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 vectorH
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
andub
the lower and upper bounds ofx
(\(size(x) \times 1\))lbA
andubA
the lower and upper bounds ofA
(\(size(x) \times 1\))
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 vectorE
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
andubC
the lower and upper bounds ofA
(\(size(x) \times 1\))