Wireless Sensor and Actuator Networks for Lighting Energy ...

developed by Dantzig [87]. The inequality constraints form a polyhedron, inside which
is the feasible region, and it has been proven that the optimal solution always occurs at
the vertex of the polyhedron. The main idea of the simplex algorithm is to find an
admissible solution at a vertex of the polyhedron, and then keep on moving along the
edges of the polyhedron to the adjacent vertex with lower objective value until no
adjacent vertex with lower objective value can be found.
It is, however, possible for the simplex algorithm to be caught in an infinite loop
and render the worst-case solution time to be infinite. The cycling problem is very rare
in practice, and variants of simplex algorithm that do not cycle exist [88]. Although
simplex algorithm is very efficient and usually converges to the optimal solution in
polynomial time, the worst-case complexity can be exponential if it does not cycle,
which inspired the development of the family of interior point methods. As opposed to
the simplex method, which travels along the edge of the polyhedron, the concept of the
interior point methods is to move through the interior of the feasible region; this
guarantees to converge to the optimal solution in polynomial time.
Linear programming is the core technique utilized in the optimal lighting
actuation system presented in Chapter 6. Given the setup of the lighting control
problem, it is very straightforward to formulate energy usage as the objective function
and users’ lighting preferences as the constraints to the linear programming problem.
Furthermore, by treating the light actuator outputs as the variables, the problem is
tractable and practical for real-time implementation since the number of variables is
limited.
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