A wrapper around Ipopt.jl to solve quadratic programming problems.
This package provides the function quadprog which calls the
Ipopt.jl library for Interior Point Nonlinear Optimization. No
other solver is required.
Install via the package manager,
Pkg.add("Quadprog")
The provided function quadprog solves the problem
min 0.5 * x' * Q * x + c' * x
s.t. A * x <= b
Aeq * x == beq
lb <= x <= ub
start optimization at x0
where Q is a symmetric matrix.
Only the parameters Q and c are required. The others are keyword
arguments that could be omitted and/or given in any order.
The full signature of quadprog with its defaults is
quadprog(Q, c;
A = Array(Float64, (0, length(c))),
b = Array(Float64, 0),
Aeq = Array(Float64, (0, length(c))),
beq = Array(Float64, 0),
lb = -Inf*ones(length(c)),
ub = Inf*ones(length(c)),
x0 = zeros(length(c)))
To solve
min 0.5 * [x1 x2]' * [3 1; 1 4] * [x1 x2] + [5 6]' * x
without any constraints you call
(x, fx, status) = quadprog([3 1; 1 4], [5, 6])
The answer you should get is:
([-1.27273,-1.18182],-6.7272727272727275,1)
That means the solution is x==[-1.27273,-1.18182], the function value at the solution
is fx=-6.7272727272727275 and the status flag is 1.
See Ipopt.jl for a description of the status flags.
To solve the dual of a linear support vector machine (SVM), i.e.
max sum(alpha) - 0.5 * alpha' * diagm(y) * X * X' * diagm(y) * alpha
s.t. 0 <= alpha <= C
y' * alpha == 0
where X contains the training locations as rows and y contains the
training labels being either 1.0 or -1.0, you call
yX = diagm(y)*X
K = yX * yX'
(alpha, val, flag) = quadprog(K, -ones(n),
lb=zeros(n), ub=C*ones(n),
Aeq=y', beq=zeros(1),
x0=zeros(n))