Part 4 Nonlinear Programming

1. Part 4 Nonlinear Programming 4.5 Quadratic Programming (QP)

2. Introduction Quadratic programming is the name given to the procedure that minimizes a quadratic (objective) function of n variables subject to m linear inequality and/or equality constraints. A number of practical optimization problems can be naturally posed as QP problems, such as • constrained least squares, • optimal control of linear system with quadratic cost functions, and • the solution of linear algebraic equations.

3. Standard QP Problems symmetrical

4. General NLP Formulation

5. Kuhn-Tucker Conditions

6. Kuhn-Tucker Conditions

11. Complementary Problem

12. Definitions

13. Basic Ideas of Complementary Pivot Method – Part 1

14. Basic Ideas of Complementary Pivot Method – Part 2 i.e., absolute value of the most negative q

15. Almost Complementary Solution

16. Example negative

17. Initial Tableau I -M w1’=w1=-6, w2’=w2=0, w3’=w3=2 By setting z1=z2=z3=z0=0 -e

18. Step 1 To determine the initialalmost elementary solution, the variable z0 is brought into the basis, replacing the basic variable with the most negative value.

19. Step 1 About to enter basis Just left basis

20. Step-2 Principles In essence, the complementary pivot algorithm proceeds to find a sequence of almost complementary solutions (w’) until z0 becomes zero. To do this, the basis changes must be done in such a way that

21. Step-2 Procedure • To satisfy (a), the nonbasic variable that enters the basis in the next tableau is always the complement of the basic variable that just left the basis in the last tableau. (Complementary Rule) • To satisfy (b), minimum ratio test is used to determine which basic variable leaves the basis.

22. Step 2.1 Just left About to enter

23. Step 2.2

24. Step 2.3

25. Termination Criteria • z0 leaves the basis, i.e., z0=0, or • The minimum ratio test fail, since all coefficients in the pivot column are nonpositive. Therefore, no solution.