EE236A (Fall 2007-08)
Lecture 12 The barrier method
brief history of interior-point methods Newtons method for smooth unconstrained minimization logarithmic barrier function central points, the central path the barrier method
121
The ellipsoid method
19
EE236A (Fall 2007-08)
Lecture 11 The simplex method
extreme points adjacent extreme points one iteration of the simplex method degeneracy initialization numerical implementation
111
Idea of the simplex method
move from one extreme point to an adjacent ex
EE236A (Fall 2007-08)
Lecture 10 Duality (part 2)
duality in algorithms sensitivity analysis via duality duality for general LPs examples mechanics interpretation circuits interpretation two-person zero-sum games
101
Duality in algorithms
many algorithms
EE236A (Fall 2007-08)
Lecture 9 Duality (part 1)
the dual of an LP in inequality form weak duality examples optimality conditions and complementary slackness Farkas lemma and theorems of alternatives proof of strong duality
91
The dual of an LP in inequa
EE236A (Fall 2007-08)
Lecture 6 FIR lter design
FIR lters linear phase lter design magnitude lter design equalizer design
61
FIR lters
nite impulse response (FIR) lter:
n1
y (t) =
=0
h u(t ),
tZ
u : Z R is input signal ; y : Z R is output signal hi R a
EE236A (Fall 2007-08)
Lecture 5 Structural optimization
minimum weight truss design truss topology design limit analysis design with minimum number of bars
51
Truss
m bars with lengths li and cross-sectional areas xi N nodes; nodes 1, . . . , n are free
EE236A (Fall 2007-08)
Lecture 4 The linear programming problem
variants of the linear programming problem LP feasibility problem examples and some general applications linear-fractional programming
41
Variants of the linear programming problem
general fo
EE236A (Fall 2007-08)
Lecture 3 Geometry of linear programming
subspaces and ane sets, independent vectors matrices, range and nullspace, rank, inverse polyhedron in inequality form extreme points the optimal set of a linear program
31
Subspaces
S Rn (S
EE236A (Fall 2007-08)
Lecture 2 Linear inequalities
vectors inner products and norms linear equalities and hyperplanes linear inequalities and halfspaces polyhedra
21
Vectors
(column) vector x Rn: x1 x x = .2 . xn
xi R: ith component or element of x al
EE236A (Fall 2007-08)
Lecture 1 Introduction and overview
linear programming example from optimal control example from combinatorial optimization history course topics software
11
Linear program (LP)
n
minimize
j =1 n
cj xj aij xj bi,
j =1 n
subject to
i
EE236A (Fall 2007-08)
Lecture 17 Integer linear programming
integer linear programming, 0-1 linear programming a few basic facts branch-and-bound
171
Denition
integer linear program (ILP) minimize cT x subject to Ax b, x Zn
c
Gx = d
mixed integer linear
EE236A (Fall 2007-08)
Lecture 16 Large-scale linear programming
cutting-plane method Benders decomposition delayed column generation Dantzig-Wolfe decomposition
161
Cutting-plane method
minimize cT x subject to Ax b A Rmn, m n general idea: solve sequenc
EE236A (Fall 2007-08)
Lecture 15 Self-dual formulations
initialization and infeasibility detection skew-symmetric LPs homogeneous self-dual formulation self-dual formulation
151
Solution of an LP
given a pair of primal and dual LPs minimize cT x subject
EE236A (Fall 2007-08)
Lecture 14 Primal-dual interior-point methods
primal-dual path-following Mehrotras corrector step computing the search directions
141
Central path and complementary slackness
s + Ax b = 0 AT z + c = 0 zisi = 1/t, z 0, i = 1, . . . ,
EE236A (Fall 2007-08)
Lecture 13 Convergence analysis of the barrier method
complexity analysis of the barrier method convergence analysis of Newtons method choice of update parameter bound on the total number of Newton iterations initialization
131
Comp