Post on 02-Oct-2020
CSE543T: Algorithms for Nonlinear Optimization
Lecture 2: Unconstrained Optimization
– Definitions and Conditions
Unconstrained Optimization
Minimize f(x)where x is defined in X
What do we know? (Assumptions)
• Minimize f(x) but f(x) is:– Unknown
• Deterministic• Stochastic
– Known• No closed form• Closed form• Closed form gradient (1st order, 2nd order, …)
What do we know?
• Minimize f(x) but x is:– Continuous– Discrete– Mixed – Categorical
A blackbox problem
• Find x that minimizes E(x)
– E(x) is a blackbox
– x is defined over a finite discrete set X
– there are n elements in X
• Are there more efficient ways than simple
enumeration?
– Deterministic vs. expected running time
Noise
Highly nonlinear function
Nonlinear function
Simulated Annealing
Simulated Annealing
Simulated Annealing
Expected complexity
of SA depends on the
smoothness of the
objective function
Stochastic search algorithms
• Simulated annealing, genetic algorithms, ant colony algorithms, controlled random walk, etc.
• Much better performance than random sampling– Why?
• “slow” but valuable – Achieves global optimality– Very relaxed assumptions
Introduction to Genetic Algorithms 1
Introduction to Genetic Algorithms
Assaf Zaritsky
Ben-Gurion University, Israel
www.cs.bgu.ac.il/~assafza
Introduction to Genetic Algorithms 2
Genetic Algorithms (GA) OVERVIEW
A class of probabilistic optimization algorithms Inspired by the biological evolution process Uses concepts of “Natural Selection” and “Genetic Inheritance” (Darwin 1859) Originally developed by John Holland (1975)
Introduction to Genetic Algorithms 3
GA overview (cont)
Particularly well suited for hard problems where little is known about the underlying search space Widely-used in business, science and engineering
Introduction to Genetic Algorithms 4
Classes of Search Techniques Search Techniqes
Calculus Base Techniqes
Guided random search techniqes
Enumerative Techniqes
BFS DFS Dynamic Programming
Tabu Search Hill Climbing
Simulated Anealing
Evolutionary Algorithms
Genetic Programming
Genetic Algorithms
Fibonacci Sort
Introduction to Genetic Algorithms 5
A genetic algorithm maintains a population of candidate solutions for the problem at hand, and makes it evolve by iteratively applying a set of stochastic operators
Introduction to Genetic Algorithms 6
Stochastic operators
Selection replicates the most successful solutions found in a population at a rate proportional to their relative quality Recombination decomposes two distinct solutions and then randomly mixes their parts to form novel solutions Mutation randomly perturbs a candidate solution
Introduction to Genetic Algorithms 7
The Metaphor
Nature Genetic Algorithm
Environment Optimization problem
Individuals living in that environment
Feasible solutions
Individual’s degree of adaptation to its surrounding environment
Solutions quality (fitness function)
Introduction to Genetic Algorithms 8
The Metaphor (cont)
Nature Genetic Algorithm
A population of organisms (species)
A set of feasible solutions
Selection, recombination and mutation in nature’s evolutionary process
Stochastic operators
Evolution of populations to suit their environment
Iteratively applying a set of stochastic operators on a set of feasible solutions
Introduction to Genetic Algorithms 9
The Metaphor (cont)
The computer model introduces simplifications (relative to the real biological mechanisms),
BUT
surprisingly complex and interesting structures have emerged out of evolutionary algorithms
Introduction to Genetic Algorithms 10
Simple Genetic Algorithm produce an initial population of individuals
evaluate the fitness of all individuals
while termination condition not met do
select fitter individuals for reproduction
recombine between individuals
mutate individuals
evaluate the fitness of the modified individuals
generate a new population
End while
Introduction to Genetic Algorithms 11
The Evolutionary Cycle
selection
population evaluation
modification
discard
deleted members
parents
modified offspring
evaluated offspring initiate & evaluate
Introduction to Genetic Algorithms 12
Introduction to Genetic Algorithms 13
Example: the MAXONE problem
Suppose we want to maximize the number of ones in a string of l binary digits
Is it a trivial problem?
It may seem so because we know the answer in advance
However, we can think of it as maximizing the number of correct answers, each encoded by 1, to l yes/no difficult questions`
Introduction to Genetic Algorithms 14
Example (cont)
An individual is encoded (naturally) as a string of l binary digits The fitness f of a candidate solution to the MAXONE problem is the number of ones in its genetic code We start with a population of n random strings. Suppose that l = 10 and n = 6
Introduction to Genetic Algorithms 15
Example (initialization)
We toss a fair coin 60 times and get the following initial population: s1 = 1111010101 f (s1) = 7
s2 = 0111000101 f (s2) = 5
s3 = 1110110101 f (s3) = 7
s4 = 0100010011 f (s4) = 4
s5 = 1110111101 f (s5) = 8
s6 = 0100110000 f (s6) = 3
Introduction to Genetic Algorithms 16
Example (selection1)
Next we apply fitness proportionate selection with the roulette wheel method:
2 1 n
3
Area is Proportional to fitness value
Individual i will have a
probability to be chosen ∑
iif
if)(
)(
4
We repeat the extraction as many times as the number of individuals we need to have the same parent population size (6 in our case)
Introduction to Genetic Algorithms 17
Example (selection2)
Suppose that, after performing selection, we get the following population: s1` = 1111010101 (s1)
s2` = 1110110101 (s3)
s3` = 1110111101 (s5)
s4` = 0111000101 (s2)
s5` = 0100010011 (s4)
s6` = 1110111101 (s5)
Introduction to Genetic Algorithms 18
Example (crossover1)
Next we mate strings for crossover. For each couple we decide according to crossover probability (for instance 0.6) whether to actually perform crossover or not
Suppose that we decide to actually perform crossover only for couples (s1`, s2`) and (s5`, s6`). For each couple, we randomly extract a crossover point, for instance 2 for the first and 5 for the second
Introduction to Genetic Algorithms 19
Example (crossover2)
s1` = 1111010101 s2` = 1110110101
s5` = 0100010011 s6` = 1110111101
Before crossover:
After crossover: s1`` = 1110110101 s2`` = 1111010101
s5`` = 0100011101 s6`` = 1110110011
Introduction to Genetic Algorithms 20
Example (mutation1) The final step is to apply random mutation: for each bit that we are to copy to the new population we allow a small probability of error (for instance 0.1)
Before applying mutation:
s1`` = 1110110101
s2`` = 1111010101
s3`` = 1110111101
s4`` = 0111000101
s5`` = 0100011101
s6`` = 1110110011
Introduction to Genetic Algorithms 21
Example (mutation2)
After applying mutation:
s1``` = 1110100101 f (s1``` ) = 6
s2``` = 1111110100 f (s2``` ) = 7
s3``` = 1110101111 f (s3``` ) = 8
s4``` = 0111000101 f (s4``` ) = 5
s5``` = 0100011101 f (s5``` ) = 5
s6``` = 1110110001 f (s6``` ) = 6
Introduction to Genetic Algorithms 22
Example (end)
In one generation, the total population fitness changed from 34 to 37, thus improved by ~9%
At this point, we go through the same process all over again, until a stopping criterion is met
Introduction to Genetic Algorithms 23
Introduction to Genetic Algorithms 24
Components of a GA
A problem definition as input, and
Encoding principles (gene, chromosome) Initialization procedure (creation) Selection of parents (reproduction) Genetic operators (mutation, recombination) Evaluation function (environment) Termination condition
Introduction to Genetic Algorithms 25
Representation (encoding)
Possible individual’s encoding Bit strings (0101 ... 1100) Real numbers (43.2 -33.1 ... 0.0 89.2) Permutations of element (E11 E3 E7 ... E1 E15) Lists of rules (R1 R2 R3 ... R22 R23) Program elements (genetic programming) ... any data structure ...
Introduction to Genetic Algorithms 26
Representation (cont)
When choosing an encoding method rely on the following key ideas
Use a data structure as close as possible to the natural representation Write appropriate genetic operators as needed If possible, ensure that all genotypes correspond to feasible solutions If possible, ensure that genetic operators preserve feasibility
Introduction to Genetic Algorithms 27
Initialization
Start with a population of randomly generated individuals, or use - A previously saved population - A set of solutions provided by a human expert - A set of solutions provided by another heuristic algorithm
Introduction to Genetic Algorithms 28
Selection
Purpose: to focus the search in promising regions of the space Inspiration: Darwin’s “survival of the fittest” Trade-off between exploration and exploitation of the search space
Next we shall discuss possible selection methods
Introduction to Genetic Algorithms 29
Fitness Proportionate Selection
Derived by Holland as the optimal trade-off between exploration and exploitation
Drawbacks Different selection for f1(x) and f2(x) = f1(x) + c Superindividuals cause convergence (that may be premature)
Introduction to Genetic Algorithms 30
Linear Ranking Selection
Based on sorting of individuals by decreasing fitness
The probability to be extracted for the ith individual in the ranking is defined as
21,11)1(21)( ≤≤
−−
−−= βββni
nip
where β can be interpreted as the expected sampling rate of the best individual
Introduction to Genetic Algorithms 31
Local Tournament Selection
Extracts k individuals from the population with uniform probability (without re-insertion) and makes them play a “tournament”, where the probability for an individual to win is generally proportional to its fitness
Selection pressure is directly proportional to the number k of participants
Introduction to Genetic Algorithms 32
Recombination (Crossover)
* Enables the evolutionary process to move toward promising regions of the search space
* Matches good parents’ sub-solutions to construct better offspring
Introduction to Genetic Algorithms 33
Mutation
Purpose: to simulate the effect of errors that happen with low probability during duplication
Results: - Movement in the search space - Restoration of lost information to the population
Introduction to Genetic Algorithms 34
Evaluation (fitness function)
Solution is only as good as the evaluation function; choosing a good one is often the hardest part Similar-encoded solutions should have a similar fitness
Introduction to Genetic Algorithms 35
Termination condition
Examples:
A pre-determined number of generations or time has elapsed A satisfactory solution has been achieved No improvement in solution quality has taken place for a pre-determined number of generations
Introduction to Genetic Algorithms 36
What is your solution?
• Consider – Min f(x), x discrete, f(x) is a blackbox– Min f(x), x continuous, f(x) is a blackbox– Min f(x), x discrete, order of f(x) is known– Min f(x), x continuous, order of f(x) is known– Min f(x), x discrete, closed form f(x) is known– Min f(x), x continuous, closed form f(x) is known
[when does discrete/continuous make difference?]
LOCAL AND GLOBAL MINIMA
f(x)
x
Strict LocalMinimum
Local Minima Strict GlobalMinimum
Unconstrained local and global minima in one dimension.
Why do we look for local optima
• Sometimes, – local optimum == global optimum
• If you can enumerate all the local optima, you can find the global optima
NECESSARY CONDITIONS FOR A LOCAL MIN
• 1st order condition: Zero slope at a localminimum x∗
∇f(x∗) = 0
• 2nd order condition: Nonnegative curvatureat a local minimum x∗
∇2f(x∗) : Positive Semidefinite
• There may exist points that satisfy the 1st and2nd order conditions but are not local minima
x xx
f(x) = |x|3 (convex) f(x) = x3 f(x) = - |x|3
x* = 0 x* = 0x* = 0
First and second order necessary optimality conditions for
functions of one variable.
Why do we need optimality conditions
• Necessary vs. sufficient• Ways that do not work
– Enumerate and check– Enumerate and compare– Solving ∂f = 0 is nontrivial
• Provide guidance for design iterative search algorithms– Convergence– Conditions for the validity of points
PROOFS OF NECESSARY CONDITIONS
• 1st order condition ∇f(x∗) = 0. Fix d ∈ �n.Then (since x∗ is a local min), from 1st order Taylor
d′∇f(x∗) = limα↓0
f(x∗ + αd) − f(x∗)α
≥ 0,
Replace d with −d, to obtain
d′∇f(x∗) = 0, ∀ d ∈ �n
• 2nd order condition ∇2f(x∗) ≥ 0. From 2ndorder Taylor
f(x∗+αd)−f(x∗) = α∇f(x∗)′d+α2
2d′∇2f(x∗)d+o(α2)
Since ∇f(x∗) = 0 and x∗ is local min, there issufficiently small ε > 0 such that for all α ∈ (0, ε),
0 ≤ f(x∗ + αd) − f(x∗)α2
= 12d
′∇2f(x∗)d +o(α2)α2
Take the limit as α → 0.
CONVEXITY
Convex Sets Nonconvex Sets
x
y
αx + (1 - α)y, 0 < α < 1
x
x
y
y
xy
Convex and nonconvex sets.
αf(x) + (1 - α)f(y)
x y
C
z
f(z)
A convex function. Linear interpolation underestimates
the function.
MINIMA AND CONVEXITY
• Local minima are also global under convexity
αf(x*) + (1 - α)f(x)
x
f(αx* + (1- α)x)
x x*
f(x)
Illustration of why local minima of convex functions are
also global. Suppose that f is convex and that x∗ is a
local minimum of f . Let x be such that f(x) < f(x∗). By
convexity, for all α ∈ (0, 1),
f(αx∗ + (1 − α)x
)≤ αf(x∗) + (1 − α)f(x) < f(x∗).
Thus, f takes values strictly lower than f(x∗) on the line
segment connecting x∗ with x, and x∗ cannot be a local
minimum which is not global.
OTHER PROPERTIES OF CONVEX FUNCTIONS
• f is convex if and only if the linear approximationat a point x based on the gradient, underestimatesf :
f(z) ≥ f(x) + ∇f(x)′(z − x), ∀ z ∈ �n
f(z)f(z) + (z - x)'∇f(x)
x z
− Implication:
∇f(x∗) = 0 ⇒ x∗ is a global minimum
• f is convex if and only if ∇2f(x) is positivesemidefinite for all x
Convex Set and Convex Function