What is quadratic programming?

Quadratic programming is a method used to optimize the multivariable quadratic function that may or may not be limited linearly. Many real world problems, such as the company's portfolio or the manufacturer's cost, can be described using a quadratic program. If the objective function is convex, there may be a feasible solution and can be solved by known algorithms such as an extended simplex algorithm. There are methods for solving some non -cellular quadratic functions, but are complicated and easily accessible.

Mathematical optimization techniques are used in quadratic programming to minimize objective function. The objective function consists of a number of decision variables that may or may not be bordered. The variable decision -making has powers of 0, 1 or 2. Objective functions can be subjected to a number of limitation of linear equality and inequalities related to variable decision -making. An example of quadratic programming is: Minimizef (x, y) = x

2 + 3Y

2 - 12y + 12, where x + y = 6 and x> 0 and y ≥ 0.

It is often interesting to use multidimensional quadratic functions to describe the real world problems. For example, using a modern portfolio theory, a financial analyst will try to optimize the company's portfolio by selecting the share of assets that minimize the risk associated with the expected yield. The quadratic equation composed of assets and correlation between assets describes the portfolio scattering that can be minimized by quadratic programming. Another example is the manufacturer that uses a quadratic equation to describe the relationship between different components of quality and product costs. The manufacturer may minimize costs while maintaining certain standards by adding linear restrictions to the quadratic program.

One of the most important conditions in solving quadratic program is the convexity of objective ROthe venue. The convexity of the quadratic function is determined by the Hessian or the matrix of its other derivatives. This feature is called convex if the Hessian matrix is ​​either positive or positive semi-define, that is, if all their own values ​​are positive or negative. If Hessian is positive and there is a feasible solution, then this local minimum is unique and is a global minimum. If the Hessian Polo-positive, a feasible solution may not be unique. Unconvex quadratic functions may have a local or global minimum, but it is more difficult to determine.

There are many approaches to solving convex quadratic function using quadratic programming. The most common approach is the extension of the Simplex algorithm. Some other methods include the inner point or barrier method, the active set method and the conjugated gradient method. These methods are integrated into certain programs such as Mathematica® and Matlab® and are available in libraries for many programming languages.