js117 / CVX

Convex Optimization code collection of solved problems and algorithms

Convex Optimization is a powerful field of applied mathematics that can be used to solve many engineering analysis and design problems. The basic idea is this: we can efficiently optimize complicated nonlinear functions subject to inequality constraints and affine equality constraints if the functions are convex, a mathematical property. There exists a convergence theory giving accurate estimates on roughly how fast we can solve these problems. For more, see http://cvxr.com/ and http://web.stanford.edu/~boyd/cvxbook/

CVX finds applications in many fields ranging from machine learning, to signal processing, to finance, statistics, circuit design, communications, and network modeling. It is the superset of problems such as quadratic programming (e.g. least squares) and linear programming. Accompanying the theory are efficient interior-point methods. For prototyping and modeling purposes, a MATLAB framework called "CVX" has been developed allowing easy programming of problem descriptions to be input to a general solver system.

The CVX software allows you to specify problems of the following form:

variable [declare variables' size/structure]

minimize [objective function]

subject to [inequality constraints]

       [affine equality constraints, i.e. Ax == b]

The software then parses your problem and converts it to a form acceptable by a general-purpose solver. Here's an example program:

cvx_begin

variable x(n);

minimize( norm(A*x-b) );

subject to

	C*x == d;
	
	norm(x,Inf) <= 1;

cvx_end

This program minimizes the norm of the difference of Ax and vector b subject to equality constraints Cx == d, and that the infinity norm (i.e. max absolute component value) of our vector variable x has to be <= 1. Running this code in your MATLAB script will invoke the CVX engine and return an optimal value of the problem, and the optimal solution vector x. The matrices A, C and vector b are problem data that can be declared outside the CVX scope. For more, see http://web.cvxr.com/cvx/doc/

In this repository are some of the practical optimization problems I've solved using CVX, as well as basic implementation of a few optimization algorithms. Examples include:

• Maximum likelihood estimation of sports team tournament results
• Fitting an ellipsoid to data points
• Rational function approximation of an exponential distribution
• Signal processing: removal of Gaussian noise
• Linear separation of trinary data
• Maximum volume rectangle inside a polyhedron (i.e. a form of "centering")
• Gradient descent algorithm template
• Newton method algorithm template

Other applications I've come across that are framed readily and solved efficiently as convex optimization problems include:

• Support vector machines (i.e. maximum margin classifier & approximate linear separation of sets)
• Network flow optimization
• Portfolio optimization using a k-factor risk model

A quick summary of convex optimization as a software tool:

1. Allows one to frame and analyze complicated problems easily,
2. Wide ranging applications encompassed by the generality,
3. Enables very fast prototyping with the CVX framework,
4. Theory extends to implementation details involving problem structure. These expose efficiency opportunities accessible through numerical linear algebra, using existing packages like LAPACK and BLAS.