Overview

The problem is written as a quadratic problem :

\[ \begin{align}\begin{aligned}\min_{x} \frac{1}{2}x^tHx + x^tg\\\text{subject to}\\lb \leq x \leq ub\\lb_A \leq Ax \leq ub_A\end{aligned}\end{align} \]
  • x the optimization vector
  • H the hessian matrix (\(size(x) \times size(x)\))
  • g the gradient vector (\(size(x) \times 1\))
  • A the constraint matrix (\(size(x) \times size(x)\))
  • lb and ub the lower and upper bounds of x (\(size(x) \times 1\))
  • lbA and ubA the lower and upper bounds of A (\(size(x) \times 1\))

Tasks are written as weighted euclidian distance function :

\[w_{task} \lVert \mathbf{E}x + \mathbf{f} \rVert_{W_{norm}}^2\]
  • x the optimization vector, or part of the optimization vector
  • E the linear matrix of the affine function (\(size(x) \times size(x)\))
  • f the origin vector (\(size(x) \times 1\))
  • w task the weight of the tasks in the overall quadratic cost (scalar \([0:1]\))
  • W norm the weight of the euclidean norm (\(size(x) \times size(x)\))

Given n_t tasks, the overall cost function is such that:

\[\frac{1}{2}x^tHx + x^tg = \frac{1}{2} \sum_{i=1}^{n_t} w_{task,i} \lVert \mathbf{E}_ix + \mathbf{f}_i \rVert_{W_{norm,i}}^2\]

Constraints are written as double bounded linear function :

\[lb_C \leq Cx \leq ub_C\]
  • C the constraint matrix (\(size(x) \times size(x)\))
  • lbC and ubC the lower and upper bounds of A (\(size(x) \times 1\))