Tagged Questions

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

37 views

Geometric Median Problem with a twist

Given two vectors $x, y$ in $\mathbb{R}^n$ and scalar $\alpha$, what is the value of $\alpha$ that minimizes $||\alpha x - y ||_1$? Give an algorithm to find the minimum. I've tried couple of ...
36 views

Subgradient of a composition w/ affine $f$

Let $f:\mathbb{R}^n \to \mathbb{R} \cup \{ \infty \}$ be convex, w/ subgradient at x in its domain $\partial f(x):=\{ d:f(y)\geq f(x)+d^T (y-x),\forall y\in \mathbb{R}^n \}$. Let $h(x'):=f(Ax'+b)$, ...
8 views

Lipschitz Continuity of the Solution Set of LP with a single additional Convex Constraint

Consider the following Linear Program with interval bounds on the decision variables: \begin{aligned} S(\mathbf b) = \ & \arg \min_{\mathbf x \in \mathbb R^n} && \mathbf ...
40 views

I was thinking to solve LASSO via vanilla subgradient methods. But,I have read people suggesting to use Proximal GD. Can somebody highlight why proximal GD instead of vanilla subgradient methods be ...
16 views

Smart penalty term for constant sparsity of matrix columns

I am looking, in an convex optimization problem, for a smart way to write a penalty term $Reg(A)$ (where $A$ is the coefficients matrix of the data $X$ w.r.t. the learned dictionary $D$, $A=D^{T}X$), ...
17 views

Quasiconvexity analog for function with an integer domain.

Suppose I have a function that is not quasiconvex, as in the graph below, but would be quasiconvex if we cared only about integer points. That is, $f:X \subset \mathbb{Z}\rightarrow \mathbb{R}$ ...
17 views

Karush-Kuhn-Tucker NLP

Consider the nonlinear program Minimize: \begin{align}f(x,y) = \frac{1}{2}x^2 - 10xy + 10y^2\end{align} Subject to: \begin{align}2x +y^2 &\le 5 \implies g_1(x,y)=2x + y^2 -5 \le0 \\ ...
97 views

46 views

Is always a convex function two times differentiable

Assume that f is twice differentiable, that is, its Hessian or second derivative $\nabla^2f$ exists at each point in dom$f$, which is open. Then f is convex if and only if dom$f$ is convex and its ...
18 views

is the Superellipse function convex or not?

I'm trying to solve an optimization problem and i need to know if the following constraint of Superellipse is convex or not $\left|\frac{x-x_o}{a}\right|^n + \left|\frac{y-y_o}{b} \right|^n \geq 1$ ...
30 views

36 views

Optimization function convex or not

I need to comment whether my optimization function is convex or non-convex. My optimization function is in the form of $(y-y_{cap})^2$. y is know. $y_{cap}$ comes out of a MATLAB pfile. So, ...
74 views

22 views

The slope and intercept of piecewise linear functions

If we have a function $f$ in the form: $$f(x)=\sum_{j}^{N}c_j \min_i(a_{ij}x+b_{ij})$$ Question: All of $a_{ij}$ and $a_{ij}$ are known. How can I find the slopes of $f$ as optimal as possible. ...
12 views

73 views

Linear Transformation of Closed Convex Cone

Given a closed convex cone $C \subset \mathbb{R}^n$ and a matrix $M \in \mathbb{R}^{m\times n}$, is the set $S = \{Mx\mid x \in C\}$ also a closed convex cone? Firstly, $S$ must be a convex cone. ...