Applied Mathematical Programming: Column Generation Algorithms and Heuristic Methods, Study notes of Operational Research

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OPR 992 - Applied Mathematical Programming

Week 9

Integer Programming: Algorithms - 3

OPR 992

Applied Mathematical Programming

Column Generation Algorithms l^ Dantzig-Wolfe Reformulation l^ Example l^ Strength of the LinearProgramming Master Heuristic Algorithms^ OPR 992 - Applied Mathematical Programming

Column Generation Algorithms

Column Generation Algorithms l^ Dantzig-Wolfe Reformulation l^ Example l^ Strength of the LinearProgramming Master Heuristic Algorithms^ OPR 992 - Applied Mathematical Programming

Example^ Solve an instance of STSP with the following distance matrix:

^ − 

Column Generation Algorithms l^ Dantzig-Wolfe Reformulation l^ Example l^ Strength of the LinearProgramming Master Heuristic Algorithms^ OPR 992 - Applied Mathematical Programming

• p. 5/

Strength of the Linear Programming Master^ z

LP M

max

K∑ k=

k c

kx

K∑ k=

A

kx

k^

b, x

k^

∈^

conv

(X

k)

, k

,... , K

LP Mz

w

LD

z

CU T

Choose the right algorithm based on speed.

Column Generation Algorithms Heuristic Algorithms l^ Introduction l^ Tabu Search l^ Simulated Annealing l^ Genetic Algorithms^ OPR 992 - Applied Mathematical Programming

Introduction^ When should we use a heuristic?^ n

A solution is required rapidly. n The instance is too large to formulate as whole problem ofreasonable size. n Once formulated, known algorithms cannot solve it in realtime. n It is much easier to find solutions by inspection than bysolving using a general-purpose algorithm.

Column Generation Algorithms Heuristic Algorithms l^ Introduction l^ Tabu Search l^ Simulated Annealing l^ Genetic Algorithms^ OPR 992 - Applied Mathematical Programming

Tabu Search^ Local search for a constrained problem of the form

min

{c

(x

g

(x

can involve a goal function

c(

x) +

αg

(x

For different values of

α

, the local search can cycle among

solutions.Solution: Make certain solution forbidden to avoid cycling.

Column Generation Algorithms Heuristic Algorithms l^ Introduction l^ Tabu Search l^ Simulated Annealing l^ Genetic Algorithms^ OPR 992 - Applied Mathematical Programming

Simulated Annealing - The Algorithm^ 1. Get an initial solution

S

2. Get an initial temperature

T

and a cooling ratio

r

with

< r <

3. While not yet frozen, do the following:

(a) Perform the following loop

L

times:

i. Pick a random neighbor

S

′^ of

S

ii. Let

f

(S

f

(S

iii. If

, set

S

S

iv. If

, set

S

S

′^ with probability

e

−∆

/T

(b) Reduce the temperature by setting

T

rT

4. Return the best solution found.

Column Generation Algorithms Heuristic Algorithms l^ Introduction l^ Tabu Search l^ Simulated Annealing l^ Genetic Algorithms^ OPR 992 - Applied Mathematical Programming

Genetic Algorithms^ Start with a finite population of solutions and evolve it from onegeneration to the next.Each iteration performs the following steps:1. Evaluation: The fitness of the individuals is evaluated.2. Parent Selection: Certain pairs of solutions (parents) are

selected based on their fitness.

3. Crossover: Each pair of parents combines to produce one

or two new solutions (offspring).

4. Mutation: Some of the offspring are randomly modified.5. Population Selection: Based on their fitness, a new

population is selected replacing some or all of the originalpopulation by an identical number of offspring.