Implementing a linear, binary SVM (support vector machine)

Are you interested in using kernels or not? Without kernels, the best way to solve these kinds of optimization problems is through various forms of stochastic gradient descent. A good version is described in http://ttic.uchicago.edu/~shai/papers/ShalevSiSr07.pdf and that has an explicit algorithm.

The explicit algorithm does not work with kernels but can be modified; however, it would be more complex, both in terms of code and runtime complexity.

Have a look at liblinear and for non linear SVM's at libsvm

Some pseudocode for the Sequential Minimal Optimization (SMO) method can be found in this paper by John C. Platt: Fast Training of Support Vector Machines using Sequential Minimal Optimization. There is also a Java implementation of the SMO algorithm, which is developed for research and educational purpose (SVM-JAVA).

Other commonly used methods to solve the QP optimization problem include:

• constrained conjugate gradients
• interior point methods
• active set methods

But be aware that some math knowledge is needed to understand this things (Lagrange multipliers, Karush–Kuhn–Tucker conditions, etc.).