0
votes
1answer
17 views

Optimality conditions in convex programming

I'm reading about Zero-order conditions in Nonlinear Programming and the following confuses me (my questions are below the theory): Consider the set $\Gamma \subset E^{n+1} = \{(r,\textbf{x}): ...
0
votes
2answers
53 views

Regularization vs. Inequality Constraint

For what values of a regularization parameter $\alpha$, there is an equivalent inequality constraint in convex optimization? In particular, in the convex optimization problems below $$ \text{ Problem ...
1
vote
1answer
60 views

Showing a function is concave

Given $F(\underline{x}) = Ax_1 + Bx_2 + \ln(a^2-(x_1^2+x^2_2))$ on $S=\{\underline{x}\in\mathbb{R}\mid x_1^2+x_2^2<a^2\}$ with $A,B,a\in\mathbb{R}$, show that $F$ is concave on $S$. Since we have ...
1
vote
1answer
23 views

Impact of constraints on convex optimization

Let me start by saying I know almost nothing about optimization so please bear with me. Basically, I am wondering whether it is possible to solve a problem with two constraints by solving the problem ...
1
vote
2answers
43 views

Minimizing a convex cost function

I'm reviewing basic techniques in optimization and I'm stuck on the following. We aim to minimize the cost function $$f(x_1,x_2) = \frac{1}{2n} \sum_{k=1}^n \left(\cos\left(\frac{\pi k}{n}\right) x_1 ...
0
votes
0answers
79 views

Calculation of the set for the polar tangent cone?

I have the following theorem in my book. Assume that $\tilde{x}$ is a local minimum from a minimization problem and that f(.) is differentible at $\tilde{x}$ Let $T_X(\tilde{x})$ be the tangent cone ...
0
votes
0answers
19 views

Implementing a projection with KL-divergence

I want to implement the following and I am looking for an easy/fast way to implement it(the programming language does not matter). Assume that $p(\mathbf{x})$ is a proper probability distribution and ...
0
votes
1answer
59 views

(pseudo-/quasi-)convexitiy of ratio between quadratic and affine function

Let $X\subseteq\mathbb{R}^n$. I have the following function $f:X\rightarrow\mathbb{R}$: $$ f({\bf x})= \frac{c_1 + \sum_{i=1}^n a_ix_i +\sum_{i=1}^n b_ix_i^2}{c_2+\sum_{i=1}^n d_i x_i}\enspace.$$ All ...
7
votes
1answer
163 views

$\max\min$ problem

Suppose there is a real valued function $f(x,y)$ where $x,y$ are vector variables $x\in\mathbb{R}^n$ $y\in\mathbb{R}^m$. Moreover $f$ is strictly/concave in $x$ and strictly/convex in $y$. $f$ is also ...
0
votes
1answer
560 views

Optimization of a function of two variables.

Suppose we have to minimize a function $f(\mathbf x,\mathbf y)$ where $\mathbf x$, $\mathbf y$ are vectors in Euclidean space. The function is convex in $\mathbf y$ when $\mathbf x$ is kept constant ...
3
votes
1answer
103 views

minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
0
votes
1answer
54 views

Distributed Robust Optimization

Consider the following constrained optimization problem $\mathcal{P}$. $$ \min_{x \in X \subseteq \mathbb{R}^n} f(x) \ \text{sub. to: } g(x,y) \leq 0 \ \forall y \in Y \subseteq \mathbb{R}^m $$ ...
1
vote
1answer
67 views

Distance between a point to a $2d$ ellipse in $3d$ ambient space

Suppose we are working in the 3D Euclidean space. We are given an arbitrary point $p$ and a 2d ellipse: $$E=\{x:x^TQx\leq1,x^Tq=0\},$$ where $Q$ is a positive definite matrix and $q$ is an ...
0
votes
1answer
261 views

Convex Functions: Property Proof

Let $f\colon S\to \mathbb R$ be a $C^1$ function on a convex domain $S \subseteq \mathbb R^n$. Show that if $f$ is convex then $(\nabla f(x) - \nabla f(y)) \cdot (x-y) \ge 0$ for all $x,y \in S$. ...
2
votes
1answer
547 views

Prove every local minimum is a global minimum

Let $Q\in\mathbb{R^{dxd}}$ and $A\in\mathbb{R^{d'xd}}$ be two matrixes and $b\in\mathbb{R^d}$, $c\in\mathbb{R^{d'}}$. Suppose $d'\lt d $. For $x\in\mathbb{R^d}$. Minimize $$f(x)= ...
5
votes
1answer
1k views

The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
2
votes
2answers
82 views

A Quadratic Problem (which looks very simple)

This arises as a part of my work. \begin{align} \min_{x^{H}x=1}~&x^{H}A_1x \\ subject~to~&x^{H}A_2x=0 \end{align} $A_1$ and $A_2$ are $N\times N$ hermitian matrices and $x$ is a unit norm ...
1
vote
0answers
94 views

minimum of the function over symmetric body

Let $X$ be a normed space. Let $K$ be centrally symmetric convex body with $M$ known vertices and with $Vol(K)=V$. We subdivide body $K$ be $M$ equal pieces, $P_i, i=1, \ldots, M$, such that each ...
0
votes
1answer
209 views

How to optimize entropy under linear constraints?

My problem is quite cumbersome. In general, it can be modelled as a non-linear programming problem, with linear constraints and non-linear objective function. The objective function is conditional ...
6
votes
7answers
2k views

Operations research book to start with

for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear ...