0
votes
1answer
46 views

Lagrangean Relaxation of quadratic assignment problem to yield $n$ knapsack problems?

Consider the assignment problem: $$ Z = \min \sum_i\sum_j\sum_k c_{jk}\cdot x_{ij}\cdot x_{ik} $$ s.t. $$ \sum_i x_{ij} = 1 \quad\forall j $$ $$ a \leq \sum_j x_{ij} \leq b \quad\forall i $$ $$ ...
1
vote
0answers
62 views

Integral Farkas Lemma

The context of this question is commutative algebra, however the question itself is more related to convex geometry. All necessary information is given. In the proof of Lemma 3.1.1 in the book ...
2
votes
0answers
56 views

Quadratic Integer Programming

Would anyone mind helping me solve this problem $$ \min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda $$ where $x$ is a vector whose entries are positive ...
1
vote
1answer
65 views

Quadratic integer Programming

Would anyone mind helping me solve this problem $$ \min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda $$ where $x$ is a vector whose entries are positive ...
1
vote
1answer
146 views

Characterization of Subset Sum via Linear Programming

I have a sample subset sum problem. Given numbers $x_1, x_2... x_N$ and a target value to sum to $x_S$ Minimize $x_S - x_1y_1 - x_2y_2 - x_3y_3 ... x_Ny_N$ such that 0 <= $y_1$ <= 1 0 <= ...
2
votes
1answer
222 views

Maximizing the number of non-crossing lines between a number of points

Suppose I have a number of points in 2-dimensional space. I want to draw as many lines between the points as possible such that no two lines cross. Hoping for a polynomial time algorithm, I ...
1
vote
2answers
447 views

LP relaxation for ILP\IP (integer linear programming)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...