Search and Optimization Strategies

Search and Optimization Strategies
Tecniche di Programmazione – A.A. 2012/2013

Summary
A.A. 2012/2013Tecniche di programmazione2
 Definitions
 Branch & Bound
 Greedy
 Local Search

Definitions

Search vs Optimization
 A search algorithm is an algorithm for finding an item
with specified properties among a collection of items
 Specified properties: usually “local” properties, defined by some
“feasibility” criteria
 Collection: explicit, or virtual (implicit)
 An optimization algorithm aims ad finding the best
possible solution, according to an objective function,
among all possible feasible solutions for a given
problem

Objective function
 Must be computable for all feasible solutions
 May be undefined for unfeasible (or incomplete) solutions
 Maximization problems
 Objective function = value, goodness
 Must find the maximum value of objective function
 Minimization problems
 Objective function = cost
 Must find the minimum value of objective function

Types of problems
 Search problems: is there (at least one) feasible solution?
 Ex: magical square
 Ex: Hamiltonian cycle
 Optimization problems: what is the best solution?
 In these problems finding feasible solutions is usually easy
 Ex: Hamiltonian cycle on a complete graph (Hamiltonan bee)
 Hybrid problems (search + optimization): are there any valid
solutions? And what is the best?
 Ex: travelling salesman problem

Representation
S: Solutions
V:Valid (feasible) solutions
M: best solutions
s
s
s
f(s)
Objective
function

Classification
 Search problems
 S V
 |V| = 0 ?
 Find a s V
 Optimization problems
 S =V
 find max(f(s))
 Hybrid problems
 S V
 find max(f(s))
 such that s V

Exact techniques
 Exhaustive generation of all possible solutions
 Iteratively
 Recursively
 Optimizing recursive techniques
 Visit order
 Generate non-useless solutions: Branch and Bound
 Recognize equivalent sub-problems: Dynamic Programming
1 2
3 4
5 6
7 8

Approximate Tecniques
 Greedy
 Pseudo-random
 Hill climber
1
2
3

Approximate solutions
 Optimum solution: the absolute best
 Optimal solution: a solution that
 approximates the optimum one (might be coincident, but we
don’t know)
 can no longer be improved with the chosen optimization
technique
 Locally optimum solution: optimum solution in a
continuous domain

Intuitively…
x
f(x)
Optimum
solution
Optimal
solution
Locally
optimum
solutions

Computational Intelligence
 Evolutionary Algorithms
 Genetic Algorithms
 Genetic Programming
 Simulated annealing
 Tabu-Search

Branch & Bound

Introduction
 Branch and Bound is an exact method for exploring all
solutions based on their implicit enumeration
 All solutions are considered, but not one-by-one. In fact
B&B:
 Considers disjoints subsets of solutions (branching)
 Evaluates them according to an estimate of the objective
function (bounding), by eliminating (pruning) those subsets that
may not contain the optimum solution

Definition
 z*: a sub-optimal estimate of the final result
 Initialized to + for minimization problems
 Initialized to - for maximization problems
 z(r)
E : the exact solution of sub-problem P(r)
 z(r)
B: a super-optimal estimate of solutions of sub-problem
P(r)
 lower bound for minimization problems
 upper bound for maximization problems

General algorithm (I)
 Branch-and-bound is a divide-et-impera method:
 Divide a given problem (and its set of feasible solutions) in sub-
problems to be analyzed separately.
 Keep a list P of open sub-problems
 At each step, select one current problem P(r)

General algorithm (II)
 If P(r) has no feasible solutions, it is closed
 If P(r) has (at least one) solution, and may be solved up to
the optimum z(r)
E
 P(r) is closed
 Its solution, with value z(r)
E, may possibly replace z*, if it
improves it

General algorithm (III)
 Se P(r) has solutions, but it’s difficult to be solved up to
the optimum, we try to show that it may not yield a
better solution than the already known ones.
 We compute a super-optimal estimate z(r)
B of its solutions:
 If z(r)
B is not better than z*, all solutions in P(r) are dominated
by the best known-solution, and P(r) may be closed
 If z(r)
B is better than z*, P(r) is broken into sub-problems that
are inserted in the list P. P(r) is closed, and the sub-problems
will be opened, eventually.

General algorithm (IV)
 Proceed until P is empty
 When P is empty, then z* is the exact optimum solution
for the original problem.

The solution tree
P0
P1 P2
P21 P22
P11 P12
P111 P112
P12
Node: sub-problem
Leaf node: closed sub-problem

Ingredients of a B&B algorithm
 Strategy of visit of the search tree: criteria to choose the
next sub-problem to be analyzed out of the list P
 Bounding technique: how to evaluate the super-optimal
estimate z(r)
B
 Branching rule: how to split the current problem P(r) into
smaller sub-problems.

Main features
 Advantages:
 Finds optimum solution
 May limit the number of visited solutions
 May give a partial (approximate) solution, if stopped earlier
 Disadvantages:
 Computation time for visiting the solution tree depends on the
strategy
 The worst case is to visit all possible nodes, anyway
 Time needed to compute z(r)
B.

Greedy

Introduction
 A greedy algorithm searches for a globally optimal
solution, by choosing locally optimum ones.
 The adopted technique is shortsighted:
 Every choice made at any step is never re-considered again
 No back-tracking
 Basic principle: define an “attractiveness” for every partial
solution
 Attractiveness (it: appetibilità) = estimate of how much, probably,
we are close to the optimum.

General structure (I)
 If the attractiveness {ai} of partial solutions are known
since the beginning, and may not be modified
Greedy ({a1, a2, …an})
S  ø
sort {ai} in decreasing attractiveness order
for each ai do
if ai may be added to S
then S  S  {ai}
return S

General structure (II)
 If attractiveness may be modified by the effects of
previous choices
Greedy2 ({a1, a2, …an})
S  ø
compute attractiveness of ai
while there are element to choose do
choose the most-attractive ai
if ai may be added to S
then S  S  {ai}
update the remaining ai
return S

Example
 Find the minimum number of coins to give change to a
customer.
 We assume we have a list of possible coin values, and
their availability is infinite

Example
 Available coins
 20, 10, 5, 2, 1
 Change to give
 55

Algorithm
 At each step, select the highest-valued coin, whose value
is less than the remaining change
Remaining change Chosen coin
55 20
35 20
15 10
5 5

Features
 Advantages
 Extremely simple algorithm
 Very fast computational time
 Disadvantages
 The solution is optimal, non necessarily optimum

Non-optimum solution
 Input:
 Coins of 6, 4, 1
 Change to give:
 9
 Greedy solution:
 4 coins: 6, 1, 1, 1
 Optimum solution :
 3 coins: 4, 4, 1

Local Search

Applicability
 Suitable for problems where:
 There is an underlying “structure” in the solution space
 There is a clear objective function
 The search space is too vast
 Exact algorithms are not knows
 “Similar solutions have similar cost”

Iterative improvement
 A family of methods
 Common basic idea
 Start from an initial (random) configuration
 Consider various possible moves starting from that configuration
 Accept or refuse such moves, depending on the objective function
 When you cannot improve anymore, start again

Example

Moves
 We must define a set of moves to lead from one
solution to another
 The set of possible moves defined the neighborhood of
a solution
 Moves are often implemented as small variations
(mutation) on the encoded solution

Example: TSP (I)
“2-opt”neighborhood

Example: TSP(II)
3-opt neighborhood

Hill Climbing (I)
 For every configuration, evaluate all possible moves
 (explore the whole neighborhood)
 Refuse all moves that worsen the objective function
 Accept the move that leads to the highest improvement
of the objective function
 If all moves worsen the objective function, quit.

Hill Climbing (II)
X = startinc configuration
E = eval(X)
do {
N = n_moves(X)
for (i=0; i<N; ++i)
Ei = eval(move(X,i))
if (all Ei  E)
return X
i* = { i | max(Ei) }
X = move(X, i*)
E = Ei*
} while(1)

Features
 Greedy
 Simple to implement
 No backtracking => no recursion, no memory
 Needs “clever” definition of moves
 Sometimes, the objective function may be incrementally
computed

Problems (I)
 Foothill: getting trapped in local maxima
Hillclimber final point
Start
Real maximum

Problems (II)
 Mesa: wide regions with null or negligible variation of the
objective function
 Needle in the haystack: isolated peak in a desolated
landscape

Variants
 First-ascent
 Accept the first move that improves the objective function
 Best-ascent
 Evaluate all moves and accept the best one
 Threshold-accept
 Accept the first move that improves the current configuration by an
amount higher that a fixed threshold
 …etc…

Example
111 fitness 10
011 fitness 12
101 fitness 20
110 fitness 5
First ascent
Best ascent
011 fitness 12
101 fitness 20
Neighborhood definition:
‘complement one bit’

Random Restart Hill Climbing
 Repeat the Hill Climbing procedure starting from many initial
starting points
X = random configuration
Best = X
do {
X’ = HillClimber(X)
if (Eval(X’) > Eval(Best))
Best = X’
X = random configuration
} while( ! Enough )

Simulated Annealing
 Stochastic approach for function minimization
 The name is taken from the physical process of
crystallization of a liquid into solid state
 The minimum of the cost function corresponds to the
crystal state of the substance (minimum energy)
 Simulated Annealing gradually “lowers” the “temperature”
until the system “freezes”

Algorithm (I)
 Start with a random solution
 Choose a solution in the neighborhood
 If the energy of the new solution is less than the current one,
accept the move
 If the energy is more than the current one, the move may be
accepted, with a probability
 The probability to accept “bad” moves depends on
 Amount of worsening (DE)
 Temperature (simulated) of the system

Algorithm(II)
current = start state
for time = 1 to forever do
T = schedule[time]
if T = 0 return current
next = a randomly chosen successor of current
dE = value(next) – value(current)
if dE > 0 then current = next
else current = next with probability edE/T
end for

Features
 The system may “jump out” of local minima by accepting
worse solutions
 At higher temperatures, worsening jumps are easier, at
lower temperature they become unlikely
 Temperature is gradually reduced
 At T=0, the algorithm converges to the locally optimum
solution (hill climbing behavior)

Boltzmann distribution
dE
P(accettazione)
1T=max
T=0

Other local search algorithms
 Tabu Search
 Ant Colony
 Genetic Algorithms

Resources

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