Comment by RobPratt on LP with n free variables can be replaced by n + 1...
@Upstart Even in the formulation with $2n$ nonnegative variables there is no "complementarity" requirement that at most one of the two nonnegative variables for $x_i$ is positive.
View ArticleComment by RobPratt on Partitioning nodes of a graph with distinct labels
Search for graph partitioning and for the variant search for multiway cut.
View ArticleComment by RobPratt on Number of words with $3n$ letters where no $3$...
OK, but then $n=2$ yields $19$, which is too large.
View ArticleComment by RobPratt on Number of words with $3n$ letters where no $3$...
@pyridoxal_trigeminus That answer is actually correct, but you generalized it incorrectly. The $2n-k$ in that answer arises from $2n-(2-1)k$, which corresponds to $3n-(3-1)k$ in your problem.
View ArticleComment by RobPratt on How to find expression for the reduced cost?
Yes, because of duality, you can call either problem the primal and the other the dual. But I changed my answer to say "primal variables in parentheses" to match your question.
View ArticleComment by RobPratt on How to calculate this sum $\sum_{n=1}^{\infty}...
@mick Watch what happens when you click the "More digits" button.
View ArticleComment by RobPratt on Find the first derivative of $F(x)=\int...
en.wikipedia.org/wiki/…
View ArticleComment by RobPratt on Maximizing sum of pairwise scores in a set selection...
stackoverflow.com/questions/17168049/…
View ArticleComment by RobPratt on Application of Farkas Lemma
Which variant of Farkas Lemma are you allowed to use?
View ArticleComment by RobPratt on You have $12$ female and $8$ male employees. How many...
@JohnDouma The $128$ you saw was intended to be $12\cdot 8$ but was typed as 12*8*. I reformatted it just now.
View ArticleComment by RobPratt on Finding the extreme points of a polyhedron
Suppose both are active: $x_{i-1}=x_i=2x_{i-1}$. Now solve for $x_{i-1}$ to obtain a contradiction.
View ArticleAnswer by RobPratt for How to symbolically (not in words) express that an...
$$m \angle Y = 0.6 = 0.6 \text{ rad} \approx 36.87 \text{ deg} = 36.87^\circ$$See also https://en.wikipedia.org/wiki/Radian#Unit_symbol
View ArticleAnswer by RobPratt for Minimising inconclusive range for binary...
You can linearize the problem and use a mixed integer linear programming solver, which will find a globally optimal solution without relying on an initial solution.Besides the decision variables $a$,...
View ArticleAnswer by RobPratt for Minimum volume sphere bounding all points as a linear...
You can solve the problem directly via second-order cone programming.Other approaches are described here: https://en.wikipedia.org/wiki/Bounding_sphere
View ArticleAnswer by RobPratt for How to efficiently solve many linear programs with...
Such a change preserves dual feasibility, so you might consider applying the dual simplex method, with a warm start from the previous optimal basis.
View ArticleAnswer by RobPratt for Linear program, find vectors that minimize equally...
Your linear programming (LP) problem is to minimize $\sum_i (x_i+y_i)$ subject to\begin{align}x_i + y_j &\ge c_{ij} &&\text{for all $i$ and $j$} \\x_i &\ge 0 &&\text{for all...
View ArticleAnswer by RobPratt for Demonstrating a Binomial Identity? #2 (The exclusion)
The second sum is\begin{align}\sum_{m=1}^{j-k}{j-k-1 \choose m-1}{n \choose m}m&= \sum_{m=1}^{j-k}{j-k-1 \choose m-1}\frac{n}{m}{n-1 \choose m-1}m \\&= n\sum_{m=1}^{j-k}{j-k-1 \choose m-1}{n-1...
View ArticleAnswer by RobPratt for Maximise $\sum_{x,y\in S} GCD(x,y)$ where $S =...
You can solve this as a maximum weight matching problem in a complete graph with node set $\{1,\dots,100\}$ and edge weight $\gcd(i,j)$. The maximum turns out to be $1187$, attained for example...
View ArticleAnswer by RobPratt for Generalization of independent set to distance at least 3
This generalization of independent set is called distance-$d$ independent set or $d$-scattered set in the literature.
View ArticleAnswer by RobPratt for Tournament organization that avoids repetitions
Six rounds are possible:{1,6,12,23} {5,11,18,24} {4,9,15,22} {2,10,13,19} {7,14,17,21} {3,8,16,20}{1,16,19,22} {2,5,12,14} {3,7,11,13} {8,9,18,23} {6,15,17,24} {4,10,20,21}{1,5,9,13} {2,11,17,23}...
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