1 / 21

The Most Important Concept in Optimization (minimization)

The Most Important Concept in Optimization (minimization). A point is said to be an optimal solution of a unconstrained minimization if there exists no decent direction. A point is said to be an optimal solution of a constrained minimization if there exists no

raymond-rosario
Download Presentation

The Most Important Concept in Optimization (minimization)

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. The Most Important Concept in Optimization (minimization) • A point is said to be an optimal solution of a unconstrained minimization if there exists no decent direction • A point is said to be an optimal solution of a constrained minimization if there exists no feasible decent direction • There might exist decent direction but move along this direction will leave out the feasible region

  2. Decent Direction of • Move alone the decent direction for a certain stepsize will decrease the objective function value i.e.,

  3. where is the feasible region. Feasible Direction of • Move alone the feasible direction from for a certainstepsize will not leave thefeasible region i.e.,

  4. Steep Decent with Exact Line Search Start with any . Having stop if (i) Steep Decent Direction : (ii) Finding Stepsize by Exact Line Search :

  5. Minimum Principle be a convex and differentiable function Let be the feasible region. Example:

  6. such that MP: KTSP: Find such that Minimization Problem vs. Kuhn-Tucker Stationary-point Problem

  7. Lagrangian Function Let and are convex then • If is convex. • For a fixed , if then • Above result is a sufficient condition if is convex.

  8. are linear functions) (Assume KTSP with Equality Constraints? such that KTSP: Find

  9. If and then is free variable KTSP with Equality Constraints Find such that KTSP:

  10. Let and are convex and is linear then • If • For fixed , if • Above result is a sufficient condition if is convex. Generalized Lagrangian Function is convex. then

  11. Lagrangian Dual Problem subject to

  12. subject to subject to Lagrangian Dual Problem where

  13. Corollary: Weak Duality Theorem be a feasible solution of the primal Let problem and a feasible solution of the dualproblem. Then

  14. where Corollary: If , then and and Weak Duality Theorem solve the primal and dual problem respectively. In this case,

  15. Saddle Point of Lagrangian satisfying Let is called Then The saddle point of the Lagrangian function

  16. Dual Problem of Linear Program Primal LP subject to Dual LP subject to • All duality theorems hold andwork perfectly!

  17. Application of LP Duality (I) Farkas’Lemma For any matrix and any vector either or but never both.

  18. Application of LP Duality (II) LSQ-Normal Equation Always Has a Solution For any matrix and any vector consider always has a solution. Claim:

  19. subject to Dual Problem of Strictly Convex Quadratic Program Primal QP subject to Withstrictly convexassumption, we have Dual QP

More Related