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 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 liear algebraic equations.

3. Standard QP Problems

4. Kuhn-Tucker Conditions

9. Complementary Problem

10. Definitions

11. Basic Ideas of Complementary Pivot Method - 1

12. Basic Ideas of Complementary Pivot Method - 2

13. Almost Complementary Solution

14. Example

15. Initial Tableau w1’w2’w3’z1z2z3z0q w1’100-42 -6 w2’0102-4 0 w3’0011102

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

17. Step 1 w1’w2’w3’z1z2z3z0q 004-2116 w2’106-6006 w3’015 108

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

19. 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.

20. Step 2.1 w1’w2’w3’z1z2z3z0q -1/3-2/3002112 z1-1/61/601001 w3’-1/6-5/6104103

21. Step 2.2 w1’w2’w3’z1z2z3z0q -1/4 -1/2001/21 z1-1/5-1/241/410 07/4 z2-1/24-5/241/401 03/4

22. Step 2.3 w1’w2’w3’z1z2z3z0q z3-1/2 00121 z1-1/12 1/2100-1/23/2 z2-1/12 1/2010-1/21/2

23. Termination Criteria 1.z0 leaves the basis, or 2.The minimum ratio test fail, since all coefficients in the pivot column are nonpositive. Therefore, no solution.