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Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes

13 votes
3 answers
828 views

Famous theorems that are special cases of linear programming (or convex) duality

The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any othe …
7 votes
0 answers
233 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's a …
Tom Solberg's user avatar
  • 4,049
6 votes
2 answers
315 views

Convex optimization with full subdifferential information

Can anyone direct me to any algorithms or theorems that describe the difficulty of solving a non-smooth convex optimization problem for the special case where the full subdifferential is available? A …
Tom Solberg's user avatar
  • 4,049
5 votes
0 answers
96 views

Non-convex optimization problem involving minimum spanning trees

Suppose I am given $n$ points $p_1,\dots,p_n\in \mathbb{R}^2$, as well as two positive coefficients $a_1$ and $a_2$, and I am trying to select $n$ points $x_1,\dots,x_n$ to solve the following optimiz …
Tom Solberg's user avatar
  • 4,049
4 votes
2 answers
269 views

Is this function always bounded below?

Is there any global constant $C$ such that $$C<\frac{\sum_{i=1}^{n}x_{i}\log x_{i}-x_{i}+(1-x_{i})\log(1-x_{i})}{\sum_{i=1}^{n}x_{i}}+\log(\sum_{i=1}^{n}x_{i})-\log(\sum_{i=1}^{n}x_{i}^{2})$$ for all …
Tom Solberg's user avatar
  • 4,049
2 votes
1 answer
123 views

Functions that are easy to compare to a norm

Let $X$ be a subset of $\mathbb{R}^d$, let $\|\cdot \|_p$ be a norm with $1\leq p\leq\infty$, and let $f:\mathbb{R}^d\to\mathbb{R}$ be a function. I'm trying to find examples of $X$, $p$, and $f$ for …
Tom Solberg's user avatar
  • 4,049
1 vote
0 answers
59 views

Minimizing square roots with the consecutive ones property

Let $A=[a_{ik}]$ be a matrix with the consecutive ones property in each column, i.e. each column consists of a single consecutive block of $1$'s (with zeros everywhere else). Is there anything at all …
Tom Solberg's user avatar
  • 4,049