Problem Difficulties

1
Problem Difficulties

• Search space
• (can the best answer be found in time?)
• 2. Complexity
• (simplification renders the solution useless)
• 3. Objective function
• (noise and time variance)
• 4. Constraints
• (difficult to find one feasible solution,
• harder to find optimum solution)
• People and Politics

2
Algorithmic Approach to Problem Solving

• Three basic concepts are common to every
  algorithmic approach to problem solving
• Representation
• Objective
• Evaluation Function
• Representation encoding of problem solution.
• Objective what is the goal?
• Evaluation (Fitness) Function quality of
  solution.

3
Problem Types
It is important to recognise different problem
types as it may suggest appropriate methods to
find the solution.

• Linear,
• SAT Boolean Satisfiability Problem
• TSP Travelling Salesman Problem
• NLP Non-linear Programming
• Proofs
• Constraint Satisfaction Problems
• Other types and subdivisions exist

4
Classical Algorithms

• Two main classes of classical algorithms
• Operate on complete solutions
• Interruptible to give potential solution
• Quality depends on the accuracy of
  procedure-problem
• Requires knowledge of problem
• Each algorithm can only tackle a class of
  problems
• Evaluate partially constructed or approximate
  solutions

Enumeration fails on real-world problems when the
number of alternative solutions is prohibitively
large
5
Classical Algorithms

• Two main classes of classical algorithms
• Operate on complete solutions
• Evaluate partially constructed or approximate
  solutions
• Utilised when complete solution methods fail
• Decompose problem into simpler sub-problems and
  then reassemble
• Or solve complex problem in a series of simpler
  stages

Methods include Greedy, Dynamic programming, A,
Branch Bound and Divide Conquer.
6
Summary of Traditional Methods

• Some problems can be solved by classical
  algorithms
• e.g. a quadratic evaluation function can be
  solved by gradient minimisation methods.
• Each algorithm is suited to a problem and may
  falter or fail on other problems
• Could use complete solutions
• (time-consuming and may get trapped in local
  optima )
• or partial solutions,
• (may be complex to implement and may not improve
  things)
• Now Consider Modern Heuristics

7
Contents

• Problems
• Basic Concepts Representation, objective,
  evaluation problem definition, neighbourhoods
  and optima
• Basic Problems Linear, SAT, TSP, NLP
  Constraint satisfaction problems
• Basic Techniques Exhaustive, Local Search
  Simplex method.
• Methods 1 Greedy, A, Branch Bound Dynamic
  programming, and Divide Conquer.
• Methods 2 Simulated Annealing, Tabu Search
• Methods 3 Evolutionary Approaches
• Constraint Handling Techniques
• Hybridise and Tune Practical tips
• Test are you sure that you have the best
  solution?

8
Escaping Local Optima

• Problem solving strategies either
• Guarantee to discover global solution, but are
  too expensive or
• May gets stuck in local optima

• We need to escape!
• Probability of moving from one point in the
  search space to another changes
• Memory, which forces exploration of new areas of
  the search space

9
Hill-climbing

• Procedure simple hill-climber
• Begin
• x some initial starting point in S
• while improve (x) ! no do
• x improve (x)
• return (x)
• End

• Simple hill-climber has several weaknesses
• Usually terminates at locally optimum solutions
• No information on difference between local
  optimum and global optimum
• Optimum obtained depends on initial conditions
• Not normally possible to provide an upper bound
  for computational time.

10
Improve function

• Procedure Improve (Hill-climber)
• Repeat
• Select all new points in neighbourhood of
  current point vc
• Select point vn with best value from evaluation
  function
• If eval(vn) gt eval(vc) then vc ? vn
• Else Local Optimum
• Until Local Optimum

• Can improve algorithm by restarting
• Maximum number of restarts (iterations) needs to
  be set

11
Simulated Annealing

• Annealing
• A heating and cooling operation implying usually
  a relatively slow cooling. Annealing is a
  comprehensive term. The process of such a heat
  treatment may be to
• remove stresses
• induce softness
• alter ductility toughness electrical
  magnetic, or other physical properties
• refine the crystalline structure
• remove gases
• produce a definite micro-structure.
• In annealing, the temperature of the operation
  and the rate of cooling depend upon the material
  being heat treated and the purpose of the
  treatment.

12
Simulated Annealing
13
Simulated Annealing

• Procedure Simulated Annealing
• Begin
• x some initial starting point in S
• while not termination-condition do x improve?
  (x,T)
• update(T)
• return (x)
• End

• Three important differences
• Requires termination condition
• Improve does not have to return a better solution
• Parameter T is updated periodically and used in
  Improve.

14
Simulated Annealing

• Procedure Improve (Simulated Annealing)
• Repeat
• Select one new point in neighbourhood of current
  point vc
• If eval(vn) Probably Better eval(vc)
• then vc ? vn
• Until halting-criterion
• Probably better?
• If eval(vn) gt eval(vc) then vc ? vn
• or
• If random0,1lt
• then vc ? vn

15
Simulated Annealing

• Effect of Temperature Diff 13 (new better)
• When Eval(vc) 107 and T 10

16
Simulated Annealing

• For any search algorithm
• What is the solution?
• What are the neighbours of the solution?
• What is the cost of the solution?
• How do we determine the initial solution?
• Specifically for simulated annealing
• How to set initial temperature T?
• How to determine cooling ratio (update of T)?
• How to stop searching at a given temperature?
• How to stop searching completely?
• cf http//exatech.com/Optimization/
• Optimization.htm

17
Simulated Annealing

• Feedback and interaction with environment
• STEP 1 T ? TMax
• select vc at random
• STEP 2 Select vn in neighbourhood vc
• If eval(vn) gt eval(vc) then vc ? vn
• then vc ? vn
• else select with probability
• repeat this step k times
• STEP 3 set T ? rT
• If T gt Tmin
• then go to STEP 2
• else go to STEP 1
• Need to set four parameters!

18
Simulated Annealing

• Kirkpatrick, S., C. D. Gelatt Jr., M. P. Vecchi,
  "Optimization by Simulated Annealing",Science,
  220, 4598, 671-680, 1983.

19
Tabu Search

• (sometimes spelt Taboo Search)
• Memory forces the search to explore new areas.
• Recently explored points become tabu (forbidden)
• Tabu search is basically deterministic
• (simulated annealing is stochastic)
• Although both tabu search and simulated annealing
  have many improvements.

Danger
High Voltage
20
Tabu Search

• Procedure Tabu Search
• Begin
• x some initial starting point in S
• while not termination-condition do x improve?
  (x,H)
• update(H)
• return (x)
• End

• Three important differences
• Requires termination condition
• Improve does not have to return a better solution
• Parameter H is updated periodically and used in
  Improve.

21
Tabu Search

• Eight-bit SAT problem x1 x8
• Tabu list
• 1 iteration
• 5 iterations
• Gives x (1,1,0,0,0,1,1,1) 36
• x1 (0,1,0,0,0,1,1,1) 37
• x2 (1,0,0,0,0,1,1,1) 32
• x3 (1,1,1,0,0,1,1,1) 31
• x4 (1,1,0,1,0,1,1,1) 31
• x5 (1,1,0,0,1,1,1,1) 30
• x6 (1,1,0,0,0,0,1,1) 28
• x7 (1,1,0,0,0,1,0,1) 30
• x8 (1,1,0,0,0,1,1,0) 31
• NB Recency-Based Memory
• Aspiration Criterion

22
Escaping Local Optima

• Tabu search and simulated annealing
• TS usually makes worse moves when stuck,
  whereas SA can at any time
• TS is deterministic, SA is stochastic
• Both work on complete solutions,

But require more parameters than classical
techniques How to choose these parameters? The
more sophisticated the method, the more human
judgment used!