I need at least 500 words Initial Post. 250 words for each question. No Plagiarism. Due in 12 hours. I will also attached the replies of other students once they are available. I need 0.5 page for each reply.
Write a minimum of 250 words for each of the discussion questions below: A quadratic programming model is an optimization model with n decision variables and m linear constraints, and of the form:
Don't use plagiarized sources. Get Your Custom Essay on
week 6 Stats Discussion
Just from $13/Page
Minimize Z = (1/2) xT Q x + CT x
Subject to: A x ≥ b
x ≥ 0
where x is the n by 1 column vector of decision variables and xT is its transpose, Q is an n by n symmetric matrix of objective parameters, C is an n by 1 vector of additional objective parameters, A is an m by n matrix of constraints parameters, and b is a m by 1 vector of constraints right hand side.
- Explain how quadratic programming is used in the real world. Provide a specific example from your own line of work, or a line of work that you find particularly interesting. Indicate explicitly and qualitatively what Z, x, Q, C, A, and b are in your example.
- Harry Markowitz, and Myron Scholes along with Robert Merton are the Nobel Laureates in Economics in 1990 (Markowitz) and 1996 (Scholes and Merton) respectively. Markowitz won the Nobel award for devising his Modern Portfolio Theory (also called: the Markowitz Portfolio Theory – MPT) in 1952. Scholes and Merton were the recipients of Nobel for their Option Pricing and Volatility models introduced in early 1980s. Explain how each of the above two models (MPT and Option Pricing) are related to quadratic programming. Describe the decision variables, the objective function, and the constraints for each model.
In your two replies to classmates describe instances, scenarios and conditions under which their quadratic programming model example may not have (i) a feasible solution; (ii) a unique solution.
Abraham Karmel
Module 6
COLLAPSE
Top of Form
An example of how quadratic programming used in the real world is in the optimal operation of a hydroelectric dam. The objective is to maximize revenue based on electricity production from the water flow, coming from the reservoir and passing through the turbine. The decision variables include the water level of the storage area of the reservoir, the volume of water to allow to pass through the turbine, and the volume to allow to pass through the spillway. The higher the water level in the storage, the greater the water pressure created when water is allowed through the turbine. The less water allowed through the spillway, the more is forced through the turbine, hence greater production of electricity. These value are all dependent on the current sale price of electricity for selling what the dam produces and the overall water level of the reservoir. One of the constraints are the maximum amount of water to allow down the stream. Through the use of quadratic programming, the operators of the dam are able to optimize the levels of water storage and water flow in order to maximize electricity production revenue.
The Markowitz Portfolio Theory is a formula used in investments to maximize stock returns while minimizing risk (Pistorius, 2017). It’s very much related to quadratic programming because the risk involved for one stock is often correlated with the risk of the other stock. This is because prices of stocks are often dependent on similar factors and what affects one would affect both, either for better or for worse. The decision variables are the percentage of each stock of the investment portfolio to invest in. The objective function is to maximize return while minimizing risk. The constraint is the maximum risk the investor is willing to tolerate.
Reference
s
DeLand, S. (2020). Optimization in MATLAB: An Introduction to Quadratic Programming. Retrieved from https://www.mathworks.com/matlabcentral/fileexchange/35856-optimization-in-matlab-an-introduction-to-quadratic-programming
Pistorius, T. Heterodox Investment Theory, DOI 10.1007/978-3-319-55005-3_2. Retrieved from https://link-springer-com.ezproxy.neu.edu/content/pdf/10.1007%2F978-3-319-55005-3_2
Bottom of Form
Shanna Mckinney
Quadratic Programming
COLLAPSE
Top of Form
Quadratic programming is a type of constrained nonlinear programming model. (Heider 2013) Constraints are limits or conditions that are imposed on the potential results. For instance, a monthly budget can be set, constraining how much can be spent toward inventory. There could be a limit on the number of hours a staff member can work during the week. The types of constraints are numerous.
Quadratic programming is the mathematical problem of finding a vector Z that minimizes a quadratic function. It can be used in just about any real-world situation where various constraints exist with varying degrees of impact. COVID 19 is very prevalent today. Something hospitals may be thinking about is the effect this pandemic is having on its medical staff. Quadratic programming could be used to determine the optimal number of staff they should have on each shift. Decision variables could include (but not limited to) the amount of hours the department of labor (DOL)has determined doctors and nurses are allowed to work in a 24 hour period, number of shifts that are in a 24 hour period (8hour or 12 hour), and the various departments that must be staffed per shift. The objective function is to have the optimal number of staff members to run a shift. Some of the constraints can be DOL limit on hours worked within 24 hours, the number of hours medical staff have already worked in 24 hours, and the specialty of staff members in relation to what units need to be staffed. Z would be the optimal number of staff per shift and x would be the number of shifts. Q is the matrix capturing the covariance of the function. C will represent the DOL hours worked constraint. A will represent the available shifts to work and B will represent the specialties that need to be staffed.
Harry Markowitz created the Modern Portfolio Theory (MPT) in 1952. “Modern Portfolio Theory is Markowitz’s theory regarding maximizing the return investors could get in their investment portfolio considering the risk involved in the investments. “ (Miller 2019) MPT is a great example of quadratic programming. The decision variable could be the level of risk the investor is willing to take. The objective function is to maximize returns while minimizing the level of risk the portfolio takes on. There are several different constraints that could exist; the number of investments the client wants to hold, whether socially responsible investments should be considered, whether they want only domestic investments or are willing to invest internationally, as well as the number of years the investor has to invest.
Fisher Black, Myron Scholes and Robert Merton created the Option Pricing and Volatility model in 1973. Purchasing options is a hedge strategy designed to limit risk and increase potential profits if one must purchase the underlying stock. In their model known as Black-Scholes, “the value of an option depends on the future volatility of a stock rather than on its expected return.” (MEZOFI, B AND SZABO, K 2018) . With options investing your initial loss is limited to the amount spent on purchasing the option. You are then required to buy or sell the underlying stock if the option is exercised in the money. The decision variable is whether the option will create an overall profit or loss. The objective function is to maximize profit spending the least amount of money. One constraint could be whether they can short the stock or not. Not all firms allow shorting (selling option without owning the underlying stock). The investor may want to limit the number of options they have open at any give time. The investor may only want to purchase the option if it is trading at a certain amount. There could also be instances when the firm limits the side of the option that can be purchased.
Reference
Heider, J. (nd 2015). Quadratic programming. Retrieved from:
https://optimization.mccormick.northwestern.edu/index
MEZOFI, B AND SZABO, K (NOVEMBER 23, 2018). BEYOND BLACK-SCHOLES: A NEW OPTION FOR OPTIONS PRICING. Advances in computing and machine learning have made it possible to employ a new, data-driven approach to pricing options. Retrieved from: https://www.weareworldquant.com/en/thought-leadership/beyond-black-scholes-a-new-option-for-options-pricing/
Miller, T. (March 22, 2019). What Is Modern Portfolio Theory (MPT) and Why Is It Important? Retrieved from:
https://www.thestreet.com/investing/modern-portfolio-theory-14903955
Bottom of Form
