?

Just had an optimization lecture. I understand unconstrained methods like Newton and Gradient descent just fine, as well as the ideas that give rise to them. I don't really understand the ideas that ...
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Showing convexity of a function with the restriction over an arbitrary line proof

Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set. $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ ...
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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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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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Maximizing a function by finding derivative

I want to find the value of $\vec{p}$, $p_s$, $p_t$ each of which is a function of the form $f:\mathbb{R}^2 \to \mathbb{R}$ that maximize the following function : $$\begin{align} \int_\mathbb{R^2} ...
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Let $B=\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Prove existence of a unique point $v_0 \in B$ that is closest to $v \notin B$?

Suppose $B=\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Consider a point $v \notin B$. How do I prove that there exist a unique point $v_0 \in B$ that is closest to $v$. Also, how do I ...
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Linear Programming - Tableau Condition

The following tableau corresponds to an iteration of the simplex method: ...
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To prove the existing and uniqueness of a solution

Let function $f$ be differentiable and convex in $R^{n}$. How can it be proved that $\forall \lambda > 0$ solution of system equations $f'(x) = -\lambda x$ exists exclusively ($\exists \hspace{3mm} ...
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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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Properties of the Cone of Positive Semidefinite Matrices

The set of positive semidefinite symmetric real matrices form a cone. We can define an order over the set of matrices by saying $X\geq Y$ if and only if $X-Y$ is positive semidefinite. I suspect that ...
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How the dual LP solves the primal LP

When I heard someone discussing LP the other day, I heard him say, "Well, we could just solve the dual." I know that both the primal LP and its dual must have the same optimal objective value ...
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What is the difference between minimum and infimum?

What is the difference between minimum and infimum? I have a great confusion about this.
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Why is convexity more important than quasi-convexity in optimization?

In the mathematical optimization literature it is common to distinguish problems according to whether or not they are convex. The reason seems to be that convex problems are guaranteed to have ...
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What numerical methods are known to solve $L_1$ regularized quadratic programming problems?

What numerical methods are suitable to solve the following problem $$\min_x \tfrac{1}{2}x^T A x + b^Tx + \lambda ||x||_1$$ where $x,b\in\mathbf{R}^n$, and $A\in \mathbf{R}^{n\times n}$ is positive ...
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Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
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2answers
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Optimize a log det function with respect to a matrix, and the saddle point analysis

Suppose I want to to find the local minima of a logdet function $\mathcal{L}$ with respect to a Matrix $\mathbf{A}$, $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
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1answer
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Show that the dual norm of spectral norm is Nuclear norm.

Could someone help to understand that the dual norm of spectral norm is Nuclear Norm ? We can focus on the real field. Given a matrix $X \in \mathbb{R}^{mn},$ then the spectral norm is defined by: ...
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Is inverse matrix convex?

I wonder a generalization of Jensen's inequality: let $\mathbf{X,Y}$ be two positive definite matrices, can we obtain the following Jensen like inequality ...
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prove this is a strongly convex function

The definition of strongly convex from Wikipedia: It is not necessary for a function to be differentiable in order to be strongly convex. A third definition for a strongly convex function, with ...
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Degeneracy in Linear Programming

Consider the standard form polyhedron, and assume that the rows of the matrix A are linearly independent. $$ \left \{ x | Ax = b, x \geq 0 \right \} $$ (a) Suppose that two different bases lead to ...
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Gradient-descent algorithm always converges to the closest local optima?

Assume $f(\vec x)$, which is Lipschitz continuous, has two local optima $\vec x_1^*$ and $\vec x_2^*$( $\vec x_1^*$ is the global minimum). We start the gradient-descent algorithm from $\vec x_0$ and ...
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If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?

If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
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Why does a positive definite matrix defines a convex cone?

I've been working on convex optimization and got stuck. What exactly does a positive definite(p.d) matrix represent geometrically ? what kind of vector space it forms ? If I have a p.d matrix which ...
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What is the time complexity of conjugate gradient method

I have been trying to figure our the time complexity of conjugate gradient method I have to solve a system of linear equations given by $$ Ax=b $$ where A is sparse and positive definite symmetrix ...
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Fourier coefficient of convex function

On $I = [0, 2π]$ consider the function $f : I → \mathbb{R}$ to be convex. Define: $$a_k\pi := \int_0^{2\pi}f(x) \cos(kx)\,dx$$ Show that the convexity of $f$ implies that $a_k ≥ 0$ when $k ≥ 1$. ...
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Newton's method vs. gradient descent with exact line search

tl;dr: When is gradient descent with exact line search preferred over Newton's method? I simply don't understand why exact line search is ever useful, and here's my reasoning. Let's say I have a ...
3
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Recovering the solution of optimization problem from the dual problem

In the context of (most of the times convex) optimization problems - I understand that I can build a Lagrange dual problem and assuming I know there is strong duality (no gap) I can find the optimum ...
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When finding root, does Newton's method fail if the function is non-differentiable?

According to wikipedia's description, the Newton's method finding a root presumes a differentiable function. Then, will it fail when encountering non-differentiable function? For example, can it find ...
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1answer
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linear approximation with respect to L1 norm

I am trying to solve this problem: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize $\int_0^1|e^x-\alpha-\beta x| dx$ any hints how to proceed
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1answer
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Hessian Related convex optimization question

My precise question is from an exercise; Let $f : \mathbb{R}^2 → \mathbb{R}$ be a twice differentiable function. Prove that there exists a $λ ∈ R$ such that $g : \mathbb{R}^2 → \mathbb{R}$ defined as ...
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1answer
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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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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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How to get the minimum and maximum of one convex function?

Condition: $h,f\in \mathbb{C}^{N\times1}, \text{where}f =\hat{f} + e \text{ and } e^H e \leq 1,\ \ \ Q=h^Hff^Hh$. The Lagrangian function of $Q$ is $\mathcal{L} = h^H(\hat{f} + e)(\hat{f} + e)^Hh + ...
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proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...
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how to find the solution of this cost function?

I have the following cost function. $J = \sum_{i=1}^N a\, Trace(W^TX_iW) - b\, Trace(W^TY_iW)$ Where $X_i$ and $Y_i$ are symmetric matrices, $a$ and $b$ are scalars. How can I find W?
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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 ...
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Dual Optimization Problem

I have the following optimization problem, $$ \text{minimize}_{X,Y} \ \lVert X\rVert_* + \lambda \lVert Y\rVert_1 \\ \text{subject to }X + Y = C$$ $C \in \mathcal{R}^{m \times n}$, $\lVert ...
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convexity of matrix “soft-max” (log trace of matrix exponential)

In convex optimization it is often convenient to use the following smooth approximation to $\max\{x_1, \ldots, x_n\}$: $$ f_\lambda(x_1, \ldots, x_n) = \frac{1}{\lambda}\log \sum_{i = 1}^n{e^{\lambda ...