Nonlinear Programming (NLP)

1. Nonlinear Programming (NLP) Operation Research December 29, 2014 RS and GISc, IST, Karachi

2. Introduction • In LP, the goal is to maximize or minimize a linear function subject to linear constraints • But in many real-world problems, either • objective function may not be a linear function, or • some of the constraints may be nonlinear • Functions having exponents, logarithms, square roots, products of variables, and so on are nonlinear

3. NLP • Optimization problems that involve nonlinear functions are called nonlinear programming (NLP) optimization • Solution methods are more complex than linear programming methods • Solution techniques generally involve searching a solution surface for high or low points requiring the use of advanced mathematics • NLPs that do not have any constraints are called unconstrained NLPs

4. Optimality Conditions: Unconstrained optimization • Can be solved using calculus • For Z=f(X), the optimum occurs at the point where f '(X) =0 and f’''(X) meets second order conditions • A relative minimum occurs where f '(X) =0 and f’''(X) >0 • A relative maximum occurs where f '(X) =0 and f’''(X) <0

5. Concavity and Second Derivative local max and global max local max f’’(x)<0 f’’(x)>0 f’’(x)<0 f‘’(x)>0 local min local min and global min

6. Example: An unconstrained problem Solution process is straightforward using calculus: f'(x) = -2x + 9 Set this equal to zero and obtain x = 4.5 f''(x) = -2 which is negative at x = 4.5 (or at any other x-value) so we have indeed found a maximum rather than a minimum point So the function is maximized when x = 4.5, with a maximum value of -4.52 + 9(4.5) + 4 = 24.25.

7. Problem • One problem is difficulty in distinguishing between a local and global minimum or maximum point Global maximum Local maximum This is trickier: a value x whose first derivative is zero and whose second derivative is negative is not necessarily the solution point! It could be a local maximum point rather than the desired global maximum point.

8. Solution point In the case of this constrained optimization problem basic calculus is of no value, as the derivative at the solution point is not equal to zero Feasible region

9. Problems • Solutions to NLPs are found using search procedures • Search can fail!!!

10. NLP Example: Searches Can Fail! • The correct answer is that the problem is unbounded. There is no solution point! • Solvers may converge to a local maximum Maximize f(x) = x3 - 30x2 + 225x + 50

11. Example Profit function, Z, with volume independent of price: Z = vp - cf - vcv where v = sales volume p = price cf = unit fixed cost cv = unit variable cost Add volume/price relationship: v = 1,500 - 24.6p Figure 1 Linear Relationship of Volume to Price

12. With fixed cost (cf = $10,000) and variable cost (cv = $8): Profit, Z = 1,696.8p - 24.6p2 - 22,000 Figure 2 The Nonlinear Profit Function

13. Optimal Value of a Single Nonlinear Function= Maximum Point on a Curve • The slope of a curve at any point is equal to the derivative of the curve’s function. • The slope of a curve at its highest point equals zero. Figure 3 Maximum profit for the profit function

14. Optimal Value of a Single Nonlinear Function Solution Using Calculus Z = 1,696.8p - 24.6p2 -22,000 dZ/dp = 1,696.8 - 49.2p = 0 p = 1696.8/49.2 = $34.49 v = 1,500 - 24.6p v = 651.6 pairs of jeans Z = $7,259.45 Figure 4 Maximum Profit, Optimal Price, and Optimal Volume

15. Constrained Optimization in Nonlinear Problems Definition • If a nonlinear problem contains one or more constraints it becomes a constrained optimization model • A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s) are nonlinear. • Solution procedures are much more complex and no guaranteed procedure exists for all NLP models.

16. Constrained Optimization in Nonlinear Problems Graphical Interpretation (1 of 3) • Effect of adding constraints to nonlinear problem: Figure 5 Nonlinear Profit Curve for the Profit Analysis Model

17. Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3)- First constrained p<= 20 Figure 6 A Constrained Optimization Model

18. Constrained Optimization in Nonlinear Problems Graphical Interpretation (3 of 3) Second constrained p<= 40 Figure 7 A Constrained Optimization Model with a Solution Point Not on the Constraint Boundary

19. Constrained Optimization in Nonlinear Problems Characteristics • Unlike linear programming, solution is often not on the boundary of the feasible solution space. • Cannot simply look at points on the solution space boundary but must consider other points on the surface of the objective function. • This greatly complicates solution approaches. • Solution techniques can be very complex.

20. Facility Location Example Problem Problem Definition and Data (1 of 2) Centrally locate a facility that serves several customers or other facilities in order to minimize distance or miles traveled (d) between facility and customers. di = sqrt[(xi - x)2 + (yi - y)2] Where: (x,y) = coordinates of proposed facility (xi,yi) = coordinates of customer or location facility i Minimize total miles d =  diti Where: di = distance to town i ti =annual trips to town i

21. Facility Location Example Problem Problem Definition and Data (2 of 2)

22. Facility Location Example Problem: Using Excel Solver

23. Excel Solver

24. Facility Location Example Problem Solution Using Excel Figure 13

25. Facility Location Example ProblemSolution Map di = sqrt[(xi - x)2 + (yi - y)2] dA=sqrt[(20- 20.668)2 + (20- 15.473)2] dA=4.57........................................ dE=6.22 d =  diti d =4.57(75)+................................+90(13.02) d =5583.8 total annual distance

26. Facility Location Example Problem Solution Map X = 20.668, Y = 15.473 Rescue Squad Facility Location

30. More Examples