Mathematical Programming: Modelling and Software - Lix - Ecole

158 Pages·2008·1.12 MB·English

Mathematical Programming:
Modelling and Software
Leo Liberti
´
LIX, Ecole Polytechnique, France
INF572/ISC610A–p.1
The course
Title: Mathematical Programming: Modelling and Software
Code: INF572 (DIX) / ISC610A (ISIC)
Teacher: Leo Liberti ([email protected])
Assistants: Sonia Cafieri ([email protected]),
Antonio Mucherino ([email protected]),
Giacomo Nannicini ([email protected]),
Fabien Tarissan ([email protected])
Timetable INF572: tue 23,30/9, 7,14/10, 4,18,25/11, 2,9/12
0830-1000 (SI34/72), 1015-1200 (SI34); exam 16/12
Timetable ISC610A: thu 6,10,13,27/11, 4,11,18/12; exam 8/1
URL INF572: http://www.lix.polytechnique.fr/~liberti/
teaching/inf572
URL ISC610A: http://www.lix.polytechnique.fr/~liberti/
teaching/isc610a
INF572/ISC610A–p.2
Contents
1. Introduction
2. AMPL basics
3. Formal definition of MP language
4. AMPL grammar
5. Solvers
6. Sudoku
7. Kissing Number Problem
8. Some useful MP theory
9. Reformulations
10. Symmetry
11. The final attack on the KNP
INF572/ISC610A–p.3
Introduction
INF572/ISC610A–p.4
Example: Set covering
There are 12 possible geographical positions A , . . . , A where some
1 12
discharge water filtering plants can be built. These plants are supposed to
service 5 cities C , . . . , C ; building a plant at site j (j 1, . . . , 12 ) has
1 5
∈ { }
cost c and filtering capacity (in kg/year) f ; the total amount of discharge
j j
water produced by all cities is 1.2 1011 kg/year. A plant built on site j can
×
serve city i if the corresponding (i, j)-th entry is marked by a ‘*’ in the
table below.
A A A A A A A A A A A A
1 2 3 4 5 6 7 8 9 10 11 12
C * * * * * *
1
C * * * * * *
2
C * * * * *
3
C * * * * * *
4
C * * * * * * *
5
c 7 9 12 3 4 4 5 11 8 6 7 16
j
f 15 39 26 31 34 24 51 19 18 36 41 34
j
What is the best placement for the plants?
INF572/ISC610A–p.5
Example: Sudoku
Given the Sudoku grid below, find a solution or prove that
no solution exists
2 1
4 1 9 2 8 6
5 8 2 7
5 1 3
9
7 8 6
3 2 6 4 9
1 9 4 5 2 8
8 6
INF572/ISC610A–p.6
Example: Kissing Number
How many unit balls with disjoint interior can be placed
d
adjacent to a central unit ball in R ?
In R2
2
1
0
-1
-2
2 1 0 -1 -2
-2 -1 0 1 2
In R3
(D = 3: problem proposed by Newton in 1694, settled by
[Schütte and van der Waerden 1953] and [Leech 1956])
INF572/ISC610A–p.7
Mathematical programming
The above three problems seemingly have in
nothing
common!
Yet, there is a that can be used to
formal language
describe all three: (MP)
mathematical programming
Moreover, the MP language comes with a rich supply of
solution algorithms so that problems can be solved right
away
Problem Reformulation
formulation and choice of so- Solution process
→ →
in MP lution algorithm
Human intelligence
AMPL Solver
→ (for now) →
INF572/ISC610A–p.8
Modelling questions
Asking yourself the following questions should help you get started with
your MP model
The given problem is usually a particular of a
instance
; you should model the whole class, not just
problem class
the instance (replace given numbers by parameter
symbols)
What are the decisions to be taken? Are they logical,
integer or continuous?
What is the objective function? Is it to be minimized or
maximized?
What constraints are there in the problem? Beware —
some constraints may be “hidden” in the problem text
If expressing objective and constraints is overly difficult, go
back and change your variable definitions
INF572/ISC610A–p.9
Set covering 1
Let us now consider the Set Covering problem
What is the problem class?
We replace the number 12 by the parameter symbol n,
the number 5 by m and the number 1.2 1011 by d
×
We already have symbols for costs (c ) and capacities
j
(f ), where j n and i m
j
≤ ≤
We represent the asterisks by a 0-1 matrix A = (a )
ij
where a = 1 if there is an asterisk at row i, column j of
ij
the table, and 0 otherwise
INF572/ISC610A–p.10