1. Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery Lecture 9

2. Optimization of thermal processes2007/2008 Overview of the lecture Constrained nonlinear programming problems Characteristics of a constrained problem Direct methods −Random search −Sequential linear programming −Methods of feasible directions Indirect methods −Tansformation techniques −Penalty function method

3. Nonlinear functions Optimization of thermal processes2007/2008 Nonlinear programming (constrained optimization problem) Find which optimizes subject to the constraints: Direct methods – the constraints are handled in explicit manner Indirect methods – the constrained problem is solved as a sequence of unconstrained minimization problems

4. Optimization of thermal processes2007/2008 Characteristics of a constrained problem The constraints may have no effect on the optimum point. In this case we can ignore the constraints and just solve the unconstrained problem. In practice it’s hard to identify such situation beforehand. Another local minimum Feasible region Positive definite Necessary and sufficient Condition:

5. Optimization of thermal processes2007/2008 Characteristics of a constrained problem Minimum value but not in the feasible region The optimum solution may occur on a constraint boundary. Active constraint In this case the constraint(s) determine the posistion of the optimum point. Feasible region

6. Optimization of thermal processes2007/2008 Characteristics of a constrained problem The constrained problem may have more local extreme points then the unconstrained problem. Active constraints Feasible region A constrained optimization technique must be able to locate the minimum in all mentioned situations.

7. Optimization of thermal processes2007/2008 Random search method (direct methods) 1.Generate a trial desing vector using one random number of each design variable. 2.Verify whether the constraints are satisfied (within a specified tolerance). If not, generate new trial vectors until you find a vector that satisfies all the constraints. 3.Check if the value of the objective function is reduced. In such a case take current design vector as the best design. Otherwise, discard the trial vector and go to step 1. 4.After specified number of iterations stop the procedure and take the last best design vector as the solution of your constrained problem. This method is very simple. Unfortunetely, it is as simple as inefficient.

8. Optimization of thermal processes2007/2008 Sequential linear programming - SLP (direct methods) In this method the solution of the nonlinear problem is found by solving a series of linear programming problems (other name: cutting plane method). 1.Start with an initial point X 1 and set the iteration number as i=1. 2.Linearize the objective function and constraint functions about the point X i as: 3.Formulate the approximating linear problem as: minimize subject to First-order Taylor expansion

9. Optimization of thermal processes2007/2008 Sequential linear programming - SLP (direct methods) 4.Solve the approximating LP problem to obtain the solution vector X i+1. 5.Evaluate the original constraints at X i+1 : if stop the procedure and take X opt = X i+1. 6.Otherwise, find the most violated constraint, for example, as and relinearize this constraint about the point X i+1 and add this as the (m+1)th inequality constraint. Prescribed tolerance

10. Optimization of thermal processes2007/2008 Sequential linear programming - SLP (direct methods) 6.Set the new iteration number as i=i+1, the total numver of constraints as m+1 inequalities and p equalities, and to to step 4. SLP method has several advantages, e.g.: It is an efficient technique for solving convex programming problems with nearly linear objective and constraint function Each of the approximating problems will be a LP problem and hence can be solved quite efficiently SLP method can be illustrated with the help of a one- variable problem. Let’s see...

11. Optimization of thermal processes2007/2008 Geometrical interpretaion of SLP method (direct methods) Minimize subject to Nonlinear function Feasible region (interval) minimum We start with initial constraint and proceed with consequent linearizations of

12. Optimization of thermal processes2007/2008 SLP method - example (direct methods) Minimize subject to the constraint Due to the nonlinear constraint the problem is nonlinear Steps 1,2,3: To avoid possible unbounded solution, we first take the bounds on the design variables, and solve the LP problem: Initial constraints The solution of this problem can be obtained as:

13. Optimization of thermal processes2007/2008 SLP method - example (direct methods) Step 4 Since we have solved one LP problem, we can take: Step 5 As we linearize about point X 2 : Thus: and we can add this constraint to the previous LP problem.

14. Optimization of thermal processes2007/2008 SLP method - example (direct methods) The new LP problem becomes: Added constraint Step 6 Set the iteration number as i=2 and go to step 4. Step 4 Solve the approximating LP problem and obtain the solution: This procedure is continued until the specified convergence criterion is satisfied:

15. Optimization of thermal processes2007/2008 Other direct methods Methods of feasible directions Rosen’s gradient projection method Generalized reduced gradient method Sequential quadratic programming Objective function contours Feasible region optimum Feasible direction

16. Optimization of thermal processes2007/2008 Transformation techniques (indirect methods) If the constraints are explicit functions of the design variables and have certain simple forms, the independent variables may be transformed such that the constraints are satisfied automatically. For instance: 1.Lower and upper bounds on x i : 2.If the variable x i is constrained to take only positive values: New independent unconstrained variable 1.The constraints have to be very simple functions. 2.For certain constraints such transformation may be not possible 3.If it is not possible to eliminate all the constraints it may be better not to use the transormation at all Note:

17. Optimization of thermal processes2007/2008 Transformation techniques - example (indirect methods) Maximize subject to the constraints By introducing new variables as the constraints can be restated as The constraints will be satisfied automatically if we define new variables: What will be the form of the objective function in the new variables?

18. Optimization of thermal processes2007/2008 Penalty function method – basic approach (indirect methods) Suppose we have an optimization problem with equality constraints: Find which optimizes subject to the constraints: The idea is to solve optimization problem in which we include the constraints in the objective function: Constant, positive for minimization and negative for maximization. New objective function

19. Optimization of thermal processes2007/2008 Penalty function method – basic approach (indirect methods) Feasible region optimum Exterior method Feasible region optimum Interior method

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