# Complexity of a quadratic program

• Socal93
In summary, the conversation discusses the topic of computing the complexity of a quadratic program. The problem is defined as minimizing a quadratic objective function subject to linear constraints. The quadratic program is solved using the interior point method and the Wolf algorithm is mentioned as a method for reducing the problem to a sequence of linear programs. The complexity is said to refer to the computational complexity or the time it takes to solve the problem.
Socal93
I'm trying to compute the complexity of the quadratic program: $$\displaystyle\min_{\mathbf{X}} (\mathbf{X^TQX +C^TX}) \quad{} \text{subject to} \quad{} \mathbf{A X \leq Y}$$
A is MxN and X is Nx1. Q is positive definite and I'm using the interior point method. Any help in computing the complexity would be appreciated.

You have caught my curiosity. What is meant by the complexity of a quadratic program?

I'm trying to determine the computational complexity or the time it takes to solve the above problem.

Socal93 said:
I'm trying to determine the computational complexity or the time it takes to solve the above problem.

Don't know anything about that. I do know that the Wolf algorithm reduces the quadratic program to a finite sequence of linear programs.

The complexity of a quadratic program, such as the one described in the content, can be determined by analyzing the algorithm used to solve it. In this case, the interior point method is being used, which is a popular and efficient method for solving quadratic programs. The complexity of this method is typically O(n^3), where n is the number of decision variables.

In this specific problem, there are N decision variables and M constraints, so the complexity would be O(N^3 + MN^2). This is because the interior point method requires solving a linear system of equations in each iteration, which has a complexity of O(n^3), and the number of iterations is determined by the number of decision variables and constraints.

Furthermore, the complexity can also be affected by the specific characteristics of the problem, such as the positive definiteness of the matrix Q and the size of the matrices A and Y. Generally, the larger these matrices are, the higher the complexity will be.

In summary, the complexity of a quadratic program can vary depending on the specific problem and the algorithm used to solve it. However, in this case, with the interior point method and the given parameters, the complexity can be estimated to be O(N^3 + MN^2).

## What is a quadratic program?

A quadratic program is a mathematical optimization problem that involves maximizing or minimizing a quadratic function subject to linear constraints. It is commonly used in fields such as economics, engineering, and computer science.

## What makes a quadratic program complex?

A quadratic program can be complex due to the presence of multiple variables and constraints, as well as the non-linear nature of the objective function. Additionally, the solution to a quadratic program may not always be unique, making it more difficult to find an optimal solution.

## What is the difference between a linear program and a quadratic program?

The main difference between a linear program and a quadratic program is the shape of the objective function. A linear program has a linear objective function, while a quadratic program has a quadratic (or non-linear) objective function. This can significantly impact the complexity of the problem and the methods used to solve it.

## How can the complexity of a quadratic program be measured?

The complexity of a quadratic program can be measured by the number of variables and constraints, as well as the degree of the objective function. Additionally, the presence of non-convexity in the problem can also increase its complexity.

## What are some common methods for solving a quadratic program?

Some common methods for solving a quadratic program include the simplex method, interior point methods, and quadratic programming algorithms. These methods use mathematical techniques to iteratively find the optimal solution to the problem.

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