# Bivariate Time Series Analysis of Simulated Annealing Data

### B.B. Sorkin

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/ERL M90/6

1990

Simulated annealing (SA) is a recent technique for finding good solutions to a wide variety of combinatorial optimization problems. Given is a graph with an energy E assigned to each node. In simulated annealing parlance, the nodes are called `states', the arcs represent `moves' from one state to a neighboring state, and the energy is sometimes called `cost'. The simulated algorithm then proceeds as follows: state = random initial state repeat (until done) { T = new temperature repeat (until inner-loop criterion) { newstate = random neighbor(state) Delta E = E(newstate) - E(state) If Delta E < or = 0, state = newstate. Else state = newstate with probability e to the power of (- delta E/T) } } The temperature T is a parameter of the algorithm. It is always decreased gradually as the annealing proceeds, but the optimal control of T is not understood. It can easily be shown that this process is equivalent to a random walk on a graph containing self-loops, with random selection of an edge out of a node in proportion to the weight of that edge, and with the edge weights determined by the temperature T. A typical application is a placement problem, say that of placing 100 circuits on a 10 x 10 grid. Here a state is a permutation of the numbers from 1 to 100, representing a placement. A move could consist of interchanging any two circuits. Thus each state would have C100 = 2 4950 neighbors. We also define the distance between two states as the length of a shortest path connecting them. While annealing works well on a wide variety of practical problems, it cannot work well on arbitrary graphs, nor on any graph with an arbitrary energy function. The goal of our research has been to characterize the energy landscape itself (the graph and energy function) and the behavior of the annealing algorithm. Because the energy landscape itself is terrifically complex, we have focused on the energy trajectory, i.e. the sequence of energies following each move accepted.

BibTeX citation:

@techreport{Sorkin:M90/6, Author = {Sorkin, B.B.}, Title = {Bivariate Time Series Analysis of Simulated Annealing Data}, Institution = {EECS Department, University of California, Berkeley}, Year = {1990}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/1392.html}, Number = {UCB/ERL M90/6}, Abstract = {Simulated annealing (SA) is a recent technique for finding good solutions to a wide variety of combinatorial optimization problems. Given is a graph with an energy E assigned to each node. In simulated annealing parlance, the nodes are called `states', the arcs represent `moves' from one state to a neighboring state, and the energy is sometimes called `cost'. The simulated algorithm then proceeds as follows: state = random initial state repeat (until done) { T = new temperature repeat (until inner-loop criterion) { newstate = random neighbor(state) Delta E = E(newstate) - E(state) If Delta E < or = 0, state = newstate. Else state = newstate with probability e to the power of (- delta E/T) } } The temperature T is a parameter of the algorithm. It is always decreased gradually as the annealing proceeds, but the optimal control of T is not understood. It can easily be shown that this process is equivalent to a random walk on a graph containing self-loops, with random selection of an edge out of a node in proportion to the weight of that edge, and with the edge weights determined by the temperature T. A typical application is a placement problem, say that of placing 100 circuits on a 10 x 10 grid. Here a state is a permutation of the numbers from 1 to 100, representing a placement. A move could consist of interchanging any two circuits. Thus each state would have C100 = 2 4950 neighbors. We also define the distance between two states as the length of a shortest path connecting them. While annealing works well on a wide variety of practical problems, it cannot work well on arbitrary graphs, nor on any graph with an arbitrary energy function. The goal of our research has been to characterize the energy landscape itself (the graph and energy function) and the behavior of the annealing algorithm. Because the energy landscape itself is terrifically complex, we have focused on the energy trajectory, i.e. the sequence of energies following each move accepted.} }

EndNote citation:

%0 Report %A Sorkin, B.B. %T Bivariate Time Series Analysis of Simulated Annealing Data %I EECS Department, University of California, Berkeley %D 1990 %@ UCB/ERL M90/6 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/1392.html %F Sorkin:M90/6