## Lagrangian Support Vector Machines

*O. L. Mangasarian, David R. Musicant*;
1(Mar):161-177, 2001.

### Abstract

An implicit Lagrangian for the dual of a simple reformulation of the
standard quadratic program of a linear support vector machine is
proposed. This leads to the minimization of an unconstrained
differentiable convex function in a space of dimensionality equal to
the number of classified points. This problem is solvable by an
extremely simple linearly convergent Lagrangian support vector machine
(LSVM) algorithm. LSVM requires the inversion at the outset of a
single matrix of the order of the much smaller dimensionality of the
original input space plus one. The full algorithm is given in this
paper in 11 lines of MATLAB code without any special optimization
tools such as linear or quadratic programming solvers. This LSVM code
can be used "as is" to solve classification problems with millions of
points. For example, 2 million points in 10 dimensional input space
were classified by a linear surface in 82 minutes on a Pentium III 500
MHz notebook with 384 megabytes of memory (and additional swap space),
and in 7 minutes on a 250 MHz UltraSPARC II processor with 2 gigabytes
of memory. Other standard classification test problems were also
solved. Nonlinear kernel classification can also be solved by LSVM.
Although it does not scale up to very large problems, it can handle
any positive semidefinite kernel and is guaranteed to converge. A
short MATLAB code is also given for nonlinear kernels and tested on a
number of problems.

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