Exam 8: Nonlinear Optimization Models

The problem of maximizing a concave quadratic function over a linear constraint set is relatively difficult to solve.

Investment manager Max Gaines wishes to develop a mutual fund portfolio based on the Markowitz portfolio model. He needs to determine the proportion of the portfolio to invest in each of the five mutual funds listed below so that the variance of the portfolio is minimized subject to the constraint that the expected return of the portfolio be at least 4%. Formulate the appropriate nonlinear program. $\text { Annual Returns (Flanning Scenarios) }$ Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 26.73 22.37 6.46 -3.19 Large-Cap Blend 18.61 14.88 10.52 5.25 Mid-Cap Blend 18.04 19.45 15.91 -1.94 Small-Cap Blend 11.33 13.79 -2.07 6.85 Intermediate Bond 8.05 7.29 9.18 3.92

A feasible solution is a global optimum if there are no other feasible points with a better objective function value in the feasible region.

Any feasible solution to a blending problem with pooled components is feasible to the problem with no pooling.

Investment manager Max Gaines has several clients who wish to own a mutual fund portfolio that matches, as a whole, the performance of the S&P 500 stock index. His task is to determine what proportion of the portfolio should be invested in each of the five mutual funds listed below so that the portfolio most closely mimics the performance of the S&P 500 index. Formulate the appropriate nonlinear program. $\text { Annual Returns (Flanning Scenarios) }$ Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 26.73 22.37 6.46 -3.19 Large-Cap Blend 18.61 14.88 10.52 5.25 Mid-Cap Blend 18.04 19.45 15.91 -1.94 Small-Cap Blend 11.33 13.79 -2.07 6.85 Intermediate Bond 8.05 7.29 9.18 3.92 S\&P 500 Index 21.00 19.00 12.00 4.00

A convex function is

MegaSports, Inc. produces two high-priced metal baseball bats, the Slugger and the Launcher, that are made from special aluminum and steel alloys. The cost to produce a Slugger bat is $100, and the cost to produce a Launcher bat is $120. We can not assume that MegaSports will sell all the bats it can produce. As the selling price of each bat model -- Slugger and Launcher -- increases, the quantity demanded for each model goes down. Assume that the demand, S, for Slugger bats is given by S = 640 - 4P_S and the demand, L, for Launcher bats is given by L = 450 - 3P_L where P_S is the price of a Slugger bat and P_L is the price of a Launcher bat. The profit contributions are P_S S - 100S for Slugger bats and P_L L - 120L for Launcher bats. Develop the total profit contribution function for this problem.

Explain how the parameters required for the Bass new-product adoption model can be estimated when no historical data are available for the new product.

Many linear programming algorithms such as the simplex method optimize by examining only the extreme points of the feasible region.

The Markowitz mean-variance portfolio model presented in the text is a convex optimization problem.

Each point on the efficient frontier is the maximum possible risk, measured by portfolio variance, for the given return.

Components that share a storage facility are called

A feasible solution is a global optimum if there are no other feasible solutions with a better objective function value in the immediate neighborhood.

When components (or ingredients) in a blending problem must be pooled, the number of feasible solutions is reduced.

Explain how the local minimum, local maximum, local optimum, global minimum, global maximum, and global optimum relate to one another in nonlinear optimization problems.

Any feasible solution to a blending problem without pooled components is feasible to the problem with pooled components.

The function f (X, Y) = X ² + Y ² has a single global minimum and is relatively easy to minimize.

Financial planner Minnie Margin has a substantial number of clients who wish to own a mutual fund portfolio that matches, as a whole, the performance of the Russell 2000 index. Her task is to determine what proportion of the portfolio should be invested in each of the five mutual funds listed below so that the portfolio most closely mimics the performance of the Russell 2000 index. Formulate the appropriate nonlinear program. $\text { Annual Returns (Flanning Scenarios) }$ Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 22.37 26.73 4.86 2.17 Large-Cap Value 15.48 19.64 11.50 -5.25 Mid-Cap Value 17.42 20.07 -4.97 -1.69 Small-Cap Growth 23.18 12.36 3.25 3.81 Short-Term Bond 9.26 8.81 6.15 4.04 Russell 2000 Index 20.00 22.00 8.00 -2.00

Which of the following is not true regarding a concave function?

A function is quadratic if its nonlinear terms have a power of 4.