Optimization of thermal processes 2007/2008

1. Optimization of thermal processes 2007/2008 Optimization of thermal processes Lecture 9 Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery

2. Optimization of thermal processes 2007/2008 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 processes 2007/2008 Nonlinear programming (constrained optimization problem) which optimizes Find • Direct methods – the constraints are handled in explicit manner • Indirect methods – the constrained problem is solved as a sequence of unconstrained minimization problems subject to the constraints:

4. Another local minimum Necessary and sufficient Condition: Positive definite Optimization of thermal processes 2007/2008 The constraints may have no effect on the optimum point. Characteristics of a constrained problem In this case we can ignore the constraints and just solve the unconstrained problem. In practice it’s hard to identify such situation beforehand. Feasible region

5. Optimization of thermal processes 2007/2008 The optimum solution may occur on a constraint boundary. Characteristics of a constrained problem In this case the constraint(s) determine the posistion of the optimum point. Minimum value but not in the feasible region Active constraint Feasible region

6. Active constraints Optimization of thermal processes 2007/2008 Characteristics of a constrained problem The constrained problem may have more local extreme points then the unconstrained problem. A constrained optimization technique must be able to locate the minimum in all mentioned situations. Feasible region

7. Optimization of thermal processes 2007/2008 Random search method (direct methods) • Generate a trial desing vector using one random number of each design variable. • 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. • 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. • 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 processes 2007/2008 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). • Start with an initial point X1 and set the iteration number as i=1. • Linearize the objective function and constraint functions about the point Xi as: • Formulate the approximating linear problem as: First-order Taylor expansion minimize subject to

9. Prescribed tolerance Optimization of thermal processes 2007/2008 Sequential linear programming - SLP (direct methods) • Solve the approximating LP problem to obtain the solution vector Xi+1 . • Evaluate the original constraints at Xi+1 :ifstop the procedure and take Xopt = Xi+1 . • Otherwise, find the most violated constraint, for example, asand relinearize this constraint about the point Xi+1 and add this as the (m+1)th inequality constraint.

10. Optimization of thermal processes 2007/2008 Sequential linear programming - SLP (direct methods) • 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. Nonlinear function minimum Feasible region (interval) We start with initial constraint and proceed with consequent linearizations of Optimization of thermal processes 2007/2008 Geometrical interpretaion of SLP method (direct methods) Minimize subject to

12. 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: Optimization of thermal processes 2007/2008 SLP method - example (direct methods) Minimize subject to the constraint

13. Step 5 As we linearize about point X2: Thus: and we can add this constraint to the previous LP problem. Optimization of thermal processes 2007/2008 SLP method - example (direct methods) Step 4 Since we have solved one LP problem, we can take:

14. Added constraint Set the iteration number as i=2 and go to step 4. Step 6 Solve the approximating LP problem and obtain the solution: Step 4 This procedure is continued until the specified convergence criterion is satisfied: Optimization of thermal processes 2007/2008 The new LP problem becomes: SLP method - example (direct methods)

15. Feasible region optimum Feasible direction Objective function contours Optimization of thermal processes 2007/2008 Other direct methods • Methods of feasible directions • Rosen’s gradient projection method • Generalized reduced gradient method • Sequential quadratic programming

16. New independent unconstrained variable Note: • The constraints have to be very simple functions. • For certain constraints such transformation may benot possible • If it is not possible to eliminate all the constraints it may be better not to use the transormation at all Optimization of thermal processes 2007/2008 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: • Lower and upper bounds on xi: • If the variable xi is constrained to take only positive values:

17. By introducing new variables as the constraints can be restated as The constraints will be satisfied automatically if we define new variables: Optimization of thermal processes 2007/2008 Transformation techniques - example (indirect methods) Maximize subject to the constraints What will be the form of the objective function in the new variables?

18. New objective function Constant, positive for minimization andnegative for maximization. Optimization of thermal processes 2007/2008 Suppose we have an optimization problem with equality constraints: Penalty function method – basic approach (indirect methods) which optimizes Find The idea is to solve optimization problem in which we include the constraints in the objective function: subject to the constraints:

19. optimum Feasible region Interior method Optimization of thermal processes 2007/2008 Penalty function method – basic approach (indirect methods) Feasible region optimum Exterior method

20. Optimization of thermal processes 2007/2008 Thank you for your attention