1. Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery Lecture 5

2. Optimization of thermal processes2007/2008 Overview of the lecture Optimization with inequality constraints −Discussion of Kuhn-Tucker conditions Convex programming Example of problem with inequality constraints A design problem: optimal parameters for drying of sugar

3. Optimization of thermal processes2007/2008 Optimization with inequality constraints Minimize subject to Feasible region Constraint surfaces Free stationary point Bound sationary point Active constraint

4. Optimization of thermal processes2007/2008 Kuhn-Tucker conditions for minimization problems (necesary conditions) Lagrange function We can use constraints themselves instead of the slack variables For minimization These conditions are also sufficient for convex programming problems.

5. Non-convex (concave) region Optimization of thermal processes2007/2008 Convex programming problem Optimization problem is called convex programming problem if −the objective function is convex −and the constraint functions are convex For any x 1 and x 2 the line is above the graph of the function Convex region A given function f is convex if the Hessian matrix is positive semidefinite

6. Optimization of thermal processes2007/2008 Convex programming problem The matrix H is positive semidefinite when: each of the principle minors H 1, H 2,..., H n is non-negative...

7. Optimization of thermal processes2007/2008 Convex programming problem Notes: If the constraint functions are linear then the feasible region is convex (every component of the Hessian is equal to zero) Linear programming problems (linear objective function and linear constraints) are convex programming problems It is very often difficult to ascertain whether the objective and constraint functions involved in a practical problem are convex For convex functions: relative extreme point = global extreme point

8. Optimization of thermal processes2007/2008 Kuhn-Tucker conditions (methodology) j-th constraint functionj-th Lagrange multiplier Example: suppose we have two constraints For each of the equations assume the constraint is active or inactive Try all possibilities, solve the equations and make sure the constraints are satisfied in each case active - inactive case 1 2 3 4 - - -- Then there are four cases.

9. Optimization of thermal processes2007/2008 Example of optimization with inequality constraints Find the point in the area D described by inequalities which is the nearest to the point A=[3,5]. First, transform the inequalities into the standard form: 3 5 Feasible region D

10. Optimization of thermal processes2007/2008 Example of optimization with inequality constraints What is the objective function? The distance from point A to This form is inconvenient. Let’s see if there is other objective function with the same extreme point. For non-negative function So, finally we choose the following objective function: Is it convex function?

11. Optimization of thermal processes2007/2008 Example of optimization with inequality constraints The Hess matrix So, f is a convex function. And the constraints are linear. Thus, the problem is convex.

12. Optimization of thermal processes2007/2008 Example of optimization with inequality constraints Lagrange function Constraint functions Kuhn-Tucker conditions (in this case also sufficient): ConstraintsMinimum Let’s make some order:

13. Optimization of thermal processes2007/2008 Example of optimization with inequality constraints A1 A2 B2 B3 B4 C1 C2 C3 D1 C1 Active constraint Inactive constraint Let’s employ the following notation:

14. Optimization of thermal processes2007/2008 Example of optimization with inequality constraints C1 (C2) (C3) Thus C2 is inactive C2 (A2) This violates (C2) We reject C1 C2C3 (A2) (A1) which violates (D1) We reject C1 C2 C3

15. Optimization of thermal processes2007/2008 Example of optimization with inequality constraints C1 (A1) as C3 has to be active C3 (A1) (A2) with (A1) and (A2) give Which violates (C2) We reject C1C2 C3 C2 We get: And all conditions are satisfied. We accept C1 C2C3 Solution: point [-1,-1]

16. Optimization of thermal processes2007/2008 Optimal parameters for drying of sugar Sugar that leaves the dryer has temperature T [ o C] and humidity W [kg/kg]. The sugar is then stored in a silo, where some amount of the sugar may lump, if the conditions (T,W) are not appropriate. The total cost of: drying, storage and loss of sugar because of lumping (caking) is described by the function [PLN/kg]: The function takes large values for: too large humidity (considerable lumping) too large temperature (high cost of drying) This is the objective function we want to minimize. What are the constraints?

17. Optimization of thermal processes2007/2008 Optimal parameters for drying of sugar The thermodynamic parameters of sugar (T,W) cannot be above the melting curve. It was experimentally verified that the following inequality should be fulfilled: Lumped sugar Loose sugar Melting curve The lowest humidity that can be achieved in a dryer and the lowest temperature of sugar are, respectively: Find the optimum parameters of the sugar.

18. Optimization of thermal processes2007/2008 Optimal parameters for drying of sugar Let’s introduce the following variables: Our optimization problem can be stated as: Minimize subject to Now, we can apply Kuhn-Tucker conditions.

19. Optimization of thermal processes2007/2008 Optimal parameters for drying of sugar C1 C2C3

20. Optimization of thermal processes2007/2008 Optimal parameters for drying of sugar C1 C2C3 + So, we reject the case C1 C3 We reject the case.We take the positive value. C2

21. Optimization of thermal processes2007/2008 Optimal parameters for drying of sugar For values: all conditions are satisfied. Thus, the solution is: The minimum cost is:

22. Optimization of thermal processes2007/2008 Thank you for your attention