Part 4 Nonlinear Programming 4 5 Quadratic Programming

Part 4 Nonlinear Programming 4. 5 Quadratic Programming (QP)

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.

Standard QP Problems

Kuhn-Tucker Conditions

Complementary Problem

Definitions

Basic Ideas of Complementary Pivot Method - 1

Basic Ideas of Complementary Pivot Method - 2

Almost Complementary Solution

Example

Initial Tableau w 1’ w 2’ w 3’ z 1 z 2 z 3 z 0 q w 1’ 1 0 0 -4 2 -1 -1 -6 w 2’ 0 1 0 2 -4 -1 -1 0 w 3’ 0 0 1 1 1 0 -1 2

Step 1 To determine the initial almost elementary solution, the variable z 0 is brought into the basis, replacing the basic variable with the most negative value.

Step 1 w 1’ w 2’ w 3’ z 1 z 2 z 3 z 0 q z 0 -1 0 0 4 -2 1 1 6 w 2’ -1 1 0 6 -6 0 0 6 w 3’ -1 0 1 5 -1 1 0 8

Step-2 Principles In essence, the complementary pivot algorithm proceeds to find a sequence of almost complementary solutions until z 0 becomes zero. To do this, the basis changes must be done in such a way that

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.

Step 2. 1 w 1’ w 2’ w 3’ z 1 z 2 z 3 z 0 q z 0 -1/3 -2/3 0 0 2 1 1 2 z 1 -1/6 0 1 -1 0 0 1 w 3’ -1/6 -5/6 1 0 4 1 0 3

Step 2. 2 w 1’ w 2’ w 3’ z 1 z 2 z 3 z 0 q z 0 -1/4 -1/2 0 0 1/2 1 1/2 z 1 -1/5 -1/24 1/4 1 0 1/4 0 7/4 z 2 -1/24 -5/24 1/4 0 1 1/4 0 3/4

Step 2. 3 w 1’ w 2’ w 3’ z 1 z 2 z 3 z 0 q z 3 -1/2 -1 0 0 1 2 1 z 1 -1/12 1/2 1 0 0 -1/2 3/2 z 2 -1/12 1/2 0 1 0 -1/2

Termination Criteria 1. z 0 leaves the basis, or 2. The minimum ratio test fail, since all coefficients in the pivot column are nonpositive. Therefore, no solution.