Optimization Techniques
Sources used
Gang Quan
Van Laarhoven, Aarts
Scheduling using
Simulated Annealing
Reference:
Devadas, S.; Newton, A.R.
Algorithms for hardware allocation in data
path synthesis.
IEEE Transactions on Computer-Aided
Design of Integrated Circuits and Systems,
July 1989, Vol.8, (no.7):768-81.
Iterative Improvement 1
•
General method to solve combinatorial optimization
problems
Principles:
1. Start with initial configuration
2. Repeatedly search neighborhood and select a neighbor as
candidate
3. Evaluate some cost function (or fitness function) and
accept candidate if "better"; if not, select another
neighbor
4. Stop if quality is sufficiently high, if no improvement
can be found or after some fixed time
Iterative Improvement 2
Needed are:
1. A method to generate initial configuration
2. A transition or generation function to find
a neighbor as next candidate
3. A cost function
4. An Evaluation Criterion
5. A Stop Criterion
Iterative Improvement 3
Simple Iterative Improvement or Hill Climbing:
• Candidate is always and only accepted if cost is lower (or
fitness is higher) than current configuration
• Stop when no neighbor with lower cost (higher fitness) can
be found
Disadvantages:
• Local optimum as best result
• Local optimum depends on initial configuration
• Generally, no upper bound can be established on the number
of iterations
Hill climbing
Simulated Annealing
Cost function
Local Search
?
Solution space
How to cope with
disadvantages
1.
Repeat algorithm many times with different initial
configurations
2.
Use information gathered in previous runs
3.
Use a more complex Generation Function to jump out of local
optimum
4.
Use a more complex Evaluation Criterion that accepts
sometimes (randomly) also solutions away from the (local)
optimum
Annealing
Use a more complex Evaluation Function:
• Do sometimes accept candidates with
higher cost to escape from local optimum
• Adapt the parameters of this Evaluation
Function during execution
• Based upon the analogy with the simulation
of the annealing of solids
Simulated
Annealing
Other Names
•
•
•
•
•
Monte Carlo Annealing
Statistical Cooling
Probabilistic Hill Climbing
Stochastic Relaxation
Probabilistic Exchange Algorithm
Optimization Techniques
•
•
•
•
•
•
Mathematical Programming
Network Analysis
Branch & Bound
Genetic Algorithm
Simulated Annealing Algorithm
Tabu Search
Simulated Annealing
• What
– Exploits an analogy between the
annealing process and the search for
the optimum in a more general
system.
Annealing Process
• Annealing Process
– Raising the temperature up to a very high
level (melting temperature, for example), the
atoms have a higher energy state and a high
possibility to re-arrange the crystalline
structure.
– Cooling down slowly, the atoms have a lower
and lower energy state and a smaller and smaller
possibility to re-arrange the crystalline structure.
Statistical Mechanics
Combinatorial Optimization
State {r:} (configuration -- a set of atomic
position )
weight e-E({r:])/K BT
distribution
-- Boltzmann
E({r:]): energy of configuration
KB: Boltzmann constant
T: temperature
Low temperature limit ??
Analogy
Physical System
Optimization Problem
State (configuration)
Solution
Energy
Cost function
Ground State
Optimal solution
Rapid Quenching
Iteration improvement
Careful Annealing
Simulated annealing
Simulated Annealing
• Analogy
–
–
–
–
Metal  Problem
Energy State  Cost Function
Temperature  Control Parameter
A completely ordered crystalline structure
 the optimal solution for the problem
Global optimal solution can be achieved as long
as the cooling process is slow enough.
Other issues related to simulated
annealing
1. Global optimal solution is possible, but near
optimal is practical
2. Parameter Tuning
1.
Aarts, E. and Korst, J. (1989). Simulated Annealing and
Boltzmann Machines. John Wiley & Sons.
3. Not easy for parallel implementation, but was
implemented.
4. Random generator quality is important
Analogy
• Slowly cool down a heated solid, so that all particles arrange
in the ground energy state
• At each temperature wait until the solid reaches its thermal
equilibrium
• Probability of being in a state with energy E :
Pr { E = E } = 1 / Z(T) . exp (-E / kB.T)
E
T
kB
Z(T)
Energy
Temperature
Boltzmann constant
Normalization factor (temperature dependant)
Simulation of cooling
(Metropolis 1953)
•
•
•
•
•
At a fixed temperature T :
Perturb (randomly) the current state to a new state
E is the difference in energy between current and new state
If E < 0 (new state is lower), accept new state as current state
If E  0 , accept new state with probability
Pr (accepted) = exp (- E / kB.T)
• Eventually the systems evolves into thermal equilibrium at
temperature T ; then the formula mentioned before holds
• When equilibrium is reached, temperature T can be lowered
and the process can be repeated
Simulated Annealing
• Same algorithm can be used for combinatorial optimization
problems:
• Energy E corresponds to the Cost function C
• Temperature T corresponds to control parameter c
Pr { configuration = i } = 1/Q(c) . exp (-C(i) / c)
C
c
Q(c)
Cost
Control parameter
Normalization factor (not important)
Metropolis Loop
• Metropolis Loop is the essential characteristic of
simulated annealing
• Determining how to:
– randomly explore new solution,
– reject or accept the new solution
at a constant temperature T.
• Finished until equilibrium is achieved.
Metropolis Criterion
• Let :
– X be the current solution and X’ be the new solution
– C(x) be the energy state (cost) of x
– C(x’) be the energy state of x’
• Probability Paccept = exp [(C(x)-C(x’))/ T]
• Let N = Random(0,1)
• Unconditional accepted if
– C(x’) < C(x), the new solution is better
• Probably accepted if
– C(x’) >= C(x), the new solution is worse .
– Accepted only when N < Paccept
Simulated Annealing Algorithm
Initialize:
– initial solution x ,
– highest temperature Th,
– and coolest temperature Tl
T= Th
When the temperature is higher than Tl
While not in equilibrium
Search for the new solution X’
Accept or reject X’ according to Metropolis Criterion
End
Decrease the temperature T
End
Components of Simulated
Annealing
• Definition of solution
• Search mechanism, i.e. the definition of a
neighborhood
• Cost-function
Control Parameters
1. How to define equilibrium?
2. How to calculate new temperature for next step?
1.
Definition of equilibrium
1.
2.
2.
Definition is reached when we cannot yield any significant
improvement after certain number of loops
A constant number of loops is assumed to reach the equilibrium
Annealing schedule (i.e. How to reduce the temperature)
1.
2.
A constant value is subtracted to get new temperature, T’ = T - Td
A constant scale factor is used to get new temperature, T’= T * Rd
•
A scale factor usually can achieve better performance
Control Parameters: Temperature
•
Temperature determination:
–
–
Artificial, without physical significant
Initial temperature
1.
–
Final temperature
1.
2.
•
Selected so high that leads to 80-90% acceptance rate
Final temperature is a constant value, i.e., based on the total
number of solutions searched. No improvement during the entire
Metropolis loop
Final temperature when acceptance rate is falling below a given
(small) value
Problem specific and may need to be tuned
Example of Simulated
Annealing
• Traveling Salesman Problem (TSP)
– Given 6 cities and the traveling cost between
any two cities
– A salesman need to start from city 1 and travel
all other cities then back to city 1
– Minimize the total traveling cost
Example: SA for traveling
salesman
• Solution representation
– An integer list, i.e., (1,4,2,3,6,5)
• Search mechanism
– Swap any two integers (except for the first one)
• (1,4,2,3,6,5)  (1,4,3,2,6,5)
• Cost function
Example: SA for traveling salesman
• Temperature
1.
Initial temperature determination
1. Initial temperature is set at such value that there is around 80%
acceptation rate for “bad move”
2. Determine acceptable value for (Cnew – Cold)
2.
Final temperature determination
•
•
•
Stop criteria
Solution space coverage rate
Annealing schedule (i.e. How to reduce the
temperature)
–
A constant value is subtracted to get new temperature, T’
= T – Td
–
For instance new value is 90% of previous value.
•
Depending on solution space coverage rate
Homogeneous Algorithm of
Simulated Annealing
initialize;
REPEAT
REPEAT
perturb ( config.i  config.j, Cij);
IF Cij < 0 THEN accept
ELSE IF exp(-Cij/c) > random[0,1) THEN accept;
IF accept THEN update(config.j);
UNTIL equilibrium is approached sufficient closely;
c := next_lower(c);
UNTIL system is frozen or stop criterion is reached
In homogeneous algorithm the value of c is kept constant in the
inner loop and is only decreased in the outer loop
Inhomogeneous Algorithm
•
Previous algorithm is the homogeneous variant:
c is kept constant in the inner loop and is only
decreased in the outer loop
•
Alternative is the inhomogeneous variant:
1. There is only one loop;
2. c is decreased each time in the loop,
3. but only very slightly
Selection of Parameters for
Inhomogeneous variants
1.
Choose the start value of c so that in the beginning nearly all perturbations
are accepted (exploration), but not too big to avoid long run times
2.
The function next_lower in the homogeneous variant is generally a simple
function to decrease c, e.g. a fixed part (80%) of current c
3.
At the end c is so small that only a very small number of the perturbations
is accepted (exploitation)
4.
If possible, always try to remember explicitly the best solution found so far;
the algorithm itself can leave its best solution and not find it again
Markov Chains for use in Simulation Annealing
Markov Chain:
Sequence of trials where the outcome of each trial
depends only on the outcome of the previous one
• Markov Chain is a set of conditional probabilities:
Pij (k-1,k)
Probability that the outcome of the k-th trial is j,
when trial k-1 is i
This example is
1/4
optimal
1/2
solution
circuit
1/4
Stage k-1
algorithm
Stage k
just a particular
application in
natural language
analysis and
generation
Markov Chains for use in Simulation
Annealing
Markov Chain:
Sequence of trials where the outcome of each trial depends only on the outcome of the
previous one
•
Markov Chain is a set of conditional probabilities:
Pij (k-1,k)
Probability that the outcome of the k-th trial is j, when trial k-1 is i
• Markov Chain is homogeneous when
the probabilities do not depend on k
Homogeneous and
inhomogeneous Markov Chains in
Simulated Annealing
• When c is kept constant (homogeneous variant),
the probabilities do not depend on k and for each c
there is one homogeneous Markov Chain
• When c is not constant (inhomogeneous variant),
the probabilities do depend on k and there is one
inhomogeneous Markov Chain
Performance of Simulated
Annealing
• SA is a general solution method that is easily applicable to a
large number of problems
• "Tuning" of the parameters (initial c, decrement of c, stop
criterion) is relatively easy
• Generally the quality of the results of SA is good, although it
can take a lot of time
Performance of Simulated
Annealing
• Results are generally not reproducible: another run can give a
different result
• SA can leave an optimal solution and not find it again
(so try to remember the best solution found so far)
• Proven to find the optimum under certain conditions; one of
these conditions is that you must run forever
Basic Ingredients for S.A.
1.
Solution space
2.
Neighborhood Structure
3.
Cost function
4.
Annealing Schedule
Optimization Techniques
•
•
•
•
•
•
Mathematical Programming
Network Analysis
Branch & Bond
Genetic Algorithm
Simulated Annealing
Tabu Search
Tabu
Search
Tabu Search
• What
– Neighborhood search + memory
• Neighborhood search
• Memory
– Record the search history – the “tabu list”
– Forbid cycling search
Main idea of tabu
Algorithm of Tabu Search
1.
Choose an initial solution X
2.
Find a subset of N(x) the neighbors of X which are not in the
tabu list.
3.
Find the best one (x’) in set N(x).
4.
If F(x’) > F(x) then set x=x’.
5.
Modify the tabu list.
6.
If a stopping condition is met then stop, else go to the second
step.
Effective Tabu Search
•
Effective Modeling
–
–
Neighborhood structure
Objective function (fitness or cost)
•
Example:
1. Graph coloring problem:
– Find the minimum number of colors needed such that no two
connected nodes share the same color.
•
Aspiration criteria
–
The criteria for overruling the tabu constraints and
differentiating the preference of among the neighbors
Effective Tabu Search
• Effective Computing
– “Move” may be easier to be stored and
computed than a completed solution
• move: the process of constructing of x’ from x
– Computing and storing the fitness difference
may be easier than that of the fitness function.
Effective Tabu Search
• Effective Memory Use
– Variable tabu list size
• For a constant size tabu list
– Too long: deteriorate the search results
– Too short: cannot effectively prevent from cycling
– Intensification of the search
• Decrease the tabu list size
– Diversification of the search
• Increase the tabu list size
• Penalize the frequent move or unsatisfied constraints
Eample of Tabu Search
• A hybrid approach for graph coloring
problem
– R. Dorne and J.K. Hao, A New Genetic Local
Search Algorithm for Graph Coloring, 1998
Problem
• Given an undirected graph G=(V,E)
– V={v1,v2,…,vn}
– E={eij}
• Determine a partition of V in a minimum number
of color classes C1,C2,…,Ck such that for each
edge eij, vi and vj are not in the same color class.
• NP-hard
General Approach
• Transform an optimization problem into a
decision problem
• Genetic Algorithm + Tabu Search
– Meaningful crossover
– Using Tabu search for efficient local search
Encoding
• Individual
– (Ci1, Ci2, …, Cik)
• Cost function
– Number of total conflicting nodes
• Conflicting node
– having same color with at least one of its adjacent nodes
• Neighborhood (move) definition
– Changing the color of a conflicting node
• Cost evaluation
– Special data structures and techniques to improve the
efficiency
Implementation
• Parent Selection
– Random
• Reproduction/Survivor
• Crossover Operator
– Unify independent set (UIS) crossover
• Independent set
– Conflict-free nodes set with the same color
• Try to increase the size of the independent set to improve the
performance of the solutions
Unify independent set (UIS)
crossover
It can be made very similar to Simulated
Annealing or Genetic Algorithm
Implementation of Tabu Search
• Mutation
– With Probability Pw, randomly pick neighbor
– With Probability 1 – Pw, Tabu search
• Tabu search
– Tabu list
• List of {Vi, cj}
– Tabu tenure (the length of the tabu list)
• L = a * Nc + Random(g)
• Nc: Number of conflicted nodes
• a,g: empirical parameters
Summary on Tabu Search
1. Neighbor Search
2. TS prevent being trapped in the local
minimum with tabu list
3. TS directs the selection of neighbor
4. TS cannot guarantee the optimal result
5. Sequential
6. Adaptive