1 OR II GSLM 52800

2 Outline  equality constraint  tangent plane  regular point  FONC  SONC  SOSC

3 Problem Under Consideration  min f(x)  s.t.g i (x) = 0 for i = 1, …, m, (which can be put as g(x) = 0)  x  S   n

4 Equality Constraint, Tangent Plane, and Gradient at a Point g(x) = 0 x*x* Tg(x*)Tg(x*) any vector on the tangent plan of point x * is orthogonal to  T g(x * ) y1y1 y2y2

5 Regular Point  the collection of constraints  g 1 (x) = 0, …, g m (x) = 0  x 0 is a regular point if  g 1 (x 0 ), …,  g m (x 0 ) 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  T g(x * )y = 0   T f(x * )y = 0  y on tangent planes of g 1 (x * ), …, g m (x * )

7 Interpretation of Lemma 4.1 g 1 (x) = 0 g 2 (x) = 0 Tg2(x*)Tg2(x*) x*x* Tg1(x*)Tg1(x*) What happens if  T f is not orthogonal to the tangent plane?  Tf(x*)Tf(x*)

8 FONC for Equality Constraints (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 * ) + T  g(x * ) = 0  (v) + g(x * ) = 0  FONC

9 FONC for Equality Constraints in Terms of Lagrangian Function (for max & min) The FONC can be expressed as:

10 Example 4.1  min 3x+4y,  s.t.g 1 (x, y)  x 2 + y 2 – 4 = 0, g 2 (x, y)  (x+1) 2 + y 2 – 9 = 0. Check the FONC for candidates of local minimum

11 Algebraic Form of Tangent Plane  M: the tangent plane of the constraints  M = {y|  T g(x * )y = 0}

12 Hessian of the Lagrangian Function  Lagrangian function  gradient of L  L   f (x * ) + T  g (x * )  Hessian of L L(x * )  F(x * ) + T G(x * )

13 SONC 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 * )+ T  g(x * ) = 0 and g(x * ) = 0) + L(x * ) is positive semi-definite on M

14 SOSC for Equality Constraints  (i) x * a regular point  (ii) g(x * ) = 0  (iii)  f(x * ) + T  g(x * ) = 0 for some   m  (iv) L(x * ) = F(x * ) + T G(x * ) (+)ve def on M  then x * being a strict local min

15 Examples  Examples 4.2 to 4.6