1. 1 Chapter 8 Nonlinear Programming with Constraints

2. 2 Chapter 8

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4. 4 Methods for Solving NLP Problems

5. 5 Chapter 8 ; see Fig. E 8.1a

6. 6 Chapter 8

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10. 10 Chapter 8

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12. 12 Chapter 8 Note that there are n + m equations in the n + m unknowns x and λ

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17. 17 By the Lagrange multiplier method. Solution: The Lagrange function is The necessary conditions for a stationary point are Chapter 8

18. 18 Chapter 8

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20. 20 Chapter 8

21. 21 Chapter 8 Penalty functions for handling equality constraints

22. 22 Chapter 8

23. 23 Chapter 8 for handling inequality constraints Note g must be >0 ; r 0

24. 24 Chapter 8 The logarithmic barrier function formulation for m constraints is

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29. 29 Chapter 8 Use x c = 2 y c = 2 for linearization (step bounds)

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32. 32 Chapter 8

33. 33 Chapter 8 Quadratic Programming (QP)

34. 34 Chapter 8 8.3 QUADRATIC PROGRAMMING

35. 35 Use of Quadratic Programming to Design Multivariable Controllers (Model Predictive Control) Targets (set points) selected by real-time optimization software based on current operating and economic conditions Minimize square of deviations between predicted future outputs and specific reference trajectory to new targets using QP Framework handles multiple input, multiple output (MIMO) control problems with constraints on manipulated and controlled variables. Dynamics obtained from transfer function model. Chapter 8

36. 36 Successive Quadratic Programming Considered by some to be the best general nonlinear programming algorithm Repetitively approximates nonlinear objective function with quadratic function and nonlinear constraints with linear constraints Uses line search rather than QP step for each iteration Inequality constrained Quadratic Programming (IQP) keeps all inequality constraints Equality constrained Quadratic Programming (EQP) only keeps equality constraints by utilizing and active set strategy SQP is an Infeasible Path method Chapter 8

37. 37 Chapter 8

38. 38 Chapter 8 solve for

39. 39 Chapter 8 Generalized Reduced Gradient (GRG)

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51. 51 sequential simplex conjugate gradient Newton’s method Quasi-Newton Chapter 8

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62. 62 Chapter 1 we have to supplement blast furnace gas with fuel oil, but we want to minimize the purchase of fuel oil. Chapter 8

63. 63 Chapter 1 define: X 1 = amount of fuel oil used in generator 1 X 2 = amount of fuel oil used in generator 2 X 3 = amount of BFG used in generator 1 X 4 = amount of BFG used in generator 2 P 1 = mw output of generator 1 P 2 = mw output of generator 2 range of operation of generator 1 and generator 2 18 ≤ P 1 ≤ 30 14 ≤ P 2 ≤ 25 Fuel effects in the generators are additive (can operate on either BFG or fuel oil) Chapter 8

64. 64 Chapter 1 10 units of BFG are available (on the average): 1 unit BFG = Btu equivalent of 1 ton/hr. fuel oil. We need 50 mw power at all times. Experimental data needed? Chapter 8

65. 65 Chapter 1 Mathematical Statement a.operating ranges18 ≤ P 1 ≤ 30 & requirements14 ≤ P 2 ≤ 25 P 1 + P 2 = 50 b. availability of x 3 + x 4 blast furnace gas c. operatingP 11 (x 1 )fuel oil characteristicsP 12 (x 3 )BFG P 21 (x 2 )fuel oil P 22 (x 4 )BFG gen 1 gen 2 Chapter 8

66. 66 Chapter 1 fcn of burners, heat transfer characteristics (convex functions) Chapter 8

67. 67 Chapter 1 Solution NLP4 ineq. const. piece-wise LP6 eq. const. No fuel oil is used in generator 1. In generator 2, fuel oil provides 58% of the power (rest is BFG). heat transfer characteristics may change, or BFG may vary w.r.t. time (on-line solution) Chapter 8