1. General Formulation of Optimization Problems

1. 1. General Formulation of Optimization Problems

2. Problem Structure • Objective Function: It is usually formulated on the basis of economic criterion, e.g. profit, cost, energy and yield, etc., as a function of key variables of the system under study. • Process Model: It is used to describe the interrelations of the key variables, which are representedwithequality and inequality constraints.

3. EXAMPLE OF OBJECTIVE FUNCTION: OPTIMUM THICKNESS OF INSULATION

4. Example of Objective Function – Optimum Thickness of Insulation Objective function Key variable

5. Essential Features of Optimization Problems At least one objective function, usually an economic model; Equality and inequality constraints. Part 2 is the mathematical formulation of the process model. A feasible solution satisfies both the equality and inequality constraints. An optimal solution is a feasible solution that optimizes (minimizes or maximizes) the objective function.

6. The dashed lines represents the side of inequality constraint in the plane that forms part of the infeasible region.

7. Mathematical Notation Objective function Equality constraints Inequality constraints

8. Economic Objective Function Objective function = annual profit = annual income - annual operating costs - annualized capital costs

9. EXAMPLE OF OBJECTIVE FUNCTION: OPTIMUM THICKNESS OF INSULATION

11. IF (1) the insulation has a lifetime of 5 years, and (2) the fund to purchase and install the insulation can be borrowed from a bank and paid back in 5 annual installments, THEN r can be considered as the fraction of the installed cost to be paid each year to the bank (r>0.2), i.e., annualization factor or repayment multiplier

12. How can we determine r? …We need to understand time value of money. • The economic analysis of any project that incurs incomes and expenses over time should incorporate the concept of the time value of money, i.e., the value of money is a function of time. • “One unit of money on hand NOW is worth more than the same unit of money in the future.”

13. Investment Time-Line Diagram for an Individual or company

14. Examples of Investment Time Line Diagram • You deposit $1000 now (the present value P) in a bank saving account that pays 5% annual interest compounded monthly. • You plan to deposit $100 per month at the end of each month for the next year • What will the future worth F of your investment be at the end of next year?

16. Present Value and Future Worth

17. PresentValue of a Series of (not Necessarily Equal) Payments or Incomes

18. Present Value of a Series of Uniform Future Paymentsor Incomes

19. Repayment Multiplier

20. Future Value of a Series of (not Necessarily Equal) Future Payments or Incomes

21. Future Value of a Series of Uniform Future Payments or Incomes

22. Cash Flows for a Typical Capital Investment Project(without borrowing from a bank) I F1 F2 Fn

23. Measures of profitability for a projectwithout considering time value of money

24. Measures of profitability for a project by considering time value of money • Net present value (NPV) is calculated by adding the initial investment (represented as a negative cash flow) to the present value of the anticipated future positive (and negative) cash flows • Internal rate of return (IRR) is the rate of return (i.e. interest rate or discount rate) at which the future cash flows (positive plus negative) would equal the initial cash outlay (a negative cash flow).

25. 2. Basic Mathematical Concepts in Optimization Problems

26. Continuity of a Function

27. Functions with Discontinuities

28. Stationary Point in 1-D

30. Unimodal and Multimodal Functions • A unimodal function has exactly one extremum. • A multimodal function has more than one extrema. • A global extremum is the biggest (or smallest) among a set of extrema. • A local extremum is just one of the extrema.

31. Definition of Unimodal 1-D Function

32. A 2-D constrained minimum. The minimum is located at (2,3) where f=2.

33. Another 2-D constrained minimum.

34. Two local extrema: f=3 f=4

35. Two extrema: a and b

36. Convex and Concave Single-Variable Functions A 1-D function is called convex over a region R, if, for any two values of x in R, the following inequality holds

38. Hessian Matrix

39. 2-D Hessian Matrix and Gradient Vector

40. Positive and Negative Definiteness

41. Remarks • An n-dimensional function is convex (strictly convex) iff its Hessian matrix is positive semi-definite (definite). • An n-dimensional function is concave (strictly concave) iff its Hessian matrix is negative semi-definite (definite).

42. Tests for Strictly Convexity • All eigenvalues of Hessian matrix must be positive. • All diagonal elements of Hessian matrix must be positive. Also, the determinants of Hessian matrix and all its leading principal minors must all be positive. (Note that the kth leading principal minor of a matrix M is the determinant of its upper-left k-by-k sub-matrix.)

43. Convex Region

45. Convex Region

47. Proposition

48. Why do we need to discuss convexity and concavity? • Determination of convexity can be used to establish whether a local optimal solution is also a global optimal solution. • If the objective function is known to be convex or concave, computation of optimum can be acceleratedby using appropriate algorithm.

49. Convex Programming Problem

50. A NLP is generally not a convex programming problem!