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

learn more… | top users | synonyms

0
votes
0answers
14 views

Polar cones' property [duplicate]

I am trying to prove: $A \subseteq B \implies B^\circ \subseteq A^\circ$ where $A^\circ$ is polar cone of $A$ ($A$ convex cone) and $B^\circ$ is polar cone of $B$ ($B$ convex cone)
1
vote
0answers
30 views

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 ...
1
vote
1answer
29 views

What does 'the level set is bounded' exactly want to tell?

'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in ...
0
votes
0answers
25 views

Are these functions convex functions?

$f_i(z) = f_i(x,y) = y_i^\alpha - x_i$, where $x,y \in R^n$, $z = (x, y)$, $\alpha \in R$ and $\alpha > 1$, $i = 1, \dots, n$. Are these functions convex functions ? For $f_i$, the gradient is $ ...
0
votes
1answer
42 views

How to show these two problems have equivalent solutions

I have two problems, where $A$ is positive definite: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\} \ (1)$$ and $$ max_\lambda \ q(\lambda) = -0.25b^T(A+\lambda I)^{-1}b - \lambda : ...
0
votes
0answers
24 views

Property of monotone operator (Positive definite)

I would like to prove this statement: "$F$ is monotone if and only if $\nabla F$ is positive semidefinte." I only know $F$ is monotone with respect to $\Omega$ if and only if ...
1
vote
2answers
42 views

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 ...
0
votes
1answer
39 views

Find those values 'a' which belongs to the Convex Hull

Find those values of 'a' for which (1,a,1) belongs to the convex hull of $$\{(0,0,0), (1,1,2),(2,4,-6), (1,3,8)\}$$ Give me hints as much as you can, I would like to understand the mindset rather ...
2
votes
1answer
77 views

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 ...
0
votes
1answer
37 views

Conditions for unique solution of a maximization problem?

Let $S\subseteq \mathbb{R}^2$, $d:=(d_1,d_2) \in S$, and $s:=(s_1,s_2)$ a generic point of $S$. Assume that there exists $s \in S$ such that $s_1>d_1$ and $s_2 >d_2$. Consider the following ...
2
votes
1answer
30 views

Is there Lipschitz property for subdifferential?

I'm trying to bound the quantity $\langle \nabla \Psi(x),\bar{x}-x \rangle$ above, with the bound depending on $\|x-\bar{x}\|$ and perhaps also of $\|x-y\|$ for fixed (but not varying) points $y$. ...
0
votes
0answers
15 views

How to work with difference-of-elements penalty in optimization

I am trying to solve the optimization problem $$\min_{H,S>0} \|W(H+S)-X\|^2_F+Q(H)+\eta\|S G\|_F^2$$ where $X\in\mathbf{R}_+^{m\times T}$, $W\in\mathbf{R}_+^{m\times k}$, ...
2
votes
2answers
85 views

Efficiency of a max-min problem for $\sum_{j=1}^m |b_j-a_j|$ with $a_i$, $b_j$ restricted to convex sets

Consider the following optimization problem: $$\max_{\{a=(a_1,a_2,\ldots,a_m)\in A\}}\min_{\{b:=(b_1,\ldots,b_m)\in B\}} \sum_{j=1}^m |b_j-a_j|.$$ Is computing the optimal value of this problem ...
2
votes
1answer
28 views

What is the solution for this quadratic program?

Given scalars $p_1\geq p_2\geq \cdots \geq p_r > 0$, can we find a solution for following problem? \begin{align} \text{minimize} & & & \sum_{j=1}^{r} p_j (1-t_j)^2 \\ \text{s.t.} \\ ...
2
votes
0answers
41 views

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 $\mathbb D$ ...
1
vote
1answer
22 views

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 ...
1
vote
1answer
24 views

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 ...
0
votes
0answers
15 views

reference request for solution to SDP is an extreme point

I'm looking for a reference which establishes that the optimal value for a standard SDP is attained at an extreme point. For instance, this is noted below Theorem 1.2 of ...
0
votes
0answers
23 views

convexity of function built from piecewise linear convex function?

Let $B(x):[0,1] \rightarrow [0,1]$ be piecewise linear increasing convex function with $B(0)=0$ and $B(1)=1$. (Think of the power of the neyman-persron test). Let $E(x)=-log(B(1-e^{-x}))$ a logaritmic ...
0
votes
1answer
33 views

Quadratic Program reformulation

I have the quadratic program $$\max\quad \mu^Tx+r_fx_0-\gamma \sum\limits_{i=1}^n |x_i-y_i|-\frac{\lambda}{2}x^TVx$$ $$\text{s.t. }\quad \mathbb{1}^Tx+x_0=1$$ where $\mu$, $r_f$, $\gamma$, $\lambda$, ...
2
votes
1answer
60 views

Sum of k-largest eigenvalues of a symmetric matrix as an SDP

I found the following statement from a google search. If $S_k(\mathbf{X})$ is the sum of the $k$ largest eigenvalues of a symmetric $m\times m$ matrix $\mathbf{X}$, then,$$S_k(\mathbf{X}) \leq t$$ is ...
1
vote
1answer
27 views

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. ...
0
votes
2answers
47 views

Prove convexity of function over space of positive definite matrices

I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. What I have done: It seems like this problem could be tackled by considering the ...
0
votes
0answers
24 views

Feasible solutions in semidefinite programming - beyond Slater's theorem

Suppose a semidefinite program is given by: \begin{align} \text{Maximize:}&\quad \text{Tr}\left[AX\right],\\ \text{subject to:}&\quad \Phi\left(X\right)=B,\\ &\quad X\geq 0. \end{align} ...
0
votes
1answer
46 views

Type of convex function?

I want a convex function $f:\mathbb{R} \to \mathbb{R}$ with the following property: given points $x,d \in \mathbb{R}$, and $\alpha \in (0,1)$, we have $$f(x + \alpha d) \geq \alpha f(x + d).$$ Is ...
0
votes
1answer
35 views

Distributed Newton methods for large scale problems

I am keen to know about the literature landscape for distributed convex optimization methods which use second order information like the Newton step. This is as such a less evolved area compared to ...
1
vote
1answer
33 views

Analytical solution to the first PCA direction

It is known that the first PCA direction for a dataset of $n$ points is the unit vector with max variance after projecting the points onto this vector. I wonder whether there are some analytical ...
0
votes
2answers
26 views

convexity of piece wise function

I have a piece-wise function defined on the real line. The pieces are connected continuously, and the second derivative of each piece is strictly positive. Does this means that the function is convex? ...
1
vote
1answer
15 views

Weighted Norm Minimization

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where ...
0
votes
2answers
57 views

Looking for numerical methods for finding roots of convex vector function ${\bf f}({\bf x})={\bf 0}$

Consider the function ${\bf f}:\mathbb{R}^n\to\mathbb{R}^m$ defined as ${\bf f} = (f_1,f_2,\ldots,f_m)$ where each $f_i:\mathbb{R}^n\to\mathbb{R}$ is twice-continuously differentiable convex in ${\bf ...
0
votes
1answer
22 views

Properties of functions having the form $g(x,t) = t f(\frac{x}{t})$

I have been frequently coming across the function $g(x,t) = t f(\frac{x}{t})$ in my course on convex optimization. A friend of mine mentioned that it is the perspective function, but the book on ...
0
votes
0answers
38 views

Low rank approximation using CVX toolbox in Matlab

I try to use CVX toolbox to do "low rank approximation" work. The code is as follows: ...
0
votes
1answer
78 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 ...
0
votes
0answers
31 views

SOCP or SDP optimization problem

I am studying an optimization problem \begin{equation} \mathbf{w}^* = \text{argmax} \sum_{d=1}^D \log \bigg( \frac{|\mathbf{f}_d^H\mathbf{w}|^2+c_1}{|\mathbf{f}_d^H\mathbf{w}|^2+c_2} \bigg)\\ \\ ...
2
votes
0answers
29 views

proving that the ewma is convex

Hi: I know it's true but I don't know how to prove that exponentially weighted moving average when view ed as a function of $\lambda$, is strictly convex. The exponentially weighted moving average ...
0
votes
0answers
27 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 ...
0
votes
1answer
33 views

Geometric interpretation of duality and Slater's condition

I am trying to study about optimization problems, Lagrange duality and related topics. I came across some presentation on the net, which claims to show the geometric interpretation of the duality and ...
0
votes
0answers
48 views

Which is the better way to optimize a function with 3 variables

I have an optimization function depends on 3 parameters a, b, and c. Which is the better way to optimize it? ...
1
vote
0answers
25 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 ...
1
vote
1answer
69 views

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 $x_1$. ...
2
votes
2answers
40 views

Every concave function that is nonnegative on its domain is log-concave?

This is a statment from Wiki. I'm not sure why this is true: If: $f(\theta x+(1-\theta y) \geq \theta f( x) + (1-\theta)f(y)$ And $f(\cdot) \geq 0$ then: $$f(\theta x+(1-\theta y) \geq f( ...
0
votes
1answer
24 views

Huber penalty function in Linear programming form.

I am trying to solve problem 6.3(b) of Convex optimization by stephen boyd and Vandenberghe. The problem is this Express the following minimization as a QP,LP,SOCP or SDP $$ ...
0
votes
0answers
20 views

What's wrong with the following trace optimization?

I'm reading a paper that has used the augmented Lagrange function for optimization. I've tried to derive one subproblem but got a different answer from that in the paper. Could you help check it ...
0
votes
1answer
24 views

Optimization problem $L(R, PQ) \rightarrow \min$

Suppose we have some $n \times m$ matrix $R$ and we want to find non-negative decomposition on matrices $P$ of dimension $n \times d$ and $d \times m$-matrix $Q$. But since exact decomposition usually ...
3
votes
1answer
47 views

Upper bound on maximum absolute value of all subdeterminants of a matrix

Let $A \in \mathbb{R}^{m \times n}$ and let $\Delta(A)$ be the maximal absolute value of the determinants of the square submatrices of $A$. A simple lower bound would be $$ \Delta(A) \geq ...
2
votes
1answer
39 views

Faster gradient descent convergence by transforming the gradient?

If we modify the gradient descent update for a convex objective function $f(\boldsymbol{\theta})$ from $\boldsymbol{\theta}_{t+1} = \boldsymbol{\theta}_t - \nabla f(\boldsymbol{\theta}_t)$ to ...
1
vote
1answer
19 views

Is it right for chain rule in trace function?

The objective function is $$ f(X)=\min_X trace(B^TX^TCXBD) $$ we know the following derivatives from Matrix Cookbook, $$ \frac{\delta{trace(B^TX^TCXB)}}{\delta X}=C^TXBB^T+CXBB^T \\ \frac{\delta ...
1
vote
1answer
30 views

What is affine hull of conv(A)

Consider the set $A = \{(1,0),(0,1),(-1,0),(0,-1)\}$. The convex hull of $A$, i.e. $conv(A)$, should look like the following: (This is also a $l_1$-norm unit ball.) My question is what is the ...
0
votes
0answers
36 views

Uniform Sampling over Convex Polytope (not full-dimensional)

I want to simulate a uniform distribution on a convex polytope that is not full-dimensional for optimization purposes (to generate random points on the set I want to minimize over). The polytope is ...
0
votes
0answers
35 views

Relative interior

I'm having trouble solving the following equivalence. Let $A \in K^{m,n}, b \in K^{m}$ and $P := \{x \in K^m | Ax \le b\}$ Show that: a) There exists $x^1, ..., x^m \in P$ such that $A_{j*}x^j < ...