Lecture 1 An Introduction to Optimization -- …qf-zhao/TEACHING/MH/Lec01.pdfUn-constrained...

Lecture 1An Introduction to Optimization –

Classification and Case Study

An Introduction to Metaheuristics: Produced by Qiangfu Zhao (Since 2012), All rights reserved (C) Lec01/1

Un-constrained Optimization

• Generally speaking, an optimization problem has an objective function f(x).

• The problem is represented by

min(max) 𝑓(𝑥), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥

• This is called an un-constrained optimization problem (無制約最適化問題).


Un-constrained Optimization

• Usually, 𝑥 is a “point” in an N-dimensional Euclidean space 𝑅𝑁, and 𝑓(𝑥) is a point in 𝑅𝑀.

• In this course, we study only the case in which 𝑀 = 1. That is, we have only one objective to optimize.

• Some special considerations are needed to extend the results obtained here to “multiple objective” cases.

• Interested students may also study optimization in “non-Euclidean” spaces (i.e. manifolds).


Constrained Optimization

• The domain can be a sub-space 𝐷 of 𝑅𝑁.

• We have constrained optimization problem:

• 𝐷 again can be defined by some functions

– 𝑥𝑖 > 0, 𝑖 = 1,2, …

– 𝑔𝑗(𝑥) > 0, 𝑗 = 1,2, …

An Introduction to Metaheuristics: Produced by Qiangfu Zhao (Since 2012), All rights reserved (C)

Subject to

Lec01/4

• min 𝑚𝑎𝑥 𝑓 𝑥

• 𝑠. 𝑡. 𝑥 ∈ 𝐷

Linear programming(線型計画法)

• If both 𝑓(𝑥) and 𝑔𝑗(𝑥) are linear functions, we have linear optimization problem, and this is usually called linear programming (LP).

• For LP, we have very efficient algorithms already, and meta-heuristics are not needed.


Non-linear programming(非線形計画法)

• If 𝑓(𝑥) or any 𝑔𝑗(𝑥) is non-linear, we have non-linear optimization problem, and this is often called non-linear programming (NLP).

• Many methods have been proposed to solve this class of problems.

• However, conventional methods usually finds local optimal solutions. Meta-heuristic methods are useful for finding global solutions.


Local optimal and global optimal

• For minimization problem, – A solution 𝑥∗ is local optimal if 𝑓(𝑥∗) < 𝑓(𝑥) for all 𝑥 in

the 𝜀-neighborhood of 𝑥∗, where 𝜖 > 0 is a real number, and is the radius of the neighborhood.

– A solution 𝑥∗ is global optimal if 𝑓(𝑥∗) < 𝑓(𝑥) for all 𝑥in the search space (problem domain).

• Meta-heuristics are useful for obtaining global optimal solutions efficiently.


Example 1: Linear Programming

• 2 materials are used for making two products. • The prices of the products are 25 and 31 (in million yen), and those

of the materials are 0.5 and 0.8 (in million yen). • Suppose that we produce x1 units for product1, and x2 units for

product2.• We can get 25*x1+31*x2 million yen by selling the products.• On the other hand, we must pay (7*x1+5*x2)*0.5 +

(4*x1+8*x2)*0.8 million yen to buy the materials.


Material used in Product1 Material used in Product2

Material 1 7 5

Material 2 4 8

Lec01/8

Example 1: Linear Programming

• The problem can be formulated as follows:

max 𝑓(𝑥1, 𝑥2) = 18.3𝑥1 + 22.1𝑥2𝑠. 𝑡. 𝑥1 > 0; 𝑥2 > 0;

6.7𝑥1 + 8.9𝑥2 < 𝐵

• The first set of constraints means that both products should be produced to satisfy social needs; and the second constraint is the budget limitation.

• This is a typical linear programming problem, and can be solved efficiently using the well-known simplex algorithm.


Example 2: Non-linear programming

• Given 𝑁 observations: (𝑥1, 𝑝(𝑥1)), (𝑥2, 𝑝(𝑥2)), … ,(𝑥𝑁, 𝑝(𝑥𝑁)) of an unknown function 𝑝(𝑥).

• Find a polynomial 𝑞(𝑥) = 𝑎0+ 𝑎1𝑥 + 𝑎2𝑥2, such that

min 𝑓 𝑎0, 𝑎1, 𝑎2 =

𝑖=1

𝑁

𝑝 𝑥𝑖 − 𝑞 𝑥𝑖 + 𝜆 𝑞(𝑥)

• Note that in this problem 𝑞(𝑥) is also a function of 𝑎0, 𝑎1, 𝑎𝑛𝑑 𝑎2.

• The first term is the approximation error, and the second term is regularization factor that can make the solution better (e.g. smoother).


Combinatorial optimization problems

• If 𝑓(𝑥) or 𝑔𝑗(𝑥) cannot be given analytically (in closed-form), we have combinatorial problems.

• For example, if 𝑥 takes 𝑘 discrete values (e.g. integers), and if there are 𝐾 variables, the number of all possible solutions will be 𝑘𝐾.

• It is difficult to check all possible solutions in order to find the best one(s).

• In such cases, meta-heuristics can provide efficient ways for obtaining good solutions using limited resources (e.g. time and memory space).


Example 3: Traveling salesman problem (TSP)

• Given 𝑁 users located in 𝑁different places (cities).

• The problem is to find a route so that the salesman can visit all users once (and only once), start from and return to his own place (to find the Hamiltonian cycle).


From Wikipedia

Lec01/12

http://commons.wikimedia.org/wiki/File:Example_The_travelling_salesman_problem_(TSP).gif

Example 3: Traveling salesman problem (TSP)

• In TSP, we have a route map which can be represented by a graph.

• Each node is a user, and the edge between each pair of nodes has a cost (distance or time).

• The evaluation function to be minimized is the total cost of the route.


From Wikipedia

Lec01/13

For TSP, the number of all possible solutions is 𝑁!, and this is a well-known NP-hard combinatorial problem.

http://commons.wikimedia.org/wiki/File:Example_The_travelling_salesman_problem_(TSP).gif

NP-hard and NP-complete

• Problems that can be solved by a deterministic algorithm in polynomial time is called class P.

• NP is a class of decision problems that can be solved by a non-deterministic algorithm in polynomial time.

• A problem H is NP-hard if it is at least as hard as any NP problem.

• NP-hard decision problems are NP-complete.

• NP-complete and NP-hard problems can be solved more efficiently if we use meta-heuristics.


P

NP-complete

NP

NP-hard

Example 4: The Knapsack problem

• Knapsack problem is another NP-hard problem defined by:– There are 𝑁 objects;

– Each object has a weight and a value;

– The knapsack has a capacity;

– The user has a quota (minimum desired value);


The problem is to find a sub-set of the objects that can be put into the knapsack and can maximize the total value.

Example 4: The Knapsack problem KNAPSACK (in OS : set of objects; QUOTA : number; CAPACITY : number;

out S : set of objects; FOUND : boolean) Begin S := empty;

total_value := 0; total_weight := 0; FOUND := false; pick an order L over the objects; loop

choose an object O in L; add O to S; total_value:= total_value + O.value; total_weight:= total_weight + O.weight; if total_weight > CAPACITY then fail

else if total_value > = QUOTA FOUND:= true; succeed;

end enddelete all objects up to O from L;

end end


This is a non-deterministic algorithm. Each time we run the program, we get a different answer. By chance, we may get the best answer.

Example 5: Learning problems

• Many optimization problems related to machine learning (learning from a given set of training data) are NP-hard/complete.

• Examples included:– Finding the smallest feature sub-set;– Finding the most informative training

data set;– Finding the smallest decision tree;– Finding the best clusters;– Finding the best neural network;– Interpret a learned neural network.


Homework

• Try to find some other examples of optimization problems (at least two) from the Internet.

• Tell if the problems are NP-hard, NP-complete, NP, or P.

• Provide a solution (not necessarily the best one) for each of the problems.

• Summarize your answer using a pdf-file, and submit the printed copy before the class of next week.


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