IWCMC14/argmin.tex

   1 The approach detailed previously requires to compute
   2 the variable that minimizes four convex functions
   3 on constraint domains, one
   4 for each primal variable $q_i$, $P_{sh}$, $R_h$, and $x_{hl}$,
   5 Among state of the art for this optimization step, there is
   6 the  L-BFGS-B algorithm~\cite{byrd1995limited},
   7 the truncated Newton algorithm,
   8 the Constrained Optimization BY Linear Approximation (COBYLA) method~\cite{ANU:1770520}.
   9 However, all these methods suffer from being iterative approaches
  10 each iteration including  many steps of computation to obtain an approximation
  11 of the minimal value.
  12 This approach is dramatic since the objective is to
  13 reduce all the computation steps to increase the network lifetime.
  14
  15 A closer look to each function that has to be minimized shows that it is
  16 differentiable and the minimal value can be computed in only one step.
  17 The table~\ref{table:min} presents these minimal value for each primal
  18 variable.
  19 Thanks to this formal calculus, computing the
  20 new iterate of each primal variable  only requires
  21 one computation step.
  22
  23 \begin{table*}[t]
  24 $$
  25 \begin{array}{|l|l|l|}
  26 \hline
  27 q_i^{(k+1)} &
  28  \arg\min_{q>0}
  29 \left(
  30 q^2 + q.
  31 \left(
  32 \sum_{l \in L } a_{il}w_l^{(k)}-
  33 \lambda_i^{(k)}B_i
  34 \right)
  35 \right)
  36  &
  37 \max \left\{\epsilon,\dfrac{\sum_{l \in L } a_{il}w_l^{(k)}-
  38 \lambda_i^{(k)}B_i}{2}\right\} \\
  39 \hline
  40 P_{sh}^{(k+1)}&
  41 \arg \min_{p > 0}
  42 \left(
  43 v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p
  44 + \delta_p p^{8/3}
  45 \right)
  46 &
  47 \max \left\{\epsilon,
  48 \left(
  49 \dfrac{
  50 -3\lambda_h^{(k)} + \sqrt{(3\lambda_h^{(k)})^2 + 64\delta_p/\gamma.\ln(\sigma^2/D_h)}
  51 }{16\delta_p}
  52 \right)^{\frac{3}{5}}
  53 \right\} \\
  54 \hline
  55 R_h^{(k+1)}
  56 &
  57 \arg \min_{r \geq 0 }
  58 \left(
  59 \delta_r r^2
  60 -v_h^{(k)}.r - \sum_{i \in N} u_{hi}^{(k)} \eta_{hi}
  61 \right) &
  62 \max\left\{0,\dfrac{v_h^{(k)}}{2\delta_r}\right\}
  63 \\
  64 \hline
  65 x_{hl}^{(k+1)} &
  66 \begin{array}{l}
  67 \arg \min_{x \geq 0}
  68 \left(
  69 \delta_x.x^2  \right.\\
  70 \qquad \qquad + x.
  71 \sum_{i \in N} \left(
  72 \lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
  73 c^r. a_{il}^{-} ) \right.\\
  74 \qquad \qquad\qquad \qquad +
  75 \left.\left. u_{hi}^{(k)} a_{il}
  76 \right)
  77 \right)
  78 \end{array}
  79 &
  80 \max\left\{0,\dfrac{-\sum_{i \in N} \left(
  81 \lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
  82 c^r. a_{il}^{-} ) + u_{hi}^{(k)} a_{il}
  83 \right)}{2\delta_x}\right\}
  84 \\
  85 \hline
  86 \end{array}
  87 $$
  88 \caption{Primal Variables: Argmin and Direct Calculus}\label{table:min}
  89 \end{table*}
  90
  91
  92 This improvement