Lecture 2 Notes: Convex Sets
ane and convex sets
some important examples
operations that preserve convexity
generalized inequalities
separating and supporting hyperplanes
dual cones and generalized inequalities
Ane set
line through x1, x2: all point
Lecture 1 Notes: Introduction
mathematical optimization
least-squares and linear programming
convex optimization
example
course goals and topics
nonlinear optimization
brief history of convex optimization
Mathematical optimization
(mathematical) op
Lecture 3 Notes: Convex functions
basic properties and examples
operations that preserve convexity
the conjugate function
quasiconvex functions
log-concave and log-convex functions
convexity with respect to generalized inequalities
Denition
f : Rn R
Lecture 5 Notes: Duality
Lagrange dual problem
weak and strong duality
geometric interpretation
optimality conditions
perturbation and sensitivity analysis
examples
generalized inequalities
Lagrangian
standard form problem (not necessarily convex)
Lecture 6 Notes: Approximation and tting
norm approximation
least-norm problems
regularized approximation
robust approximation
Norm approximation
minimize Ax b
(A Rmn with m n, is a norm on Rm)
interpretations of solution x = argminx Ax
b: geometric:
Lecture 8 Notes: Geometric problems
extremal volume ellipsoids
centering
classication
placement and facility location
Minimum volume ellipsoid around a set
Lowner-John ellipsoid of a set C: minimum volume ellipsoid E s.t. C E
parametrize E as E = cfw
Lecture 9 Notes: Filter design
FIR lters
Chebychev design
linear phase lter design
equalizer design
lter magnitude specications
FIR lters
nite impulse response (FIR) lter:
n1
y(t) =
h u(t ),
=0
(sequence) u : Z R is input signal
(sequence) y : Z R
Lecture 7 Notes: Statistical estimation
maximum likelihood estimation
optimal detector design
experiment design
Parametric distribution estimation
distribution estimation problem: estimate probability density p(y) of a
random variable from observed va