Local search algorithms
Chapter 4, Sections 3–4
Chapter 4, Sections 3–4 1
♦ Hill-climbing
♦ Simulated annealing
♦ Genetic algorithms (briefly)
♦ Local search in continuous spaces (very briefly)
Outline
Chapter 4, Sections 3–4 2
Iterative improvement algorithms
In many optimization problems, path is irrelevant; the goal state itself is the solution
Then state space = set of “complete” configurations;
find optimal configuration, e.g., TSP
or, find configuration satisfying constraints, e.g., timetable
In such cases, can use iterative improvement algorithms; keep a single “current” state, try to improve it
Constant space, suitable for online as well as offline search
Chapter 4, Sections 3–4 3
Example: Travelling Salesperson Problem
Start with any complete tour, perform pairwise exchanges
Variants of this approach get within 1% of optimal very quickly with thou- sands of cities
Chapter 4, Sections 3–4 4
Example: n-queens
Put n queens on an n × n board with no two queens on the same
row, column, or diagonal
Move a queen to reduce number of conflicts
h =5 h = 2 h = 0
Almost always solves n-queens problems almost instantaneously for very large n, e.g., n = 1million
Chapter 4, Sections 3–4 5
Hill-climbing (or gradient ascent/descent)
“Like climbing Everest in thick fog with amnesia”
function Hill-Climbing( problem) returns a state that is a local maximum inputs: problem, a problem
local variables: current, a node
neighbor, a node
current ← Make-Node(Initial-State[problem]) loop do
end
neighbor ← a highest-valued successor of current
if Value[neighbor] ≤ Value[current] then return State[current] current ← neighbor
Chapter 4, Sections 3–4 6
Useful to consider state space landscape
objective function
global maximum
shoulder
Hill-climbing contd.
current state
state space
Random-restart hill climbing overcomes local maxima—trivially complete Random sideways moves escape from shoulders loop on flat maxima
local maximum
“flat” local maximum
Chapter 4, Sections 3–4 7
Simulated annealing
Idea: escape local maxima by allowing some “bad” moves
but gradually decrease their size and frequency
function Simulated-Annealing( problem, schedule) returns a solution state inputs: problem, a problem
schedule, a mapping from time to “temperature” local variables: current, a node
next, a node
T, a “temperature” controlling prob. of downward steps
current ← Make-Node(Initial-State[problem]) for t ← 1 to ∞ do
T ← schedule[t]
if T = 0 then return current
next ← a randomly selected successor of current ∆E ← Value[next] – Value[current]
if ∆E > 0 then current ← next
else current ← next only with probability e∆ E/T
Chapter 4, Sections 3–4 8
Properties of simulated annealing
At fixed “temperature” T, state occupation probability reaches Boltzman distribution
kT p(x) = αeE(x)
T decreased slowly enough =⇒ always reach best state x∗ kT kT kT
because eE(x∗)/eE(x) = eE(x∗)−E(x) ≫ 1 for small T
Is this necessarily an interesting guarantee??
Devised by Metropolis et al., 1953, for physical process modelling Widely used in VLSI layout, airline scheduling, etc.
Chapter 4, Sections 3–4 9
Local beam search
Idea: keep k states instead of 1; choose top k of all their successors Not the same as k searches run in parallel!
Searches that find good states recruit other searches to join them Problem: quite often, all k states end up on same local hill
Idea: choose k successors randomly, biased towards good ones Observe the close analogy to natural selection!
Chapter 4, Sections 3–4 10
Genetic algorithms
= stochastic local beam search + generate successors from pairs of states
24748552 24 31% 32752411 32748552 32748152 32752411 23 29% 24748552 24752411 24752411 24415124 20 26% 32752411 32752124 32252124 32543213 11 14% 24415124 24415411 24415417
Fitness Selection Pairs Cross−Over Mutation
Chapter 4, Sections 3–4 11
Genetic algorithms contd.
GAs require states encoded as strings (GPs use programs) Crossover helps iff substrings are meaningful components
+=
GAs ̸= evolution: e.g., real genes encode replication machinery!
Chapter 4, Sections 3–4 12
Continuous state spaces
Suppose we want to site three airports in Romania:
– 6-D state space defined by (x1, y2), (x2, y2), (x3, y3) – objective function f (x1, y2, x2, y2, x3, y3) =
sum of squared distances from each city to nearest airport Discretization methods turn continuous space into discrete space,
e.g., empirical gradient considers ±δ change in each coordinate Gradient methods compute
∇ f = ∂ f , ∂ f , ∂ f , ∂ f , ∂ f , ∂ f ∂x1 ∂y1 ∂x2 ∂y2 ∂x3 ∂y3
to increase/reduce f , e.g., by x ← x + α∇f (x)
Sometimes can solve for ∇f(x) = 0 exactly (e.g., with one city).
Newton–Raphson (1664, 1690) iterates x ← x − H−1(x)∇f(x) 2f
to solve ∇f(x) = 0, where Hij =∂ f/∂xi∂xj
Chapter 4, Sections 3–4 13