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

Licenza d’uso
 Queste diapositive sono distribuite con licenza Creative Commons
“Attribuzione - Non commerciale - Condividi allo stesso modo (CC
BY-NC-SA)”
 Sei libero:
 di riprodurre, distribuire, comunicare al pubblico, esporre in pubblico,
rappresentare, eseguire e recitare quest'opera
 di modificare quest'opera
 Alle seguenti condizioni:
 Attribuzione — Devi attribuire la paternità dell'opera agli autori originali
e in modo tale da non suggerire che essi avallino te o il modo in cui tu
usi l'opera.
 Non commerciale — Non puoi usare quest'opera per fini commerciali.
 Condividi allo stesso modo — Se alteri o trasformi quest'opera, o se la
usi per crearne un'altra, puoi distribuire l'opera risultante solo con una
licenza identica o equivalente a questa.
 http://creativecommons.org/licenses/by-nc-sa/3.0/

Search and Optimization Strategies

Recommended

Recommended

More Related Content

What's hot

What's hot (16)

Similar to Search and Optimization Strategies

Similar to Search and Optimization Strategies (20)

Recently uploaded

Recently uploaded (20)

Search and Optimization Strategies