Optimization algorithms for Engineering Application
CSE 691

Spring 2016
Optimality criterion for differentiable f0
x is optimal if and only if it is feasible and
f0(x)T (y x) 0
for all feasible y
f0(x)
X
x
if nonzero, f0(x) defines a supporting hyperplane to feasible set X at x
Convex optimization problems
49
unconstrained p
Optimization algorithms for Engineering Application
CSE 691

Spring 2016
Operations that preserve convexity
practical methods for establishing convexity of a function
1. verify definition (often simplified by restricting to a line)
2. for twice differentiable functions, show 2f (x) 0
3. show that f is obtained from simple conv
Optimization algorithms for Engineering Application
CSE 691

Spring 2016
Convex Optimization Boyd & Vandenberghe
3. Convex functions
basic properties and examples
operations that preserve convexity
the conjugate function
quasiconvex functions
logconcave and logconvex functions
convexity with respect to generalized ineq
Optimization algorithms for Engineering Application
CSE 691

Spring 2016
Separating hyperplane theorem
if C and D are disjoint convex sets, then there exists a 6= 0, b such that
aT x b for x C,
aT x b for x D
aT x b
aT x b
D
C
a
the hyperplane cfw_x  aT x = b separates C and D
strict separation requires additional assumptions
Optimization algorithms for Engineering Application
CSE 691

Spring 2016
Affine function
suppose f : Rn Rm is affine (f (x) = Ax + b with A Rmn, b Rm)
the image of a convex set under f is convex
S Rn convex
=
f (S) = cfw_f (x)  x S convex
the inverse image f 1(C) of a convex set under f is convex
C Rm convex
=
f 1(C) = cfw_
Optimization algorithms for Engineering Application
CSE 691

Spring 2016
Convex Optimization Boyd & Vandenberghe
1. Introduction
mathematical optimization
leastsquares and linear programming
convex optimization
example
course goals and topics
nonlinear optimization
brief history of convex optimization
11
Mathematical o