15.053 Thursday, April 18

1. 15.053 Thursday, April 18 • Nonlinear Programming (NLP) • – Modeling Examples • – Convexity • – Local vs. Global Optima • Handouts: Lecture Notes

2. Linear Programming Model • Maximize ..... c1x1 +c2x2 +……+cnxn • subject to • a11x1+a12x2 +…+a1nxn≤ b1 • a21x1+a22x2 +…+a2nxn≤ b2 • . • . • . • am1x1+am2x2 +…+amnxn≤ bm • x1,x2,…,xn ≥ 0 • ASSUMPTIONS: • Proportionality • Assumption • – Objective function • – Constraints • Additivity Assumption • – Objective function • – Constraints

3. What is a non-linear program? • maximize 3 sin x + xy + y3 - 3z + log z • subject to x2 + y3 = 1 • x + 4z ≥ 2 • z ≥ 0 • A non-linear program is permitted to have • non-linear constraints or objectives. • A linear program is a special case of non- • linear programming!

4. Nonlinear Programs (NLP) Let x = (x1,x2,…,xn) Max f(x) gi (x) ≤ bi Nonlinear objective function f(x) and/or Nonlinear constraints gi(x) Could include xi ≥ 0 by adding the constraints xi= yi 2 for i=1,…,n.

5. Unconstrained Facility Location This is the warehouse location problem with a single warehouse that can be located anywhere in the plane. Distances are “Euclidean.” Loc. Dem. A: (8,2) 19 B: (3,10) 7 C: (8,15) 2 D: (14,13) 5 P: ?

6. An NLP • Costs proportional to distance; • known daily demands d(P,A) = … d(P,D) = minimize 19 d(P,A) + … + 5 d(P,D) subject to: P is unconstrained

7. Here are the objective values for 55 different locations. Objective value values for y

8. Facility Location. What happens if P must be within a specified region?

9. The model Minimize Subject to

10. 0-1 integer programs as NLPs minimize Σj cj xj subject to Σj aij xj = bi for all i xj is 0 or 1 for all j is “nearly” equivalent to minimize Σj cj xj + 106 Σj xj (1- xj). subject to Σj aij xj = bi for all i 0 ≤ xj ≤ 1 for all j

11. Some comments on non-linear models • The fact that non-linear models can model • so much is perhaps a bad sign • – How can we solve non-linear programs if we • have trouble with integer programs? • – Recall, in solving integer programs we use • techniques that rely on the integrality. • Fact: some non-linear models can be • solved, and some are WAY too difficult to • solve. More on this later.

12. Variant of exercise from Bertsimas and Freund • Buy a machine and keep it for t years, and • then sell it. (0 ≤ t ≤ 10) • – all values are measured in $ million • – Cost of machine = 1.5 • – Revenue = 4(1 - .75t) • – Salvage value = 1(1 + t)

13. Machine values revenue Millions of dollars salvage total Time

14. How long should we keep the machine? • Work with your partner on how long we • should keep the machine, and why?

15. Nonlinearities Because of Time • Discount rates • decreasing value of equipment over time • – wear and tear, improvements in • technology • Tax implications (Depreciation) • Salvage value • Secondary focus of the previous model(s): • Finding the right model can be subtle

16. Nonlinearities in Pricing • The price of an item may depend on the • number sold • – quantity discounts for a small seller • – price elasticity for monopolist • Complex interactions because of • substitutions: • – Lowering the price of GM automobiles will • decrease the demand for the competitors

17. Non-linearities because of congestion • The time it takes to go from MIT to Harvard • by car depends non-linearly on the • congestion. • As congestion increases just to its limit, • the traffic sometimes comes to a near halt.

18. Portfolio Optimization • In the following slides, we will show how • to model portfolio optimization as NLPs • The key concept is that risk can be • modeled using non-linear equations • Since this is one of the most famous • applications of non-linear programming, • we cover it in much more detail

19. Risk vs. Return • In finance, one trades of risk and return. • For a given rate of return, one wants to • minimize risk. • For a given rate of risk, one wants to • maximize return. • Return is modeled as expected value. • Risk is modeled as variance (or standard • deviation.)

20. Portfolio Selection: The value of diversification. Suppose that the following investments all have an expected return of 10% per year, and have similar variance. You can choose any of the following 3 pairs. Penguin Umbrellas, and Bay Watch Sunglasses (negatively correlated) Cogswell Cogs and Gilligan’s Cruise Tours (no correlation) CSX Railroad, Burlington Northern Railroad (positively correlated)

21. On Correlations These variables have a correlation of .998 These Variables have a correlation of -.866

22. More on correlations • Finding the best linear fit is itself a • nonlinear program. • Regression programs do this • “automatically” using a least squares fit.

23. The best fit regression line minimizes the sum of the squares of the residuals. The vertical red lines are the residuals. The goal is to select the line the minimizes the sum of the residuals squared. It is a non- linear program.

24. Correlations that are 0 (or close to 0). Correlation is related to the best linear fit. These Variables have a correlation of -.026 These variables are dependent but have a correlation of 0

25. Key Formula for Expected Values • Let X and Y be random variables, and E( ) • denote the expected value. • Expected values act in a linear manner. • For all constants a and b, E(aX + bY) = a E(X) + b E(Y) e.g., E(.3X + .7Y) = .3 E(X) + .7 E(Y)

26. Mixing distributions Expected Values Suppose that E(X) = 5 and E(Y) = 10. What is the expected value of pX + (1-p)Y as p varies from 0 to 1? E(pX + (1-p)Y)

27. Key Formula for Variances • Let X and Y be random variables, Var(X) and • Var(Y) denote their variances. (risk ~ variance) • The variance of aX + bY depends on the • covariance of X and Y, which depends on how • correlated the two variables X and Y are. • For all constants a and b • Var(aX + bY) = a2 Var(X) + b2 Var(Y) + 2ab Cov(X,Y) • For example, • Var(.3X + .7Y) = .09 Var(X) + .49 Var(Y) + .42 Cov(X,Y)

28. On Reducing Variance if X and Y • are independent • If two variables X and Y are independent, • then their covariance is 0. • Var(pX + (1-p)Y) = p2 Var(X) + (1-p)2 Var(Y) • ≤ p Var(X) + (1-p) Var(Y).

29. Mixing Uncorrelated Distributions Here X and Y both have a standard deviation of 5, and they have a correlation of 0. Let W = pX + (1-p)Y, as p goes from 0 to 1.

30. On reducing variance if X and Y are negatively correlated • If two variables X and Y are negatively • correlated then their covariance is • negative. Var(pX + (1-p)Y) = p2 Var(X) + (1-p)2 Var(Y) + 2p(1-p) Cov(X,Y) < p Var(X) + (1-p) Var(Y). The most extreme example is if the correlation is –1.

31. Mixing Negatively Correlated Distributions Suppose X and Y both have a standard deviation of 5, and they have a correlation of –1. Standard Deviation of W Standard Deviation of W Let W = pX + (1-p)Y, as p goes from 0 to 1.

32. On reducing variance if X and Y are • positively correlated • If two variables X and Y are positively correlated • then their covariance is positive. • If 0 < p < 1, and if the positive correlation is less • than 1, then • Var(pX + (1-p)Y) = • p2 Var(X) + (1-p)2 Var(Y) + 2p(1-p) Cov(X,Y) • < p Var(X) + (1-p) Var(Y). • If the correlation is 1, the above holds with • equality.

33. Mixing Positively Correlated Distributions Suppose X and Y both have a standard deviation of 5, and they have a correlation of 1. Standard Deviation Let W = pX + (1-p)Y, as a goes from 0 to 1. Conclusion: Covariances are important!

34. Summary of reducing risk • Diversification is a method of reducing • risk, even when investments are positively • correlated (which they often are). • If only two investments are made, then the • risk reduction depends on the covariance. • Diversifying over investments that are • negatively correlated has a powerful • impact on risk reduction.

35. Portfolio Selection Example • When trying to design a financial portfolio • investors seek to simultaneously minimize risk • and maximize return. • Risk is often measured as the variance of the • total return, a nonlinear function. • FACT: • var (x1+x2+…xn)= • var (x1 )+ …+var(x2) + Σcov (xi,xj) i ≠j

36. Portfolio Selection (cont’d) • Two Methods are commonly used: • – Min Risk • s.t. Expected Return ≥ Bound • – Max Expected Return - θ (Risk) • where θ reflects the tradeoff between return • and risk.

37. Portfolio Selection Example • There are 3 candidate assets for out portfolio, • X, Y and Z. The expected returns are 30%, 20% • and 8% respectively (if possible we would like • at least a 12% return). Suppose the covariance • matrix is: • What are the variables? • Let X,Y,Z be percentage of portfolio of each asset.

38. Portfolio Selection Example Min 3X2+2Y2+Z2+2XY−XZ−0.8YZ st 1.3X+1.2Y+1.08Z ≥ 1.12 X+Y+Z=1 X≥ 0, Y≥ 0, Z≥ 0 Max 1.3X+1.2Y+1.08Z -θ(3X2+2Y2+Z2+2XY-XZ-0.8YZ) st X+Y+Z=1 X≥ 0, Y≥ 0, Z≥ 0

39. More on Portfolio Selection • There can be institutional constraints as well, • especially for mutual funds. • No more than 15% in the energy sector • Between 20% to 25% high growth • At most 3% in any one firm • etc. • We end up with a large non-linear program. • The unconstrained version becomes the • “CapM model” in finance.

40. Determining best linear fits • A famous application in Finance of determining the • best linear fit is determining the β of a stock. • CAPM assumes that the return of a stock s in a • given time period is rs = a + βrm + ε, rs = return on stock s in the time period rm = return on market in the time period β = a 1% increase in stock market will lead to a β% increase in the return on s (on average)

41. Regression, and estimating β Return on Stock A vs. Market Return Stock What is the best linear fit for this data? What does one mean by best? Market

42. Regression. The vertical red lines are the residuals. The goal is to select the line the minimizes the sum of the residuals squared. It is a non- linear program.

43. Regression, and estimating β Return on Stock A vs. Market Return Stock Market The value β is the slope of the regression line. Here it is around .6 (lower expected gain than the market, and lower risk.)

44. Difficulties of NLP Models Linear Program: Nonlinear Programs:

45. Difficulties of NLP Models (contd.) Def’n: Let x be a feasible solution, then – x is a global maxif f(x) ≥ f(y) for every feasible y. – x is a local maxif f(x) ≥ f(y) for every feasible y sufficiently close to x (i.e. xj-ε ≤ yj ≤ xj+ ε for all j and some small ε). There may be several locally optimal solutions.

46. Convex Functions Convex Functions: f(λ y + (1- λ)z) ≤ λ f(y) + (1- λ)f(z) for every y and z and for 0≤ λ ≤1. e.g., f((y+z)/2) ≤ f(y)/2 + f(z)/2 We say “strict” convexity if sign is “<” for 0< λ <1. Line joining any points is above the curve

47. Convex Functions Convex Functions: f(λ y + (1- λ)z) ≥ λ f(y) + (1- λ)f(z) for every y and z and for 0≤ λ ≤1. e.g., f((y+z)/2) ≥f(y)/2 + f(z)/2 We say “strict” convexity if sign is “<” for 0< λ <1. Line joining any points is above the curve

48. Classify as convex or concave or both or neither.

49. Recognizing convex functions • For functions of one variable, if the 2nd • derivative is always positive, then the • function is convex . • The sum of convex functions is convex • – e.g., f(x,y) = x2 + ex + 3(y-7)4 - log2 y

50. Recognizing convex feasible regions • If all constraints are linear, then the • feasible region is convex • The intersection of convex regions is • convex • If for all feasible x and y, the midpoint of x • and y is feasible, then the region is • convex (except in totally non-realistic • examples. )