Optimization for Learning, Planning and Problem Solving

Optimization for Learning, Planning andProblem SolvingIntroduction and Concepts

Joshua KnowlesSchool of Computer Science

The University of Manchester

COMP60342 - Week 1 2.15, Mar 13th 2015

TRAVELING SALESPERSONInstance: Complete graph G = (V,E) and set of edge weights wi,j

for each i, j ∈ E. Number k.Question: Is there a tour starting and ending at v0 and visiting everyother vertex of V exactly once, of total edge weight less than k ?

COMP60342-Introduction 2 2.15, Mar 13th 2015

This Course

Aim: a practical introduction to solving optimization problems computationally.

• Develop skill at classifying and formulating problems

• Practise programming skills

• Develop algorithm design skills

• Learn:

- a few general-purpose algorithms for optimization

- how to adapt these to different problems

• Learn to reason about complexity of a problem and methods

Less important: deeply mathematical approaches to optimization.


Structure of the Course

Week 1: Intro to OptimizationBasic exact methods: Enumeration and Greedy

Week 2: Branch-and-BoundComplexity/NP-Completeness

Week 3: Dynamic ProgrammingStochastic DP Problems

Week 4: Evolutionary Algorithms and Local SearchWeek 5: Multiobjective Optimization

Assessments:1st Lab (two weeks) Knapsack Problems 10%2nd Lab (one week) Dynamic Programming Applications 10%3rd Lab (one week) Weighted Graph Matching Problem 10%4th Lab (one week) Multiobjective MAX-SAT 10%Examination (2 hours) 20 Multiple choice questions 40%

2 Long Questions from 4 60%


The Course Text

Relevant Chapters / Sections:

1. Optimization Problems

1.A Mathematical Revision Notes

8. Algorithms and Complexity

12. Minimum Spanning Tree

15. NP-Complete Problems

16.2 Strongly NP-hard

17. Approximation Algorithms

18. Branch-and-Bound; Dynamic Programming

19. Local Search


This Lecture


This Lecture

• Problems, and Optimization Problems

• Applications of optimization (a brief tour)

• Formulation of optimization problems

• Algorithms and basic complexity concepts


The Meaning of Problem

In mathematics, one general way to define a problem is

a problem is a function that partitions a set of candidatesolutions into two sets, solutions and non-solutions.

For example

2 + 2 = ?

partitions the set of candidate solutions (say the natural numbers) intothe set of solutions {4} and non-solutions {0,1,2,3,5,. . .}.


Equally,

10 + 10 > ?

partitions the set of candidate solutions (the natural numbers) into theset of solutions {0,1,2,. . . ,19} and the set of non-solutions{20,21,. . .}.

To solve a problem is to find one or more (or all) of the solutions.


The Meaning of OptimizationProblem

An optimization problem is just a mathematical problem concernedwith finding an optimum (= minimum or maximum) of a set ofcandidate solutions.


The Meaning of OptimizationMaximize a measure of utility over a set of candidate solutions.

E.g. In a given graph, find a clique of greatest total weight.

A set of candidate solutionsand their utility values

As in this problem, the problems we will be concerned with on thiscourse have a discrete and finite set of candidate solutions.


The Meaning of OptimizationAim: find the best from a set of possible alternatives.

The set of all possible alternatives is the search space or searchdomain.

Alternatives are often (though not always) described as vectors ofvariables. They are known as solutions, candidates orconfigurations.

The “best” is defined as that which maximizes some well-definedmeasure of performance, profit, fitness, utility or quality, orminimizes some cost or penalty.

Alternatives have to be feasible. Feasibility is defined as meeting anumber of constraints.


The Difficulty of Optimization

As described, optimization seems mathematically trivial — eveneasier than say sorting. (Why?)

However, in most optimization problems the input objects only definethe feasible alternatives in a non-direct way. We must search overmany or all the combinations (or permutations) of the input objects.

Hence, while in sorting there are only N objects to sort, inoptimization there are often 2N or N ! candidates to search and test(i.e., an exponential function of the input size N ).

The problems are called combinatorial optimization because of thissearch over combinations.


The very large number of alternatives that are associated with suchproblems is known as a combinatorial explosion.

Optimization is about how to organize the searching of this set toobtain results efficiently in time (and sometimes space).

Another difficulty of optimization is that the evaluation of even a singlecandidate may be time-consuming.


Applications of Optimization


Machine Learning...Chess

Simple: What move wouldmaximize value of capturedpieces?

Tricky: What move nowwould force a win?


Machine Learning...Poker

Given the state of play, how much should I bet this round?


Machine Learning...Simulated CarRacing

What can we optimize in a shoot-em-up or real-time game?


Machine Learning...Feature Selection

Red: genes up-regulated indiseased tissueGreen: genes up-regulatedin healthy tissue

What selection of features (genes) makes it easy to detect the presence (orprogression) of a disease?


Planning ... the Shortest Tour

Circuit boards must be drilled.

What is the shortest route to move the drill from its ‘home’ position, to visit each hole,and return home?


Planning ... the London Olympics

Athletes need rest, spectators need to travel, finals must followsemi-finals, ...


Planning ... where to point Hubble

• Hubble is a sharedresource so need toschedule teams fairly

• Targets for observationmay specify time windows


Planning ... where the pipes should go


Planning ... which vehicles, on which routes


Problem Solving

What is the minimum number of moves (from here) to obtain a solved cube?

What about from ANY configuration?


Time for a Break and a Think


Exercise 1: Problem Formulation

In pairs or threes:

Select a problem (given above or one ofyour own) and prepare answers to thesequestions.

• What is the measure of optimality?(Could there be more than one?)

• What are the variables?

• What are the constraints?

• Are there any notable features to thisproblem, or difficulties in formulating it?


More on Applications...Codebreaking was the first use of early computers. This is a kind ofoptimization/search process.

Operations Research: Military operations, mining industries, finance/investment,logistics and transportation, large-scale infrastructure decisions, floor-planning,scheduling,...

Used by:

• Governments / think-tanks

• Economists and Bankers

• Evolutionary biologists

• Pharmaceutical companies

• Engineers and Designers

• AI and Machine Learning Scientists...and many more


Optimization Techniques(Development)

And who studies/develops optimization methods?

• Mathematicians

• Business/management scientists and economists

• Computer scientists

They each have a different purpose and approach (though with muchoverlap).

Computer scientists focus on algorithms.


Optimization Problems: Formal Definitions


The General Nonlinear Optimization Problem

A general form is the following.

minimize f (x)subject to gi(x) ≥ 0 i = 1, . . . ,m

hj(x) = 0 j = 1, . . . , p(1)

where x is a vector of variables ∈ Rn, known as a (candidate)solution, f is the cost or objective function to be minimized, and gand h are sets of constraint functions.

When f , g and h are general1, the problem is the general non-linearoptimization problem.

1i.e., they are not restricted to linear functions


←− Nonlinear(curvy) cost surface

←− Nonlinear feasiblespace


Combinatorial OptimizationProblem

General verbal definition:

Given a definition of a finite set of objects, find an object from that set which isoptimal, in some given sense.

E.g.:Traveling Salesperson Problem. Given a complete edge-weighted graph, find atour in the graph that has minimum total weight.

A tour is an object and there are certainly a finite number of distinct tours within anyfinite graph. An optimal tour has been defined as one with minimum total weight.


Combinatorial Optimization -Alternative DefinitionsCombinatorial Optimization Problem Given some finite setN = {1, . . . , n}, weights cj for each j ∈ N , and a set F of feasiblesubsets of N , find a minimum weight feasible subset

minS⊆N

∑j∈S

cj : S ∈ F

.

(paraphrased from Laurence Wolsey “Integer Programming”)

For example, the change-making problem: given a set of coins ofdifferent denominations, find a minimum cardinality set that has totalvalue exactly V .


Combinatorial Optimization -Alternative DefinitionsPapadimitriou and Steiglitz (course text) define first an instance of aproblem, where an instance specifies a set F of feasible points, and acost function c that maps F to <.

The problem associated with this instance is to find a set f ∈ F suchthat c(f ) is a minimum among all c(y), y ∈ F .

A problem (or problem class) is then just defined as a set I ofinstances.

e.g. An instance of a traveling salesman problem refers to a particularset of cities to visit, and their distances from one another. Theproblem, TSP, in general, consists of a set of instances (possibly aninfinite set), but usually all generated in the same way (Papadimitriouand Steiglitz, page 4).


Hierarchy of Problems

We will concern ourselves with Matching and especially NP-completeInteger Programming Problems.


Algorithms and Complexity


Algorithms

Recall: the course is about algorithms for optimization.

What is an algorithm?


Algorithms

Recall: the course is about algorithms for optimization.

What is an algorithm?

...intimately tied up with concepts Turing, Post and others developed: the concept ofa “machine” or computer.

An algorithm is an effective procedure for achieving a specific well-defined effect(the output) on a given specific type of input. It mechanically solves any giveninstance of a problem.

It must halt . A non-halting program is not an algorithm.

−→ There is no algorithm for running the economy perfectly, or purchasing an ideal family home, etc, sincethese problems are not mathematically well-defined.


Algorithms — Historical Note

These are algorithms (high-school maths)

109 66

901 + 24 x

---- ----

1010 264

1320

----

1584

These arithmetic procedures were invented by the Persianmathematician al-Khwarizmi in c. 800-820AD. His name becameassociated with any ”mechanical” or step-by-step procedure:al-Khwarizmi −→ Algorithm.


Complexity Hierarchy of Problems

Shortest Path

Spanning Tree

Non−computable /Undecidable

Post’s Translation

Halting Problem

Easy

Tiling the Plane

Integer additionSatisfiability

Traveling Salesman

Knapsack

NP−completeNP−hard /

Factorization

?Polynomial−time

Fairly Hard

Integer

ImpossibleHard

NB: This is a crude division of problems based on their difficulty. There are manymore classes than these.


Analysis of Algorithms — Big-O

Algorithm running time grows with input size.

The running time can be expressed as a function f (n), where n is input size.

Some functions grow faster thanothers.Constant: no growthLogs: slow growthRoots: a bit quickerPolynomials: n10 g.f. n9 g.f. n8...Exponentials: 10n g.f. 2n

Factorials: grow faster than all theabove

Linear

Polynomial

50.n

n3

Exponential2n

0

500

1000

1500

2000

2500

0 2 4 6 8 10


O(f (n)) expresses that the rate of growth is of the order of f (n) (or less). Thismeans that it is the same up to a constant factor for sufficiently large n(asymptotically).

Space complexity: We may also be concerned with the amount of space (memory)an algorithm uses.


Understanding Big-O as a Set

A function f (n) is in the set O(g(n)) if f (n) grows more slowly or at the same rateas g(n).

In particular, f (n) ∈ O(g(n)) if and only if there is some positive constant k andnumber n0 such that f (n) ≤ k.g(n) for all n > n0.

Here are a few members of the set O(n2).

2n2

n nlog

2)

25

n+100

2n5 n+6

n(O


2n2

n nlog

2)

25

n+100

2n5 n+6

n(O

Not in the set

n3 is an example of a function not in the set O(n2). It is not there because n3

always ‘overtakes’ n2 for sufficiently large n. And n3 will be bigger than even say106.n2 for sufficiently large n.


Complexity — The Class P

The class P or Polynomial is the class of problems for which there exists apolynomial-time bounded algorithm.

Polynomial-time bounded includes any algorithm with big-oh a polynomial. e.g.O(n4).

But also includes any algorithm with big-oh bounded by a polynomial. E.g. O(n2.5)and O(n2 log n), since these are both bounded by O(n3).

N.B: the program must be able to solve every instance in this time bound.Complexity is worst-case by default.

Why is the class P practically important? (see pp.163–165 of the core text).


The class P represents problems for which solutions are tractable. This means thatreasonably large instances of these problems can be solved reasonably quickly oncomputers.

An important fact about P is the fact that a problem in P gets exponentially quicker tosolve as computer speed increases (exponentially). So problems that were slow afew years ago can now be solved very quickly.

However, for problems not in P , in particular problems with exponential timecomplexity or worse, the growth in computer power only causes a polynomialspeed-up. So a problem (not in P ) that was intractable to solve on a computer a fewyears ago is still intractable to solve today. And it will remain intractable for many,many years to come.

For these problems, we may need to satisfy ourselves with approximate solutionsonly.


Summary

You’ve:

• seen a range of optimization problems (applications)

• had a go at formulating one or two of them

• recalled what an algorithm is, and

• began to appreciate algorithmic complexity .

NEXT: a short break, then Graphs


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Optimization for Learning, Planning and Problem Solving