1 15.083J/6.859J Integer Optimization Lecture 12: Lattices III
2
6.4
The approximate nearest vector problem
In this section, we use Algorithm 6.3 to produce a reduced basis of a lattice, and then use this reduced basis to solve approximately the nearest v
15.083J/6.859J Integer Optimization
Lecture 11: Lattices II
1 Outline
2 GS orthogonalization
n
ij
~ = (b ) b2 for j = 1 b jj~ jj
j i
~ b =b
i
;
1 X
i; j
=1
2.1 Intuition
0
0
2.2 Properties
i0 j n
j
j
j
n
i
i
i
=1
i
n
j
1
(~ ) b = 0 for all i = j: b~ 6
15.083J/6.859J Integer Optimization
Lecture 9: Solving Relaxations
1 Outline
The key geometric result behind the ellipsoid method The ellipsoid method for the feasibility problem The ellipsoid method for optimization Problems with exponentially many const
15.083J/6.859J Integer Optimization
Lecture 8: Duality II + E cient Algorithms
1 Outline
Solution of Lagrangean dual Geometry and strength of the Lagrangean dual E cient algorithms and computational complexity
X
Slide 1
2 The TSP
e2 (fig)
X
s:t:
e2 (
15.083J/6.859J Integer Optimization
Lecture 7: Duality I
1 Outline
Duality from lift and project Lagrangean duality
Slide 1
2 Duality from lift and project
Z
IP = max
:
cx s t Ax = b
0
Slide 2
x
f 2 <n j
Without of loss of generality i + i+n = 1 are inclu
15.083J/6.859J Integer Optimization
Lecture 6: Ideal formulations III
1 Outline
Minimal counterexample Lift and project
Slide 1
2 Matching polyhedron
P
matching = x
X
e2
(f X ig)
xe
=1 1 1
e
Slide 2
i
2V
V
e2 (S )
xe
S
jS j
odd
jS j
3
0
F
xe G
2E
:
set of
15.083J/6.859J Integer Optimization
Lecture 5: Ideal formulations II
1 Outline
Randomized rounding methods
Slide 1
2 Randomized rounding
Solve c x subject to x 2 P for arbitrary c. x be optimal solution. From x create a new random integer solution x, feas
15.083J/6.859J Integer Optimization
Lecture 4: Ideal formulations I
1 Outline
Total unimodularity Dual Methods
Slide 1
2 Total unimodularity
n S = fx 2 Z+ j Ax bg, A 2 Z mn and b 2 Z m . P = fx 2 <n j Ax bg. + When P = conv(S ) for all integral v
15.083J/6.859J Integer Optimization
Lecture 3: Methods to enhance formulations II
1 Outline
Independence set systems and Matroids Strength of valid inequalities Nonlinear formulations
Slide 1
2 Independence set systems
2.1 De nition
2.2 Examples
15.083J/6.859J Integer Optimization
Professor: Dimitris Bertsimas
1 Structure of Class
Algebra and geometry of IO, Lec. 9-14
Algorithms for IO, Lec. 15-22
2 Requirements
Midterm Exam: 30%
Final Exam: 30%
Use of CPLEX for solving IO problems
3 Todays Lectu