# Tagged Questions

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

60 views

34 views

### solving Basis pursuit denoising with nuclear norm regularization

$$\min_{S,L} \quad \left\| S \right\|_{1} + \left\| L \right\|_{*}$$ $$\text{s.t.}\quad { \left\| D-MS-L \right\| }_{2}^{2}\le \epsilon$$ S,L,D,M are all matrix, $\epsilon$ is scalar, D and M is ...
54 views

### how do i show that a function $f(x)$ is convex given that the inequality holds

So i have to prove that the inequality below is true if and only if f is convex and Lipschitz continuous. i have the first part down which is to assume f is convex and show the inequality. But i cant ...
29 views

### Minimization problem over convex compact set [duplicate]

Does anybody know a method to solve the following problem numerically (or analytically, if there is a general method, but I doubt it). I am given a matrix $A \in \mathbb{C}^{n \times n}$ and I want ...
37 views

### Minimize the product of the traces of PSD matrices

Given two positive semidefinite matrices $X,Y$, I want to minimize their product: $Tr(X)Tr(Y)$. Now as far as I understand the following holds: 1) $Tr(X)Tr(Y)$ is convex since $Tr(X)$ and $Tr(Y)$ ...
18 views

### Convex optimization: how to understand epigraphical projection

In Rockafellar's text on convex optimization: Here we can think of the epigraphical projection as if someone shined a light to $f(x,u)$ and the shadow on the $u$ plane is the projection. My ...
43 views

26 views

### Given $\{x \in \mathbb{R}^2_+: x_1 x_2 \geq 1\}$, find expression of supporting hyperplane

$\{x \in \mathbb{R}^2_+: x_1 x_2 \geq 1\}$ is difficult to sketch because the boundary of this set is the set $x = (t, 1/t)$, $t > 0$ and I don't have a good idea where this supporting hyperplane ...
53 views

### Finding a sparse convex combination of basis vectors

Real-world problem: Given $m$ registered 3D geometry meshes with some known consistent measurements (e.g. human bodies with height, hip girth, inseam etc.), I want to produce a new mesh for some ...
29 views

### Duality in Langrange Multiplier

The problem is to minimize $$f(x) = x^T.x$$ subject to condition $$Ax = b$$ With the help of Lagrange Multipliers, which gives the equation $$L(x,\lambda) = x^Tx + {\lambda}^T(Ax-b)$$ The solution ...
41 views

66 views

### Quadratic programming over nonnegative orthant?

I'm trying to solve a quadratic optimization over nonnegative orthant as below. \begin{aligned} \mbox{maximize} \quad & -\frac{1}{2} \lambda^{T} [\frac{\Sigma_{i}}{\alpha_{i}} + ...
34 views

39 views

### Solving the Lagrange dual problem

If we have a convex constraint problem $$\text{min}\quad f_0(x)\\ f_i(x)\le0\\ h_i(x)=0,$$ where $f_1,\ldots,f_n$ are convex and $h_1,\ldots,h_r$ are affine. Assuming Slater's conditions we know ...
50 views

### sparse norm for optimization problem

I want to solve an optimization problem in general form: $$\arg \min f(x) + \lambda *g(x)$$ and i want to choose / define a $g(x)$ in a way to have a sparse solution such that between two possible ...
50 views

### how to find the set of feasible optimal solutions?

Consider the following optimization problem; $\min_{M} ~\|a-M*b\|_2$ subject to $\|M\|_2<1$. where $M\in R^{5\times3}$ and a and b are constant vectors. There might be many optimal solutions ...
44 views

### Why is an affine set convex?

I wanted to know why do we say that an affine set is convex? From what I understood, if we take two points $x_1$ and $x_2$ $\in \mathbb{R}$, then, the affine set $A$ defined by these two points will ...
151 views

### Optimization problem involving, $L_2$, $L_1$ norm and constraints.

Can somebody suggest me how to solve the following optimization problem? \begin{equation*} F(\mathbf{w},\xi)= \begin{aligned} & \underset{\mathbf{w,\xi}}{\text{minimize}} & & \frac{1}{2}\|\...
75 views

34 views

### Uniqueness of saddle point solution to zero-sum game

Considering a two player zero-sum game, is a found saddle point solution unique? And if not, are there any conditions under which the saddle point solution is unique?
27 views

### Maximum overlap of two convex bodies

Let $\Delta_n=\big\{(x_1,x_2,\ldots,x_n): \sum_i x_i \leq 1, x_i \geq 0\big\}$ denote an n-dimensional simplex. I am trying to find an $y \in \mathbb{R}^n$ such that the $y-$shifted negative simplex ...
11 views

### Approximate matrix partition

This problem is similar to the Optimizing sums of log det problem that I had asked earlier, but it is not the same. I have matrix $H$, which has columns $h_1, \ldots, h_n$. I want to partition the ...