1

OR IIOR IIGSLM 52800GSLM 52800

2

OutlineOutline

equality constraint tangent plane

regular point

FONC

SONC

SOSC

3

Problem Under ConsiderationProblem Under Consideration

min f(x) s.t. gi(x) = 0 for i = 1, …, m, (which can be put as g(x)

= 0)

x S n

4

Equality Constraint, Tangent Plane, Equality Constraint, Tangent Plane, and Gradient at a Pointand Gradient at a Point

g(x) = 0

x*

Tg(x*)

any vector on the tangent plan of point x* is orthogonal to Tg(x*)

y1

y2

5

Regular PointRegular Point

the collection of constraints

g1(x) = 0, …, gm(x) = 0

x0 is a regular point if g1(x0), …, gm(x0) are linearly independent

6

Lemma 4.1

x* be a local optimal point of f and a regular point with respect to the equality constraints g(x) = 0

any y satisfying Tg(x*)y = 0 Tf(x*)y = 0

y on tangent planes of g1(x*), …, gm(x*)

7

Interpretation of Lemma 4.1Interpretation of Lemma 4.1

g1(x) = 0

g2(x) = 0

Tg2(x*)

x*

Tg1(x*)

What happens if Tf is not orthogonal to the tangent plane?

Tf(x*)

8

FONC for Equality ConstraintsFONC for Equality Constraints(for max & min)(for max & min)

(i) x* a local optimum

(ii) objective function f

(iii) equality constraints g(x) = 0

(iv) x* a regular point

then there exists m for (v) f(x*) + Tg(x*) = 0

(v) + g(x*) = 0 FONC

9

FONC for Equality Constraints in FONC for Equality Constraints in Terms of Lagrangian FunctionTerms of Lagrangian Function

(for max & min)(for max & min)

1( , ) ( ) ( )

m

i ii

L f g

x x x

1

( )( )0, 1,...,

mi

iij j j

gL fj n

x x x

xx

( ) 0, 1,...,ii

Lg i m

x

The FONC can be expressed as: The FONC can be expressed as:

10

Example 4.1Example 4.1

min 3min 3xx+4+4yy, ,

ss..tt.. gg11((xx, , yy) ) xx22 + + yy22 – 4 = 0, – 4 = 0,

gg22((xx, , yy) ) ( (xx+1)+1)22 + + yy22 – 9 = 0. – 9 = 0.

Check the FONC for candidates of local Check the FONC for candidates of local minimumminimum

11

Algebraic Form of Tangent Plane Algebraic Form of Tangent Plane

M: the tangent plane of the constraints

M = {y| Tg(x*)y = 0}

12

Hessian of the Lagrangian Function Hessian of the Lagrangian Function

Lagrangian functionLagrangian function

1( , ) ( ) ( )

m

i ii

L f g

x x x

gradient of gradient of LL LL ff (x (x**) + ) + TTgg (x (x**))

Hessian of Hessian of LL L(xL(x**)) F(xF(x**) + ) + TTGG(x(x**))

13

SONC for Equality ConstraintsSONC for Equality Constraints

(i) x* a local optimum

(ii) objective function f

(iii) equality constraints g(x) = 0

(iv) x* a regular point

SONC

= FONC (f(x*)+Tg(x*) = 0 and g(x*) = 0)

+ L(x*) is positive semi-definite on M

14

SOSC for Equality ConstraintsSOSC for Equality Constraints

(i) x* a regular point

(ii) g(x*) = 0

(iii) f(x*) + Tg(x*) = 0 for some m

(iv) L(x*) = F(x*) + TG(x*) (+)ve def on M

then x* being a strict local min

15

Examples Examples

Examples 4.2 to 4.6