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### 25.2 Quadratic Programming

Octave can also solve Quadratic Programming problems, this is

    min 0.5 x'*H*x + x'*q

subject to

         A*x = b
         lb <= x <= ub
         A_lb <= A_in*x <= A_ub

[x, obj, info, lambda] = qp (x0, H)
[x, obj, info, lambda] = qp (x0, H, q)
[x, obj, info, lambda] = qp (x0, H, q, A, b)
[x, obj, info, lambda] = qp (x0, H, q, A, b, lb, ub)
[x, obj, info, lambda] = qp (x0, H, q, A, b, lb, ub, A_lb, A_in, A_ub)
[x, obj, info, lambda] = qp (…, options)

  Solve a quadratic program (QP).

  Solve the quadratic program defined by

      min 0.5 x'*H*x + x'*q
       x

  subject to

      A*x = b
      lb <= x <= ub
      A_lb <= A_in*x <= A_ub

  using a null-space active-set method.

  Any bound (A, b, lb, ub, A_in, A_lb, A_ub) may be set to the empty matrix (`[]`) if not present. The constraints A and A_in are matrices with each row representing a single constraint. The other bounds are scalars or vectors depending on the number of constraints. The algorithm is faster if the initial guess is feasible.

  options

    An optional structure containing the following parameter(s) used to define the behavior of the solver. Missing elements in the structure take on default values, so you only need to set the elements that you wish to change from the default.

    `MaxIter (default: 200)`

      Maximum number of iterations.

  info

    Structure containing run-time information about the algorithm. The following fields are defined:

    `solveiter`

      The number of iterations required to find the solution.

    `info`

      An integer indicating the status of the solution.

      0

        The problem is feasible and convex. Global solution found.

      1

        The problem is not convex. Local solution found.

      2

        The problem is not convex and unbounded.

      3

        Maximum number of iterations reached.

      6

        The problem is infeasible.

x = pqpnonneg (c, d)
x = pqpnonneg (c, d, x0)
x = pqpnonneg (c, d, x0, options)
[x, minval] = pqpnonneg (…)
[x, minval, exitflag] = pqpnonneg (…)
[x, minval, exitflag, output] = pqpnonneg (…)
[x, minval, exitflag, output, lambda] = pqpnonneg (…)

  Minimize `1/2*x'*c*x + d'*x` subject to `x >= 0`.

  c and d must be real matrices, and c must be symmetric and positive definite.

  x0 is an optional initial guess for the solution x.

  options is an options structure to change the behavior of the algorithm (see optimset. `pqpnonneg` recognizes one option: `"MaxIter"`.

  Outputs:

  x

    The solution matrix

  minval

    The minimum attained model value, `1/2*xmin'*c*xmin + d'*xmin`

  exitflag

    An indicator of convergence. 0 indicates that the iteration count was exceeded, and therefore convergence was not reached; >0 indicates that the algorithm converged. (The algorithm is stable and will converge given enough iterations.)

  output

    A structure with two fields:

    - `"algorithm"`: The algorithm used (`"nnls"`)
    - `"iterations"`: The number of iterations taken.
  lambda

    Undocumented output

  See also: lsqnonneg, qp, optimset.

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