1 / 15

OR II GSLM 52800

OR II GSLM 52800. Outline. equality constraint tangent plane regular point FONC SONC SOSC. 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. Equality Constraint, Tangent Plane, and Gradient at a Point.

alize
Download Presentation

OR II GSLM 52800

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. OR IIGSLM 52800 1

  2. Outline • equality constraint • tangent plane • regular point • FONC • SONC • SOSC 2

  3. Problem 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 3

  4. Equality Constraint, Tangent Plane, and Gradient at a Point any vector on the tangent plan of point x* is orthogonal to Tg(x*) Tg(x*) y1 y2 x* g(x) = 0 4

  5. Regular Point • the collection of constraints • g1(x) = 0, …, gm(x) = 0 • x0 is a regular point if g1(x0), …, gm(x0) are linearly independent 5

  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*) 6

  7. Interpretation of Lemma 4.1 Tf(x*) Tg2(x*) What happens if Tf is not orthogonal to the tangent plane? g1(x) = 0  x* Tg1(x*) g2(x) = 0 7

  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 8

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

  10. Example 4.1 • min 3x+4y, • s.t. g1(x, y) x2 + y2 – 4 = 0, g2(x, y)  (x+1)2 + y2 – 9 = 0. Check the FONC for candidates of local minimum 10

  11. Algebraic Form of Tangent Plane • M: the tangent plane of the constraints • M = {y| Tg(x*)y = 0} 11

  12. Hessian of the Lagrangian Function • Lagrangian function  gradient of L L  f (x*) + Tg (x*)  Hessian of L L(x*) F(x*) + TG(x*) 12

  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 13

  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*) + TG(x*) (+)ve def on M • then x* being a strict local min 14

  15. Examples • Examples 4.2 to 4.6 15

More Related