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

learn more… | top users | synonyms

0
votes
0answers
48 views

Strong convexity of quadratic function

Assume that $Q$ is a positive definite matrix, is it true to say that the function $f(v)=v^TQv$ is strongly convex with respect to the norm $||u||=\sqrt{u^TQu}$? Thanks
0
votes
2answers
75 views

Calculation of minimum infinity norm subject to L1 norm

Can somebody tell me how to evaluate the following in MATLAB or any other programming language? \begin{equation} \min_{x \in \partial \|w\|_1} \| x+y\|_\infty \end{equation} $x,w,y \in R^n$. $w,y$ ...
0
votes
0answers
26 views

Notion of outer normal cone and supporting cone if $x \in$ relint($C$)

In my lecture we defined the outer normal cone $ N_c(x^*)= \{ c\ \in \mathbb{R^n} : \max\limits_{x \in C} \ \ c^Tx = c^Tx^* \}$ and the supporting cone $S_C(x^*)= \bigcap\limits_{c \in N_c(x^*)} ...
0
votes
0answers
43 views

On a modified least square.

Given a vector $y \in \mathbb R^n$ and real constants $x_{ij}$ ($i=1,\dots,n$, $j=1,\dots,p$), we consider a vector $\beta = (\beta_0,\dots,\beta_p)$ which minimize ...
0
votes
0answers
73 views

Computational complexity of the following quadratic program (QP)

Let $A^TA$ be a $n \times n$ matrix. I have the following quadratic program to solve: \begin{array}{rl} \min \limits_{x} & x^T A^T A x \\ \mbox{subject to} & \sum_{i=1}^{r} x_i =1, ...
0
votes
0answers
28 views

How to judge the convexity of this function?

$ f(X) = -\log \det(X^TX+I)$, $X \in \mathbb{R}^{n \times n}$, is this function convex or not? Does anybody have an idea about this problem? Thanks.
0
votes
0answers
38 views

Indicator function to zero-set of a function

Given the indicator function $I_{C}: \mathbb{R} \rightarrow \mathbb{R}$ to a convex set $C \subset \mathbb{R}$ and a function $g(x): \mathbb{R}^n \rightarrow \mathbb{R}$ $$ I_{C}(g(x)) = ...
1
vote
1answer
19 views

Can Somebody Help Me Find A Certain Paper about Hybrid Proximal Extragradient method for Bregman Functions?

I have read these two papers by Svaiter and Solodov. The first one, published in 1999 (http://pages.cs.wisc.edu/~solodov/solsva99Teps.pdf) presents an error criterion for the hybrid proximal ...
3
votes
0answers
50 views

Is minmax equivalent to maxmin?

More precisely, problem $1$ is as follows: \begin{eqnarray} &\max_{1{\le}i{\le}N}\min_{[\gamma_m^i]}\left[\lambda_i - \sum_{m=1}^M\phi_m\gamma_m^iF_m^i\right] \\ &\mbox{subject to} ...
0
votes
1answer
35 views

Matrix equation from optimization problem

I am having a problem to find the solution to the following equation which has arisen as part of the solution of a (convex) optimization problem I am considering. $$\left(\frac{a}{n ...
0
votes
1answer
30 views

Strong convexity, non-smoothness, and directional derivative

I have a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ that is (strongly) convex (say in $\mathbb{R}^n$), but not necessarily differentiable. It attains its minimum at $\mathbf{q}$. Given two ...
0
votes
1answer
14 views

concavity of functions of many variables

I have a function in many variables, the function is concave and non-increasing in each one of the variables, is the entire function concave?
0
votes
0answers
39 views

Newton-Raphson convergence for function $f(\gamma)$ with $\gamma \geq0$ constraint.

I am reading an (engineering) paper that in the part of their solution, They propose a 2-step iterative solution based on Newton-Raphson method for concave function $f$ as below : ...
1
vote
1answer
31 views

Gauss-Newton convergence for constant Hessian

If I use Gauss-Newton to solve a least square optimization problem and $\mathbf{J}^H\mathbf{J}$ is constant does it imply that I will reach the solution in one iteration?
0
votes
1answer
41 views

Non-vanishing of sub gradient near optimal solution

Consider the non-smooth optimization problem \begin{equation} \min_{x \in \mathbb{R}^n} f(x). \end{equation} To solve the above problem, I am suing subgradient descent \begin{equation} ...
0
votes
1answer
24 views

Partitioning in convex problem (variables in two subsets)

Consider the following problem from textbook Convex Optimization Algorithm p.10: \begin{equation} \begin{aligned} &{\text{min}} & & F(x)+G(y)\\ ...
5
votes
1answer
65 views

Why lower semicontinuity?

I'm reading a proof on the existence of a solution to a minimisation problem, but I'm stuck. I give a brief summary of the arguments up to the point at which I'm stuck(at the yellow box). ...
0
votes
1answer
35 views

Proving convergence of projected subgradient descent

Any idea how to sum the series $\sum_{t=1}^T \frac{1}{\sqrt{t}} (\|x_t -a\|^2 -\|x_{t+1}-a\|^2) $, where $a$ is any constant and you can assume $\|x_{T+1}-a\|=0$. This sum occured in proving ...
0
votes
1answer
35 views

Gradient descent with linear perturbation

Given a convex, differentiable function $f$ (from a Hilbert space to $\mathbb{R}$) with a minimum (say $x^*$), I know you can find $x^*$ using gradient descent. Suppose now that you apply gradient ...
0
votes
0answers
20 views

Existence of lagrange multipliers with polyhedral constraints

I am working with a paper (Exact regularization of polyhedral norms, Schöpfer 2012) which states as a well-known fact that, if $f$ is a polyhedral norm, then for some $\mu^* > 0$ \begin{equation} ...
0
votes
0answers
34 views

optimization of a function with inequality constraint

I have a function to be maximized subject to constraints. I can write the primal Lagrange function as the following: (objective function WITH two constraints in the last two terms) $$L_P = ...
1
vote
1answer
104 views

Arc length function of a helix/spiral is convex?

Given the arc-length of a parametric curve, $\int_a^b\|\gamma'(t)\|$ if the parametric curve was non-convex, can the arc length be a convex function?If the parametric curve was convex, will the arc ...
0
votes
1answer
73 views

How to computer the proximal operator of a indicator function?

For $G_{1}(x) = I_{c}(x), c =\{x|Ax=b\}$, the Proximal operator is: $Prox_{\gamma G_{1}}(x) = Proj_{c}(x) = x + A^{T}(AA^{T})^{-1}(y-Ax)$ I hope to know, how to derivative this result. And, for ...
1
vote
1answer
44 views

Is this function convex or non-convex? How do you decide?

The problem is: find $$\min⁡ \mathrm{P}\left[{\log(1+p||H^H \mathbf{w}||^2)\over 1+p||G^H \mathbf{w}||^2}<R\right]$$ constraint to: $||\mathbf{w}||^2=1$ where $H$ and $G$ are matrices of ...
0
votes
1answer
60 views

What modification is this of the notion of Renyi divergence?

Given two probability distributions $P$ and $Q$ over the same outcome and event space (assume finite if needed) one defines their Renyi divergence as $D_\alpha (P \vert \vert Q) = \frac{1}{\alpha -1} ...
0
votes
0answers
79 views

Concave optimization and corner solution

I have a optimization problem as follows: Assumptions: $f$ is an increasing and convex function on $R^+$ such that: $f(x): R^+\rightarrow R^+, \quad f(0)=0, \quad f'(x)\ge1,\quad f''(x)\ge 0 ...
0
votes
0answers
24 views

Should the object function value be decreasing during the iteration procedure in ADMM

I want to solve the following convex optimization problem: $$\operatorname{argmin}\limits_X\|Ax-b\|_2^2+\lambda\sum_{i=1}^3 \|X_i\|_{*}$$ where $X$ is a three order tensor, $X_{(i)}$ is a matrix ...
1
vote
1answer
50 views

How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
3
votes
3answers
78 views

How to show the convexity of this set

Is the set, $S=\{\bf x \in \mathbb{R}^n: \sum_{i=1}^{n} \frac{e^{x_i}}{1+e^{x_i}}=1 \}$, a convex set?
0
votes
1answer
19 views

Is it covex function?$J_{new}(u)=\int_{\Omega} \sum_{i=1}^{N} \lambda_if(x)u_i(x)dx$

I have a function such as $$J(u)=\int_{\Omega} \sum_{i=1}^{N} f(x)u_i(x)dx$$ where $f(x):\Omega \to R$, $0 \le u_i(x) \le 1,\sum_i u_i(x)=1$ Given that $J(u)$ is a convex function w.r.t $u$. Now I ...
2
votes
1answer
82 views

Least Squares Nuclear Norm Optimization

I have the following least squares nuclear norm problem, $$ \min_{\bf X} \frac{1}{2}{\left\lVert {\bf b} - {{\bf W}}vec({\bf X}) \right\rVert}^2_2 + {\lambda_*}\Arrowvert {\bf X} \Arrowvert_* $$ ...
1
vote
0answers
44 views

Convergence results for block coordinate descent methods

I am trying to solve the problem minimize $f(x)$ subject to $x_1 \in C_1, x_2\in C_2, ... x_m\in C_m$ where $x_1, ..., x_m$ are block subvectors of $x$, and $C_i$ are each closed convex sets (not ...
2
votes
0answers
51 views

Accelerated Gradient Descent V.S Nonlinear Conjugate Gradient Descent

Let's consider smooth and convex minimization problem, i.e. $min f(x)$, where $f$ is not necessarily a quadratic function. If measured by iterations, Accelerated Gradient Descend (AGD) has ...
0
votes
0answers
35 views

non-linearity and non-convexity

I am taking a course on linear regression online and it talks about the sum of square difference cost function and one of the points it makes is that the cost function is always convex i.e. it has ...
1
vote
1answer
80 views

Matrix norm in the objective of an optimization problem

I am stuck with the following optimization problem from research. The optimization problem have the following objective function: $\|Q-H\|_\infty$. Here $Q$ is a PSD matrix and $H$ is a symmetric ...
1
vote
0answers
28 views

Question about sub differentiability of convex function

I am reading this book to study sub differentiability. On page 20 it says, as $V$ is a normed linear space for simplification, "a continuous affine function $l(v)$: $V\to \mathbb R$ everywhere less ...
0
votes
1answer
173 views

Compressive sensing for complex matrix

I'm fairly new to compressive sensing, and I have been looking for a MATLAB implementation of the problem $$ A x = b $$ where $A$ is non square, $x$ is kind of sparse and all the numbers involved are ...
2
votes
0answers
46 views

How does one evaluate the derivative of a matrix with a tensor $\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}$?

I am stuck on the following: $$\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}=\text{ ?}$$ with $A$ a $d\times d^2$ matrix, $\mathrm{Id}$ the identity matrix of $d\times d$ ...
2
votes
1answer
76 views

Can I solve a problem like a combination of PCA and compressed sensing?

$$ \underset{A,x}{\text{minimize}} \quad \lambda \left\| x \right\|_{1} + \left\| A \right\|_{*} $$ $$ D = A + Mx $$ Where $M \in \mathbb{R}^{n \times m}$, $x \in \mathbb{R}^{m \times z}$, $E=Mx \in ...
0
votes
0answers
39 views

Minimizing a quadratic term

$\mathbf{x_1},\mathbf{x_2}$ are known and I need to solve the following objective wrt to one variable $\mathbf{y}$. The single constraint is $y(1,1)=1$. This is expressed as an inner product ...
3
votes
2answers
85 views

In linear programming, how to check whether a convex polyhedron is contained in another

Suppose we have two convex polyhedra $P_1=\{x\in \mathbb{R}^n \mid A_1 x \geq b_1 \}$ and $P_2=\{x\in \mathbb{R}^n \mid A_2 x \geq b_2 \}$ Is there a way to check whether $P_1 \subseteq P_2$? I was ...
0
votes
0answers
44 views

What's the difference between this proximal method and subgradient projection?

http://stanford.edu/~boyd/papers/pdf/prox_algs.pdf In the link above it is proposed that the nonsmooth separable resource allocation problem $$\min \sum f_i(x_i) \ \ \text{s.t.} \ \ \textbf{1}^Tx = ...
1
vote
1answer
26 views

Subgradient of the optimal value of a linear program with respect to its parameters

Consider the linear program $f(c)=\min\{c'x\mid x\in\mathbb{R}^n,Ax=b,x\geq0\}$. Are there any results on what is $\partial f/\partial b$?
1
vote
1answer
47 views

How to pivot to an adjacent vertex in simplex method

In the simplex method, we need to move from one vertex of the polyhedron to an adjacent one. Suppose the polyhedron is $P=\{x\in\mathbb{R}^n\mid Ax=b,x\geq0\}$ with rank$A=m<n$. For a ...
3
votes
0answers
69 views

Is $K =\{ S: \exists \text{ positive diagonal} D, D^TSD \;\text{diagonally dominant}\}$ convex?

I am doing some convex cone optimization and wonder whether the following set $K_1$ is convex or not. Assume the following matrices are all in $\mathbb{R}^{n\times n}$ and symmetric. Let the set of ...
2
votes
1answer
106 views

Is the expectation of log-concave function still log-concave?

I know the expectation preserves the concavity (or convexity), but I was wondering is it still true that the expectation of log-concave function still log-concave; to be more precise, Let ...
0
votes
0answers
15 views

Gomory's cut typical running time until the constraint is fractional

I was considering the following problem. Say we are given an linear programming problem $$ \max c^Tx $$ $$ Ax \le b $$ $$ x \ge 0$$ Where instead I consider $i^{th}$ the optimal solution $X_i$ of ...
2
votes
0answers
41 views

how to find the edges emanating from a given vertex in a polyhedron

Suppose my polyhedron $P$ is defined as $P={ x\in \mathbb{R}^n \mid Ax=b, x\geq0 }$ I have $x_0$, which is a vertex of $P$. How to find the edges emanating from $x_0$? In other words, I want to find ...
0
votes
2answers
50 views

Minimum quadratic form value within a line?

If I have $x\in R^n , C\in R^{m\times n}, d\in R^m$, $m<n$, then $Cx=d$ is a linear manifold. And $P\in R^{n\times n}$, $P>0$, the quadratic form is $y=x^TPx$ Is there an analytical expression ...
0
votes
1answer
92 views

How do I prove that objective function is not convex

Here is my objective function. \begin{equation} \begin{array}{c} \underset{\mathbf{x},\mathbf{y}}{\text{minimize}}\hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + ...