Non-Linear Programming

Posted: August 26th, 2021

Non-Linear Programming

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Non-Linear Programming

Question 1: Explain How Quadratic Programming Is Used In the Real World

Quadratic programming involves solving of special types of the optimization problem in mathematics, which specifically include linear constrained quadratic optimization problem. Quadratic programming is widely applied in the real-world to facilitate resource optimization(Fabozzi &Markowitz, 2011). In this case, optimization situations are modeled to suit real-life situations. The following is a case used by my company in managing portfolio risks.The case involves risk minimization.

Subject to;

Where C representsthe covariance matrix of rates of return and n-dimensional row vector that describes coefficients of linear terms in the objective quadratic terms. Any existing constant is dropped from the model. Decision variables are represented by n-dimensional column vector x. The constraints are defined by (m x n) A matrix and m- dimensional column vector b. These are on the right-hand side of the coefficients. Thus, based on the equations, it is assumed that there is a feasible solution, and the constraint region is bounded.

Question 2:  Explain How Markowitz Theory of Portfolio Management (MPT) and Options Pricing Models are related to Quadratic Programming

The two models are about building a framework for portfolio risks and returnanalysisas well as their inter-relationships. As such, management of portfolio encompasses choosing an optimal structure to maximize returns while minimizing risks(Fabozzi &Markowitz, 2011). As a result, the two models or theories employ the knowledge of quadratic programming to build a structure that would aid minimize risks while maximizing the risks(Dostál, 2009). For instance, if the portfolio contains n types of assets. The rate of return of the assets, y is a random variable with an expected value of my. The problem is to establish a fraction of xy that should be invested in each of y assets to help minimize risks. It is subjected to a particular minimum expected return rate. If C is the covariance matrix of asset returns, then, the problem can be formulated as:

Minimize portfolio risk, Z=

Subject to constraints. The expected return should be higher or equal to the minimum rate of portfolio return, p that an investor desires, such that;

, y = 1 such that sum of x investments should add up to 1, and they are bound as; . Thus, the objective for minimizing portfolio risk is a quadratic equation, having linear constraints. The optimization problem is a quadratic program.

References

Dostál, Z. (2009). Optimal quadratic programming algorithms: with applications to variational inequalities. New York: Springer.

Fabozzi, F. & Markowitz, H. (2011). The theory and practice of investment management: asset allocation, valuation, portfolio construction, and strategies. Hoboken, N.J: John Wiley & Sons.