# Tagged Questions

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

1answer
19 views

### What is the maximum of the following function?

Let $f(x,y) = \frac{xy^\alpha}{x+y},\alpha\in(0,\infty)$. How to compute $$\sup_{(x,y)\in[a,b]\times [0,c]}\frac{xy^\alpha}{x+y},$$ with $b>a>0$?
1answer
45 views

### Convex function (vector composition rule)

I'm looking at the Boyd & Vandenberghe slides on Convex Optimization. In slide 18, it applies the rules of vector composition on an example to say that it is convex. The example given is ...
1answer
27 views

### Does optimal solution from primal problem follow from optimal solution to dual?

In a linear programming context, does the primal optimal solution yield an explicit way to find the primal dual solution? I vaguely remember something like this from an optimization class but can't ...
1answer
52 views

### Are posynomial functions convex?

I know that you can transform a posynomial function into an exponential function, which is convex. Does this imply that all posynomial functions are convex?
0answers
11 views

### Find the dual problem to a quadratic program

Consider the quadratic program: minimize $x_1^2 + 2x_2^2 - x_1x_2 - x_1$ subject to $x_1 + 2x_2 \leq u_1, x_1 - 4x_2 \leq u_2, 5x_1 + 5x_2 \leq 1$ Could anyone explain to me how to find the dual ...
1answer
49 views

### How can L1-sparse representation be formulated as linear programming?

Problem Statement Show how the $L_1$-sparse reconstruction problem: $$\min_{x}{\left\lVert x\right\rVert}_1 \quad \text{subject to} \; y=Ax$$ can be reduced to a linear programming problem of form ...
0answers
49 views

### How to prove that unnormalized neg entropy is strongly convex with respect to 1-norm?

the unnormalized negative entropy of $\mathbf{x} \in \mathbb{R}^n_+$ is $$g(\mathbf{x}) = \sum_i (x_i \log(x_i) - x_i)$$ it is stated that $g(\mathbf{x})$ is strongly convex with respect to 1-norm, ...
0answers
14 views

### Explain the dual problem to D-optimal design problem

Given the following D-optimal design problem $$\text{minimize } \log \det (\sum_{i=1}^p x_i v_i v_i^T)^{-1}\\ \text{subject to } x \geq 0, {\bf{1}}^T x = 1$$ Find the dual problem. I don't ...
1answer
25 views

### Is my proof correct? Convex optimization

There's a theorem that says that if $C \subset \mathbb R^n$ is a convex set, then $x^* \in C$ is the closest point in $C$ to $y \notin C$ if and only if $(y-x^*)\cdot(x-x^*)\leq 0$ for all $x \in C$. ...
0answers
29 views

### What non-convex functions be written as the $\min$ of multiple convex functions?

I am working on an optimization framework that can be used to optimize objective functions that can be written as the $\min$ of several convex functions. I was thinking about the generality of this ...
0answers
27 views

### Minimize $\|\mathbf{x-y}\|^2$ subject to $x \in$ set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$

We are given the set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$ and a point $\mathbf{y} \in \mathbb{R}^n$. Our goal is to find point $\mathbf{\hat{x}}$ ...
1answer
50 views

### About a definition of “rank” of a matrix.

I am familiar with the definition of rank of a matrix as either (1) the maximal number of linearly independent rows or columns or (2) as the dimension of the image of the matrix. Another ...
1answer
14 views

### Log transformations of function domain and inequalities

If I know that for some function $f$, the following is true for $x, y \geq 0$: $f(\log (x^a y^b)) \leq f(\log x)^a f(\log y)^b$ Can I make the claim that $f(x^a y^b) \leq f(x)^a f(y)^b$ If I ...
1answer
47 views

### Probability that max cos(φ)x + sin(φ)y according to uniform distribution = (8,5)

max $x_2$ subject to $x_1 - 2x_2 \le 0$ $2x_1 - 3x_2 \le 2$ $x_1 - x_2 \le 3$ $-x_1 + 2x_2 \le 2$ $-2x_1 + x_2 \le 0$ Optimal solution: (8, 5) --> $x_2 = 5$ Now assume that the objective is ...
1answer
146 views

### Gradient of the dual function for a nonlinear program

I'm attempting to find a proof for a property from Floudas' Nonlinear and Mixed-Integer Optimization book. Consider a nonlinear optimization problem of the form \begin{align} \min_{{\bf x}}&\quad ...
1answer
19 views

### Tracking a vehicle moving with uniform velocity?

Suppose there are three cell towers at three positions $P_1$, $P_2$ and $P_3$. A vehicle is moving at uniform speed along a straight line. Three towers are pinging the vehicle at certain ...
1answer
25 views

### How to show the Hessian matrix of such functions are positive semi-definite?

Let $f:R\to R$, $g:R^n\to R$. Thus $f\circ g:R^n\to R$. Now suppose $f$ is non-decreasing and convex while $g$ is convex. In additon, $f,g$ are of $C^2$. I want to show that their composition is ...
2answers
21 views

### why Quasiconvex function is not concave?

A quasiconvex function is a function whose all sublevel set are convex. I am curious to know whether a quasiconvex function is a concave function.
0answers
27 views

### Explicit solution for minimization over unit box with total budget constraint

I am trying to solve question 4.8, part (e) from Convex Optimization by Boyd. The problem is to find an explicit solution for the minimization problem: Minimize $\textbf{c}^T \textbf{x}$ subject to ...
1answer
37 views

### Gradient of Least Squares function

I have trouble understand the gradient of equation 3.12 with respect to $W$. Tn is a scalar output variable, $\phi(x)$ and $W$ are $N \times 1$ dimensional. According to the book, the gradient ...
1answer
59 views

0answers
74 views

### KKT conditions (equations) for Generalized Assignment Problem or Binary integer programming problem

I have this formulated Generalized Assignment Problem (GAP) or it can also be considered as Binary integer programming problem. Solving this problem can be achieved through Branch and Bound Technique. ...