Nonlinear Programming: Overview of Variables, Cost, and Constraints - Prof. E. Cliff, Study notes of Aerospace Engineering

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Gill - Murray - Wright Chap 2

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Nonlinear Programming (NLP)

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ä

Cost Functional

f

R

n

→ R

ä

Constraints

g i (z)

i

= 1

,... , m

e

g j (^) (z)

≥ 0 j = m e

,... , m

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Special Cases for for Cost

Functional

ä

single real argument

ä

linear function

ä

functionssum of squares of linear

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Special Cases for Constraints

ä

none

ä

linear function

ä

simple bounds

ä

sparse linear functions

ä

smooth nonlinear functions

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ä

sparse nonlinear functions

ä

non-smooth nonlinear functions

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Real (Complex) Vector Space

ä

elements (vectors) XA vector space is a set of

ä

vector addition

x, y

∈ X → ( x + y ) ∈ X

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ä

scalar multiplication

x

X

, α

I

R

α x

X

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ä

thrdimensional subspaces are linesIn two dimensions the one

ough

the

origin

ä

S =

is the zero-dimensional

subspace

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Inner-Product Spaces (Hilbert

Spaces)

ä

An

inner-pr

o duct

space is a

Vector Space X

and an

inner-product

< x , y > X

ä

Given a set S its

ortho

gonal

16

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Special Subspaces for Linear

Operators

ä

Consider a linear map

T

: X

Y

ä

The

r ange-sp

ac

e

of

T

R

T

y

Y

y =

T

(^) x some x

X

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ä

The

nul

(^) l-sp

ac

e

of

T

is

N

T

x

X

T

(^) x = 0

Y

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write

N

T

N

T

⊥

X

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Adjoints and Transposes

ä

inner-product spacesConsider a linear map between

T

: X

Y

X

Y

ä

For fixed x

X and y

Y