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Readings. Readings. Chapter 8 Nonlinear Optimization Models. Overview. Overview. Overview. Overview. Tool Summary

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Readings

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  1. Readings • Readings • Chapter 8 • Nonlinear Optimization Models

  2. Overview • Overview

  3. Overview

  4. Overview Tool Summary • Distinguish between a local optimum and a global optimum. The former is an optimum only in the immediate neighborhood, and the latter is an optimum in the entire feasible region. • Know that the product of two binary variables is, itself, a binary variable. And the product is 1 if, and only if, each separate binary variable is 1. Thus if x is a binary variable that is 1 when you mow the yard and y is a binary variable that is 1 when you wash the car, the product of x and y is 1 when you mow the yard and wash the car.

  5. Local and Global Optima • Local and Global Optima

  6. Local and Global Optima Overview Local and Global Optima of nonlinear programming problems are the optimal solutions either over a neighborhood of the feasible set (local), or over the entire feasible set (global). Because it is so hard for a computer program to solve some nonlinear programs, you need not compute any solutions.

  7. Local and Global Optima Nonlinear optimization problems are those optimization problems in which at least one term in the objective function or a constraint is nonlinear. Nonlinear terms include

  8. Local and Global Optima • A feasible solution is a local optimum if there are no other feasible solutions with a better objective function value in the immediate neighborhood. • For a maximization problem the local optimum corresponds to a local maximum. • For a minimization problem the local optimum corresponds to a local minimum. • A feasible solution is a global optimum if there are no other feasible points with a better objective function value in the feasible region. • Obviously, a global optimum is also a local optimum.

  9. Local and Global Optima • Nonlinear optimization problems can have multiple local optimal solutions, in which case we want to find the best local optimum. • Nonlinear problems with multiple local optima are difficult to solve with optimization software. • In these cases, the software can get “stuck” and terminate at a local optimum. • There can be a severe penalty (loss of objective) for confusing a local optimum for a global optimum. • Developing software to find global optima is an active research area.

  10. Local and Global Optima • Consider the function • The shape of this function is shown on the next slide. • The hills and valleys in the graph show that this function has several local maximums and local minimums. • There are three local maximums, one of which is the global maximum.

  11. Local and Global Optima

  12. Local and Global Optima • Consider the function • The shape of this function is shown on the next slide. • A function that is bowl-shaped down is called a concave function. • The maximum value for this particular function is 0 and the point (0, 0) gives the optimal value of 0. • Functions such as this one have a single local maximum that is also a global maximum. • A nonlinear problem with that type of objective function is relatively easy to maximize.

  13. Local and Global Optima

  14. Local and Global Optima • Consider the function • The shape of this function is shown on the next slide. • A function that is bowl-shaped up is called a convex function. • The minimum value for this particular function is 0 and the point (0, 0) gives the optimal value of 0. • Functions such as this one have a single local minimum that is also a global minimum. • This type of nonlinear problem is relatively easy to minimize.

  15. Local and Global Optima 40 Z 20 4 4 2 2 0 0 -2 -2 -4 -4 X Y

  16. Approximation Errors • Approximation Errors

  17. Approximation Errors • Overview • Approximation Errors are typically measured as the sum of squares of the differences between predicted or desired values and actual values, such as the difference between the returns on an Index Fund and the past performance of a broad market index, such as the S&P 500.

  18. Approximation Errors • Index funds are a very popular investment vehicle in the mutual fund industry. • Vanguard 500 Index Fund is the largest mutual fund in the U.S. with over $70 billion in net assets in 2005. • An index fund is an example of passive asset management, which emphasizes past performance. • The key idea behind an index fund is to construct a portfolio of stocks, mutual funds, or other securities that closely matches the past performance of a broad market index such as the S&P 500. • Behind the popularity of index funds is research that basically says “you can’t beat the market”.

  19. Approximation Errors Question: LymannBrothers has a substantial number of clients who wish to own a mutual fund portfolio that closely matches the performance of the S&P 500 stock index. A manager at Lymann Brothers has selected five mutual funds (shown on the next slide) that will be considered for inclusionin the portfolio. The manager must decide what percentage of the portfolio should be invested in each mutual fund.

  20. Approximation Errors • Mutual Fund Performance in 4 Selected Years Annual Returns (Planning Scenarios) Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 25.64 27.62 5.80 -3.13 Large-Cap Blend 15.31 18.77 11.06 4.75 Mid-Cap Blend 18.74 18.43 6.28 -1.04 Small-Cap Blend 14.19 12.37 -1.92 7.32 Intermediate Bond 7.88 9.45 10.56 3.31 S&P 500 13.00 12.00 7.00 2.00

  21. Approximation Errors Formulate and solve a non-linear programming problem to determine what percentage of the portfolio should be invested in each mutual fund so that the portfolio closely matches the performance of the S&P 500 stock index. Use the sum of squared deviations to measure the closeness of the portfolio to the index.

  22. Approximation Errors Answer: Define the 9 Decision Variables IS = proportion of portfolio invested in international stock LC = proportion of portfolio invested in large-cap blend MC = proportion of portfolio invested in mid-cap blend SC = proportion of portfolio invested in small-cap blend IB = proportion of portfolio invested in intermediate bond R1 = portfolio return for scenario 1 (year 1) R2 = portfolio return for scenario 2 (year 2) R3 = portfolio return for scenario 3 (year 3) R4 = portfolio return for scenario 4 (year 4)

  23. Approximation Errors Fund Year 1 Year 2 Year 3 Year 4 IS 25.64 27.62 5.80 -3.13 LC 15.31 18.77 11.06 4.75 MC 18.74 18.43 6.28 -1.04 SC 14.19 12.37 -1.92 7.32 IB 7.88 9.45 10.56 3.31 SP 13.00 12.00 7.00 2.00 • Define the objective function Min (R1 – 13)2 + (R2 – 12)2 + (R3 – 7)2 + (R4 – 2)2 • Define the constraints (including non-negativity) 25.64IS + 15.31LC + 18.74MC + 14.19SC + 7.88IB = R1 27.62IS + 18.77LC + 18.43MC + 12.37SC + 9.45IB = R2 5.80IS + 11.06LC + 6.28MC - 1.92SC + 10.56IB = R3 - 3.13IS + 4.75LC - 1.04MC + 7.32SC + 3.31IB = R4 IS + LC + MC + SC + IB = 1 IS, LC, MC, SC, IB> 0

  24. Approximation Errors • Optimal Solution for Lymann Brothers Example • R1 = 12.51 (12.51% portfolio return for scenario 1) • R2 = 12.90 (12.90% portfolio return for scenario 2) • R3 = 7.13 ( 7.13% portfolio return for scenario 3) • R4 = 2.51 ( 2.51% portfolio return for scenario 4) • IS = 0 ( 0.0% of portfolio in international stock) • LC = 0 ( 0.0% of portfolio in large-cap blend) • MC = .332 (33.2% of portfolio in mid-cap blend) • SC = .161 (16.1% of portfolio in small-cap blend) • IB = .507 (50.7% of portfolio in intermediate bond) • Lymann Brothers Portfolio Return vs. S&P 500 Return Scenario Portfolio Return S&P 500 Return 1 12.51 13.00 2 12.90 12.00 3 7.13 7.00 4 2.51 2.00

  25. Nonlinear Assignment • Nonlinear Assignment

  26. Nonlinear Assignment • Overview • Nonlinear Assignment Problems are assignment problems where the objective is nonlinear. For example, minimizing the distance times weight of cargo assigned to loading docks.

  27. Nonlinear Assignment • The port of Long Beach has three available loading docks. The distance (in meters) between the loading docks is giving in the table below: • 1 2 3 • 1 | 0 100 150 • 2 | 100 0 50 • 3 | 150 50 0 • Three tankers currently at sea are coming into Long Beach. Each tanker needs its own dock. Currently tankers 2 and 3 are empty. Tanker 1 has cargo that must be loaded onto the other tankers: 60 tons to Tanker 2 and 80 tons to Tanker 3. • Formulate then solve a non-linear programming problem to minimize the product of tonnage times distance, added across all tankers.

  28. Nonlinear Assignment • This is an extension of the assignment problems considered earlier. Like assigning “agents” to “tasks”, now assign tankers to docks. • Using standard notation for an assignment problem, let XIJ = 1 if Tanker I is assigned to Dock J, and 0 if not. • The constraints are just like the standard assignment problems. • ! Each tanker must be assigned to a dock; • X11 + X12 + X13 = 1; !Tanker 1; • X21 + X22 + X23 = 1; !Tanker 2; • X31 + X32 + X33 = 1; !Tanker 3; • ! Each dock can be assigned to at most one tanker; • X11 + X21 + X31 < 1; !Dock 1; • X12 + X22 + X32 < 1; !Dock 2; • X13 + X23 + X33 < 1; !Dock 3;

  29. Nonlinear Assignment • The difference from the standard assignment problem is the objective function is now nonlinear. The reason is the cost of assigning Tanker I to Dock J is no longer constant. For example, the cost of assigning Tanker 3 to Dock J depends on where Tanker 1 is assigned, because that assignment affects how far the 80 tons of cargo must move from Tanker 1 to Tanker 3. • For example, consider the result of assigning Tanker 1 to Dock 2 and Tanker 3 to Dock 1. • The distance between loading Docks 1 and 2 is 100 meters. And Tanker 1 must transfer 80 tons of goods to Tanker 3. This means that 80 tons must be moved 100 meters. To capture that in the objective function, write 100*80*X12*X31. Since both X12 = 1 and X31 = 1 their product is 1, and the product of distance and tonnage moved (100*80) is the term 100*80*X12*X31.

  30. Nonlinear Assignment • The complete objective function is •  MIN = 100*60*X11*X22 + 150*60*X11*X23 • + 100*60*X12*X21 + 50*60*X12*X23 • + 150*60*X13*X21 + 50*60*X13*X22 • ! Those are the costs of moving 60 tons from Tanker 1 to 2; • + 100*80*X11*X32 + 150*80*X11*X33 • + 100*80*X12*X31 + 50*80*X12*X33 • + 150*80*X13*X31 + 50*80*X13*X32; • ! Those are the costs of moving 80 tons from Tanker 1 to 3; • For example, compute the cost of assigning Tanker 1 to Dock 2, Tanker 2 to Dock 3, and Tanker 3 to Dock 1. • Variables are X12 = X23 = X31 = 1, with all other XIJ = 0. • The objective function is then • 100*60*0*0 + 150*60*0*1 • + 100*60*1*0 + 50*60*1*1 + … = 50*60 + 100*80 = 11,000

  31. Nonlinear Assignment • Optimal assignment: • Tanker 1 to Dock 2 • Tanker 2 to Dock 1 • Tanker 3 to Dock 3 @BIN(X33); specifies X33 as a binary variable

  32. Variance • Variance

  33. Variance Overview Variance of an investment portfolio is measured as the expected value of the sum of squares of the differences between expected rates of return and actual rates of return.

  34. Variance • The key tradeoff in most portfolio optimization models is between risk and return. • The index fund model (Lymann Brothers example) presented earlier managed the tradeoff passively (ex post, or after the fact). • The Markowitz mean-variance portfolio model provides a very convenient way for an investor to trade off risk versus return actively (ex ante, or before the fact). • We demonstrate the Markowitz portfolio model by extending the Lymann Brothers example.

  35. Variance • In the Lymann Brothers example there were four scenarios and the return under each scenario was defined by the variables R1, R2, R3, and R4. • If ps is the probability of scenario s, and there are n scenarios, then the expected return for the portfolio R is • If we assume that the four scenarios in the Lymann Brothers model are equally likely, then

  36. Variance • The measure of risk most often associated with the Markowitz model is the variance of the portfolio. • For our example, the portfolio variance is • For our example, the four planning scenarios are equally likely. Thus,

  37. Variance • The portfolio variance is the average of the sum of the squares of the deviations from the mean value under each scenario. • The larger the variance value, the more widely dispersed the scenario returns are around the average return value. • If the portfolio variance equaled zero, then every scenario return Ri would have to equal.

  38. Variance • There are two alternative ways to formulate the Markowitz model: • (1) Minimize the variance of the portfolio subject to constraints on the expected return, and • (2) Maximize the expected return of the portfolio subject to a constraint on risk (we did that in a previous lesson). • We demonstrate the first (1) formulation, assuming that Lymann Brothers’ client requires the expected portfolio return to be at least 9 percent.

  39. Approximation Errors Question: LymannBrothers has a substantial number of clients who wish to own a mutual fund portfolio with minimum variance and an expected return on 9 percent. A manager at Lymann Brothers has selected five mutual funds (shown on the next slide) that will be considered for inclusionin the portfolio. The manager must decide what percentage of the portfolio should be invested in each mutual fund.

  40. Approximation Errors • Mutual Fund Performance in 4 Selected Years Annual Returns (Planning Scenarios) Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 25.64 27.62 5.80 -3.13 Large-Cap Blend 15.31 18.77 11.06 4.75 Mid-Cap Blend 18.74 18.43 6.28 -1.04 Small-Cap Blend 14.19 12.37 -1.92 7.32 Intermediate Bond 7.88 9.45 10.56 3.31

  41. Approximation Errors Formulate and solve a non-linear programming problem to determine what percentage of the portfolio should be invested in each mutual fund so that the portfolio has minimum variance and an expected return of 9 percent. Assume all scenarios are equally likely.

  42. Approximation Errors Answer: Define the 9 Decision Variables IS = proportion of portfolio invested in international stock LC = proportion of portfolio invested in large-cap blend MC = proportion of portfolio invested in mid-cap blend SC = proportion of portfolio invested in small-cap blend IB = proportion of portfolio invested in intermediate bond R1 = portfolio return for scenario 1 (year 1) R2 = portfolio return for scenario 2 (year 2) R3 = portfolio return for scenario 3 (year 3) R4 = portfolio return for scenario 4 (year 4)

  43. Variance • Define the Objective Function Minimize the portfolio variance: • Constrain the return for • each scenario: 25.64IS + 15.31LC + 18.74MC + 14.19SC + 7.88IB = R1 27.62IS + 18.77LC + 18.43MC + 12.37SC + 9.45IB = R2 5.80IS + 11.06LC + 6.28MC- 1.92SC + 10.56IB = R3 - 3.13IS + 4.75LC- 1.04MC + 7.32SC + 3.31IB = R4 Fund Year 1 Year 2 Year 3 Year 4 IS 25.64 27.62 5.80 -3.13 LC 15.31 18.77 11.06 4.75 MC 18.74 18.43 6.28 -1.04 SC 14.19 12.37 -1.92 7.32 IB 7.88 9.45 10.56 3.31

  44. Variance • Constrain that all money be invested: IS + LC + MC + SC + IB = 1 • Define the expected return for the portfolio: • Constrain the expected portfolio return to be at least 9 percent: • Constrain non-negativity:IS, LC, MC, SC, IB> 0

  45. Variance • Optimal Solution • R1 = 10.63% portfolio return for scenario 1 • R2 = 12.20% portfolio return for scenario 2 • R3 = 8.93% portfolio return for scenario 3 • R4 = 4.24% portfolio return for scenario 4 • Rbar = 9.00% expected portfolio return • IS = 0.00% of portfolio in international stock • LC = 25.10% of portfolio in large-cap blend • MC = 0.00%of portfolio in mid-cap blend • SC = 14.1% of portfolio in small-cap blend • IB = 60.8% of portfolio in intermediate bond 100.0% of portfolio

  46. Semi Variance • Semi Variance

  47. Semi Variance Overview Semi Variance of an investment portfolio modifies Variance by measuring only the times when actual rates of return fall short of expected rates of return. Those are the times when unexpected returns are bad. .

  48. Semi Variance • Most investors are happy when their returns are above average, but want to avoid returns below average. • In the previous Markowitz porfolio optimization problem, we minimized variance, which is the sum of squares of the deviations from the mean. • Now change the objective to minimize the semi-variance, which is the sum of squares of those deviations from the mean that are below the mean.

  49. Semi Variance Fund Year 1 Year 2 Year 3 Year 4 IS 25.64 27.62 5.80 -3.13 LC 15.31 18.77 11.06 4.75 MC 18.74 18.43 6.28 -1.04 SC 14.19 12.37 -1.92 7.32 IB 7.88 9.45 10.56 3.31 SP 13.00 12.00 7.00 2.00 • As before, constrain the return • for each scenario: 25.64IS+ 15.31LC + 18.74MC + 14.19SC + 7.88IB = R1 27.62IS+ 18.77LC + 18.43MC + 12.37SC + 9.45IB = R2 5.80IS+ 11.06LC + 6.28MC- 1.92SC + 10.56IB = R3 -3.13IS+ 4.75LC- 1.04MC + 7.32SC + 3.31IB = R4 • Then, define and constrain the expected return for the portfolio:

  50. Semi Variance • But instead of minimizing the variance • Decompose deviations into positive or negative terms • R1 - Rbar = DP1 – DN1, • R2 - Rbar = DP2 – DN2, • R3 - Rbar = DP3 – DN3, • R4 - Rbar = DP4 – DN4. • Then minimize the sum of the squares of the negative deviations, which is the semi-variance.

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