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

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535 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
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31 views

Tractability of a cardinality problem

I have this confusion related to the convexity and tractability of a problem. The given problem is maximize $u^TSu$ subject to $||u||_2 = 1$ and card(u) <= r This is a NP hard problem because ...
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84 views

Optimization problem in the Von Neumann Entropy

I have a constrainted optimization problem in the Von Neumann Entropy. In a CVX-like syntax the problem goes as follows: given variable $\mathtt{c(n)}$ $$\begin{align} \text{minimize} \qquad & ...
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1answer
37 views

Confusion related to explanation of convexity of a function

I was reading this paper where the define an optimization problem as where K and L are kernel matrices and $\pi$ is the permutation matrix. They have explained that the function is convex because ...
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58 views

Quadratic Integer Programming

Would anyone mind helping me solve this problem $$ \min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda $$ where $x$ is a vector whose entries are positive ...
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24 views

Analogue of Helly’s theorem for non-exact interpolation

Let $\overrightarrow{x}=(x_1,x_2, \ldots ,x_n),\overrightarrow{a}=(a_1,a_2, \ldots ,a_n)$ and $\overrightarrow{b}=(b_1,b_2, \ldots ,b_n)$ be vectors in ${\mathbb R}^n$, with $a_k \leq b_k$ for every ...
2
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192 views

$\{x:Ax\leq 0\}$ contains a subset of type $\{x:A'x=0, ax\leq 0\}$

If $C:=\{x:Ax\leq 0\}\neq\{x:Ax=0\}$, an independent set of rows of $A$ can be chosen, one denoted by $a$ and the others put as rows into a matrix $A'$, such that $\{x:A'x=0,ax\leq 0\}\subseteq C$. ...
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98 views

Nonlinear optimization of constraint parameter - subdifferential?

Disclaimer: I discovered that the FAQ suggests to post research-level to mathoverflow instead of math.stackexchange. I "moved" the question accordingly, cp. post at mathoverflow. Sorry for the ...
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525 views

SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' ...
2
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1answer
78 views

Proof for certain matrix results?

There are certain results of matrices that Stephen Boyd uses often in his book on Convex optimization. Can someone provide me proof for the results I have enumerated below: If $B \in S^n$ and $A \in ...
2
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1answer
294 views

Convex optimization and linear programming please help! :)

How would I write the following as a standard form LP? Minimizing $\sum_{i=1}^n x_i + c\max(a_i-x_i)$ for $a_i \ge 0$ and what is the optimal value for when $c=n$ How to express minimize $\frac{1}{2} ...
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163 views

Optimization problem about large matrices

I'd like to solve the following optimization problem: Find non-negative scalar $a$, $b$, $c$ to minimize $\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$ where ...
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45 views

Convex formulation of a nearly convex optimization problem

The following problem has come up in my studies of logarithmic norms. I wish to find $\mu \in \mathbb{R}$ and a positive semidefinite $B$ so as to minimize the convex function $c \mu - \log\det(B)$ ...
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113 views

Examples of functions that are Lipschitz w.r.t. Schatten p-norm?

A convex function $f$ is $R$-Lipschitz w.r.t. to a norm $\|\cdot\|$ if for all points $a, b$ we have $|f(a)-f(b)| \leq R\|a-b\|$. For a real symmetric $n\times n$ matrix $A$ with eigenvalues denoted ...
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161 views

How to minimize the supremum of two convex functions?

Given $f_1(x)$, $f_2(x)$, $x\in \mathbb{R}^d$, two convex functions, we define the following problem: $\underset{x\in C}{{\rm minimize}}\,{\rm ...
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166 views

Numerical Methods for minimizing a Non-Differentiable Convex Function of Several Variables

I have a multi-variable convex continuous function which is not differentiable. I am interested to know about different numerical techniques, possibly also references to them, used for this. Read ...
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56 views

convex optimization with inconsistent constraints

If you have a problem in convex optimization where all $N$ constraints ($N >> 0$) yield no possible solution but you are able to rank, or weight the constraint in terms of their importance are ...
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83 views

Convexity of a function

Suppose we have $F: R^n \longrightarrow R$ , $P: R^n \longrightarrow R^n$ and $G: R^n \longrightarrow R$ all nice- let's say given by polynomial and $P$ is invertible - such that $F(x) =G( P(x) )$. ...
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54 views

Duality gap in cone programming

Let $K\subset \mathbb{R}^2$ be a closed convex and pointed cone, $A$ be a $2\times 2$ square matrix and $b, c\in \mathbb{R}^2$. Consider the problem $$ (P)\quad \min\{\langle c, x\rangle: Ax\geq_K ...
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79 views

Find a vector such that its matrix product is positive in every element

Given a matrix $A$ I want to find a vector $\vec{x}$ such that every element of $A\vec{x}$ is strictly positive. Also, the columns of $A$ do not span the full space, so if I were to just naively pick ...
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366 views

A convex programming problem involving sum of logarithms of linear functions

Here is a convex programming problem I encountered while working on an estimation problem for a mixture of multinomial distributions. We have a matrix $A_{m \times n}$ containing non-negative real ...
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405 views

real-time linear programming

I'm going to implement in C a light-weight embedded lp-solver for a production system. I need to be able to sequentially solve a series of (possibly unrelated) linear programming problems with ~6-60 ...
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1answer
1k views

is nonlinear least square a non convex optimization?

linear least-squares are convex optimization. Are nonlinear least squares also convex optimization? Can someone please give some simple examples?
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111 views

Legendre transform of a norm

Let $||\cdot||$ be a norm on $\mathbb{R}^n$, with dual norm $||x||_* :=\max_\limits{y:||y||\leq 1}y^T x$. I'd like to show $$\max_{x \in \mathbb{R}^n}(x^T d-||x||)=\begin{cases} 0 & \text{ if } ...
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2k views

How to prove a set of positive semi definite matrices forms a convex set?

Let $C$ be the set of positive semi-definite matrices, how can I prove it is a convex set?
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3answers
54 views

Inequality involving a convex function

Do the points that satisfy an inequality involving a convex function constitute a convex set? Specifically if $x \in \mathbb R^n$ and I have a function $f(x)$ then is the set $\{x \mid f(x) \le 0\}$ ...
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88 views

Why is the constraint $\|w\| = 1$ non-convex?

Related: Why is this function, related to SVM derivation, non-convex? I am studying notes which cover the derivation of SVM. The intuition is the geometric margin should be maximized in order to ...
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1answer
387 views

What's the difference between interior and relative interior?

As defined in Convex Optimization written by Stephen Boyd, both interior and relative interior seems to describe a same thing: a set that peels away it's boundary points. So what on earth is the ...
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2answers
41 views

Is f(x)=-log(x) a closed function?

I am reading Convex optimization written by Stephen Boyd. In page 640, there is an example said \begin{equation} f(x)=-log(x) \end{equation} is a closed function. But this function seems does not ...
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203 views

Entropy expression optimization with Langrange multipliers

I have recently encountered variants of the following expression: \begin{equation} S = H(a,b,c,d)-H(a+b,c+d) \end{equation} where $H$ is the Shannon entropy function, that is $H(X)=\sum_{x\in X}-x\log ...
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39 views

$sup_x [ f(x)+sup_x g(x)]= sup_x[f(x)+g(x)]$?

$sup_x [ f(x)+sup_x g(x)]= sup_x[f(x)+g(x)]$ is the statement correct? Can I prove like this: $sup_x [ f(x)+sup_x g(x)] = sup_x[f(x)] + sup_x[g(x)] = sup[f(x)+g(x)]$.
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1answer
44 views

Equivalent formulation of linear discrimination problem on Boyd convex optimization slides

In Boyd's CVX slides on pg 189 he has the linear discrimination problem http://stanford.edu/class/ee364a/lectures.html Given data $\{x_1, \ldots, x_n\}$, $\{y_1, \ldots, y_n\}$ The problem of ...
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2answers
57 views

Finding Lagrange multiplier

Suppose I'm looking for $\mathbf{x} \in \mathbb{C^{M\times 1}}$ such that: $$\mathbf{x}=\text{arg}\min_\mathbf{x}\|\mathbf{a+Ax}\|_2^2+\lambda\|\mathbf{x}\|_2^2~,$$ where $\mathbf{a}\in ...
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2answers
46 views

What is the relation between these two definitions of an ellipsoid

There are two definitions of an ellipsoid in Boyd's book (Convex Optimization) $E = \{ x | (x-x_c)^T P (x-x_c) \leq 1 \}$ In the above, P is a positive semi definite matrix. $ E=\{ x_c+Au |\; ...
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4answers
85 views

Optimization of a quadratic function with qudratic constraints

I'm a Graduate student of Electrical Engineering. I have some basic knowledge on Convex Optimization. For my research, I cam across the following optimization program. With $\mu > 0$, find $\arg ...
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1answer
318 views

Sub-gradient of the “$\ell_0$ norm”

I am trying to characterize the sub-gradient of l0-norm ($f(x) = ||x||_0=\sum_{i=1}^n 1\{{x_i \neq 0}\}$). At first, I thought l0-norm is a convex non-smooth function since it satisfies the triangle ...
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3answers
69 views

Questions about coerciveness and convexity

I just have a few yes/no questions, and would really appreciate if you could correct me where I am wrong, and for what fundamental flaw I have. 1. Would the set of coercive functions a linear space? ...
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1answer
64 views

Proving convexity using the Hessian

Suppose I have $f: \mathbb{R}^n \to \mathbb{R}_\infty$ which is twice continuously differentiable, on some convex set C, which is open. How can I prove that $f$ is convex over C, iff the hessian ...
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1answer
541 views

Does every strongly convex function has a stationary point?

Say does every differentiable $\mu$-strongly convex function $f:\mathbb{R}^n\mapsto\mathbb{R}$, with $\mu>0$ have a point where its gradient is $0$? If not so which is the minimum you can impose ...
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2answers
99 views

Why is any subspace affine?

I am studying 'Convex Optimization' written by Stephen Boyd. I am confused by an assertion in the book(page 27). Any one can tell me why and give an explanation ? Any subspace is affine, and a ...
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2answers
521 views

When is the difference of two convex functions convex?

Assume that $X$ is a finite dimensional Banach space. I know that in general if two functions $f:X \mapsto \mathbb{R}$, $g:X \mapsto \mathbb{R}$ are convex then the function $(f-g):X \mapsto ...
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2answers
100 views

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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80 views

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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78 views

multi-objective optimization

I am currently encounterring a optimization problem. The goal is optimize an objective function A and B at the same time. But the problem is that optmizing A will almost always tradoff with B, such ...
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90 views

The Dual problem of a non constraints problem?

The primal problem is $min_{w\in R^d}: P(w)$ where $P(w)=\frac{1}{n}\sum_{i=1}^n\phi_i(w^Tx_i)+\frac{\lambda}{2}||w||^2$. The dual problem is $max_{\alpha\in R^n}: D(\alpha)$ where ...
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1answer
53 views

Is $f(X) = \| Y - XX^T \|_F$ convex given fixed $Y$?

In the scene of nonnegative matrix factorization, $f(X_1, X_2) = \| Y - X_1 X_2 \|_F$ is not convex, but both $f(X_1)$ given fixed $X_2$ and $f(X_2)$ given fixed $X_1$ are convex, enabling us to ...
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2answers
521 views

Maximization of a log det function

I want to solve the following optimization problem $$ \text{maximize } f(X) = - \log \mathrm{det}(X+Y) - a^T (X+Y)^{-1} a \\ \text{subject to } X \succeq W, $$ where the design variable $X$ is ...
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303 views

Composition of non-monotonic convex function

Given the following composition of functions: $h:\Bbb R^k\rightarrow\Bbb R$ $g:\Bbb R^n\rightarrow\Bbb R$ $f(x)=h(g_1(x),g_2(x),...,g_k(x))$ There are known rules which guarantee ...
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1answer
163 views

Free software or algorithm for Second-Order Cone Program

I need to solve the following optimization problem: $$ \mathbf{x}^\ast = \operatorname{argmin}_{\mathbf{x}} \Vert \mathbf{Rx} \Vert_2^2 \;\;\; \mathrm{s.t.} \;\;\; \mathbf{s}^\mathrm{H} \mathbf{x} = ...
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2answers
265 views

What is the motivation behind strong convexity

Definition : A function is said to be $\beta$-strongly convex if, $f(\theta w + (1-\theta) w') \le \theta f(w) + (1-\theta) f(w') - \frac{\beta}{2}\theta(1-\theta)(w-w')^2$ What is the motivation ...