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November 2014

Quadprog.jl

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.

Build Status

Installation

Install via the package manager,

Pkg.add("Quadprog")

Quadratic programming

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)))

Example 1

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.

Example 2

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))