Module qps_ipopt

Quadratic Program Solver based on IPOPT.

Uses IPOPT to solve the following QP (quadratic programming) problem:

   min 1/2 x'*H*x + c'*x
    x

subject to:

   l <= A*x <= u       (linear constraints)
   xmin <= x <= xmax   (variable bounds)

Inputs (all optional except H, C, A and L):

• H : matrix (possibly sparse) of quadratic cost coefficients
• C : vector of linear cost coefficients
• A, l, u : define the optional linear constraints. Default values for the elements of l and u are -Inf and Inf, respectively.
• xmin, xmax : optional lower and upper bounds on the x variables, defaults are -Inf and Inf, respectively.
• x0 : optional starting value of optimization vector x
• opt : optional options structure with the following fields, all of which are also optional (default values shown in parentheses)
  • verbose (0) - controls level of progress output displayed
    • 0 = no progress output
    • 1 = some progress output
    • 2 = verbose progress output
  • max_it (0) - maximum number of iterations allowed
    • 0 = use algorithm default
  • ipopt_opt - options struct for IPOPT, values in verbose and max_it override these options
• problem : The inputs can alternatively be supplied in a single problem dict with fields corresponding to the input arguments described above: H, c, A, l, u, xmin, xmax, x0, opt

Outputs:

• x : solution vector
• f : final objective function value
• exitflag : exit flag
  • 1 = first order optimality conditions satisfied
  • 0 = maximum number of iterations reached
  • -1 = numerically failed
• output : output struct with the following fields:
  • iterations - number of iterations performed
  • hist - dict list with trajectories of the following: feascond, gradcond, compcond, costcond, gamma, stepsize, obj, alphap, alphad
  • message - exit message
• lmbda : dict containing the Langrange and Kuhn-Tucker multipliers on the constraints, with fields:
  • mu_l - lower (left-hand) limit on linear constraints
  • mu_u - upper (right-hand) limit on linear constraints
  • lower - lower bound on optimization variables
  • upper - upper bound on optimization variables

Calling syntax options:

   x, f, exitflag, output, lmbda =             qps_ipopt(H, c, A, l, u, xmin, xmax, x0, opt)

   x = qps_ipopt(H, c, A, l, u)
   x = qps_ipopt(H, c, A, l, u, xmin, xmax)
   x = qps_ipopt(H, c, A, l, u, xmin, xmax, x0)
   x = qps_ipopt(H, c, A, l, u, xmin, xmax, x0, opt)
   x = qps_ipopt(problem), where problem is a struct with fields:
                   H, c, A, l, u, xmin, xmax, x0, opt
                   all fields except 'c', 'A' and 'l' or 'u' are optional
   x = qps_ipopt(...)
   x, f = qps_ipopt(...)
   x, f, exitflag = qps_ipopt(...)
   x, f, exitflag, output = qps_ipopt(...)
   x, f, exitflag, output, lmbda = qps_ipopt(...)

Example:

   H = [   1003.1  4.3     6.3     5.9;
           4.3     2.2     2.1     3.9;
           6.3     2.1     3.5     4.8;
           5.9     3.9     4.8     10  ]
   c = zeros((4, 1))
   A = [   1       1       1       1
           0.17    0.11    0.10    0.18    ]
   l = [1, 0.10]
   u = [1, Inf]
   xmin = zeros((4, 1))
   x0 = [1, 0, 0, 1]
   opt = {'verbose': 2)
   x, f, s, out, lam = qps_ipopt(H, c, A, l, u, xmin, [], x0, opt)

Problem from http://www.jmu.edu/docs/sasdoc/sashtml/iml/chap8/sect12.htm