Branch and Bound

Illustration of a linear-programming (LP)-based branch and bound algorithm. Not knowing how to solve a mixed-integer programming (MIP) problem directly, it first removes all integrality constraints. This results in a solvable LP called the linear-programming relaxation of the original MIP. The algorithm then picks some variable x restricted to be integer, but whose value in the LP relaxation is fractional. Suppose its value in the LP relaxation is x = 0.7. It then excludes this value by imposing the restrictions x ≤ 0 and x ≥ 1, thereby creating two new MIPs. By applying this recursively step and exploring each resulting bifurcation, the globally optimal solution satisfying all constraints can be found.

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% Illustration of a linear-programming (LP)-based branch and bound algorithm. Not knowing how to solve a mixed-integer programming (MIP) problem directly, it first removes all integrality constraints. This results in a solvable LP called the linear-programming relaxation of the original MIP. The algorithm then picks some variable x restricted to be integer, but whose value in the LP relaxation is fractional. Suppose its value in the LP relaxation is x = 0.7. It then excludes this value by imposing the restrictions x ≤ 0 and x ≥ 1, thereby creating two new MIPs. By applying this recursively step and exploring each resulting bifurcation, the globally optimal solution satisfying all constraints can be found.

\documentclass{standalone}

\usepackage{forest}

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  declare toks={left branch suffix}{},
  declare toks={right branch suffix}{},
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\begin{document}
\begin{forest}
  branch and bound,
  where level=1{
  set branch labels={x_1\leq}{}{x_1\geq}{},
  }{if level=2{set branch labels={x_2\leq}{}{x_2\geq}{}}{set branch labels={x_3\leq}{}{x_3\geq}{}}}
  [P_0[P_1:0][P_2:1[P_3:0[P_5:0][P_6:1]][P_4:1]]]
\end{forest}
\end{document}