(octave.info)Quadratic Programming

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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 (Note: 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: Note: lsqnonneg, Note: qp, Note:
     optimset.