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ConstrainedOptPack::QPSchurPack::QP Class Reference

Represents the QP to be solved by QPSchur {abstract}. More...

#include <ConstrainedOptPack_QPSchur.hpp>

Inheritance diagram for ConstrainedOptPack::QPSchurPack::QP:

[legend]
List of all members.

Public Types

typedef vector_one_based_checked<
EBounds > 		x_init_t
 
typedef vector_one_based_checked<
size_type > 		l_x_X_map_t
 
typedef vector_one_based_checked<
size_type > 		i_x_X_map_t
 
typedef QPSchurPack::Constraints Constraints
 

Public Member Functions

virtual ~QP ()
 
virtual size_type n () const =0
 
virtual size_type m () const =0
 
virtual const DVectorSlice g () const =0
 
virtual const MatrixSymOp & G () const =0
 
virtual const MatrixOp & A () const =0
 If m == 0 then don't call this, it may throw an exception or worse.
virtual size_type n_R () const =0
 
virtual const x_init_t & x_init () const =0
 Return the status of a variable initially.
virtual const l_x_X_map_t & l_x_X_map () const =0
 Map from full x(i) to initially fixed x_X(l).
virtual const i_x_X_map_t & i_x_X_map () const =0
 Map from initially fixed x_X(l) to full x(i).
virtual const DVectorSlice b_X () const =0
 
virtual const GenPermMatrixSlice & Q_R () const =0
 (Q_R().ordered_by() == BY_ROW)
virtual const GenPermMatrixSlice & Q_X () const =0
 (Q_X().ordered_by() == BY_ROW)
virtual const MatrixSymOpNonsing & Ko () const =0
 
virtual const DVectorSlice fo () const =0
 
virtual Constraints & constraints ()=0
 
virtual const Constraints & constraints () const =0
 
virtual void dump_qp (std::ostream &out)
 Dump the definition of the QP to a stream.

Detailed Description

Represents the QP to be solved by QPSchur {abstract}.

In order to solve a QP, clients must define subclasses for this interface and the Constraints interface defined later. This is where the specialized properties of the QP are exploited. For certain types of QPs, standard implementation classes can be defined.

Here the QP is of the form: 


  (1.a)	min		g'*x + 1/2*x'*G*x
  (1.b)	s.t.	A'*x = c
  (1.c)			cl_bar <= A_bar'*x <= cu_bar

  where:
    x <: R^n
    g <: R^n
    G <: R^(n x n)
    A <: R^(n x m)
    c <: R^m
    A_bar <: R^(n x m_bar)
    cl_bar, cu_bar <: R^m_bar
Above, cl_bar <= A_bar'*x <= cu_bar may represent variable bounds, general inequalities and equality constraints and these constraints are represented by the class Constraints.

When solving the above QP using the schur complement QP solver we start out with a KKT system with a subset of variables initially fixed at a bound: 


  [ G_RR    G_RX     A_F     0  ] [ x_R    ]   [ - g_R ]
  [ G_RX'   G_XX     A_X     I  ] [ x_X    ]   [ - g_X ]
  [ A_R'    A_X'      0      0  ]*[ lambda ] = [   c   ]
  [           I       0      0  ] [ mu_X   ]   [   b_X ]
We can simplify the above system by solving for the initially fixed variables and removing them from the initial KKT system to give: 

  x_X = b_X

  [ G_RR     A_R  ]   [ x_R    ]   [ - g_R - G_RX*b_X ]
  [ A_R'      0   ] * [ lambda ] = [    c - A_X'*b_X  ]
  \_______________/   \________/   \__________________/
          Ko              vo                fo

 mu_X = - g_X - G_RX'*x_R - G_X*b_X - A_X*lambda

  where:
      n_X = n - n_R
    x_R  = Q_R'*x        <: R^n_R
    g_R  = Q_R'*g        <: R^n_R
    G_RR = Q_R'*G*Q_R    <: R^(n_R x n_R)
    G_RX = Q_R'*G*Q_X    <: R^(n_R x n_X)
    G_XX = Q_X'*G*Q_X    <: R^(n_X x n_X)
    A_R  = Q_R'*A        <: R^(n_R x m)
    A_X  = Q_X'*A        <: R^(n_X x m)
    Q_R                  <: R^(n x n_R)
    Q_X                  <: R^(n x n_X)
This class is an interface for encapsulating the QP. Operations are available for accessing g, G, A, Ko, vo, and fo as well as the mapping matrices Q_R and Q_X (both ordered by row). Also, operations are available for accessing data structures that describe the set of initially fixed and free variables. See the Constraints interface for how to access the constraints in (1.c) and the matrix A_bar.

Definition at line 153 of file ConstrainedOptPack_QPSchur.hpp.

Member Typedef Documentation

typedef vector_one_based_checked<EBounds> ConstrainedOptPack::QPSchurPack::QP::x_init_t
 

Definition at line 160 of file ConstrainedOptPack_QPSchur.hpp.

typedef vector_one_based_checked<size_type> ConstrainedOptPack::QPSchurPack::QP::l_x_X_map_t
 

Definition at line 162 of file ConstrainedOptPack_QPSchur.hpp.

typedef vector_one_based_checked<size_type> ConstrainedOptPack::QPSchurPack::QP::i_x_X_map_t
 

Definition at line 164 of file ConstrainedOptPack_QPSchur.hpp.

typedef QPSchurPack::Constraints ConstrainedOptPack::QPSchurPack::QP::Constraints
 

Definition at line 167 of file ConstrainedOptPack_QPSchur.hpp.

Constructor & Destructor Documentation

virtual ConstrainedOptPack::QPSchurPack::QP::~QP (   )  [inline, virtual]
 

Definition at line 173 of file ConstrainedOptPack_QPSchur.hpp.

Member Function Documentation

virtual size_type ConstrainedOptPack::QPSchurPack::QP::n (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual size_type ConstrainedOptPack::QPSchurPack::QP::m (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const DVectorSlice ConstrainedOptPack::QPSchurPack::QP::g (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const MatrixSymOp& ConstrainedOptPack::QPSchurPack::QP::G (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const MatrixOp& ConstrainedOptPack::QPSchurPack::QP::A (   )  const [pure virtual]
 

If m == 0 then don't call this, it may throw an exception or worse.

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual size_type ConstrainedOptPack::QPSchurPack::QP::n_R (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const x_init_t& ConstrainedOptPack::QPSchurPack::QP::x_init (   )  const [pure virtual]
 

Return the status of a variable initially.

For 1 <= i <= n: 

               / FREE      : x(i) is initially free
               | LOWER     : x(i) is initially fixed at xl(i)
   x_init(i) = | UPPER     : x(i) is initially fixed at xu(i)
               \ EQUALITY  : x(i) fixed at xl(i) = xu(i) always

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const l_x_X_map_t& ConstrainedOptPack::QPSchurPack::QP::l_x_X_map (   )  const [pure virtual]
 

Map from full x(i) to initially fixed x_X(l).

For 1 <= i <= n:

                   / l : x(i) = x_X(l) = b_X(l) initially (1 <= l <= n_X)
   l_x_X_map(i) =  |
                   \ 0 : otherwise

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const i_x_X_map_t& ConstrainedOptPack::QPSchurPack::QP::i_x_X_map (   )  const [pure virtual]
 

Map from initially fixed x_X(l) to full x(i).

For 1 <= l <= n_X:

   i_x_X_map(l) = i : x(i) = x_X(l) = b_X(l) initially (1 <= i <= n)

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const DVectorSlice ConstrainedOptPack::QPSchurPack::QP::b_X (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const GenPermMatrixSlice& ConstrainedOptPack::QPSchurPack::QP::Q_R (   )  const [pure virtual]
 

(Q_R().ordered_by() == BY_ROW)

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const GenPermMatrixSlice& ConstrainedOptPack::QPSchurPack::QP::Q_X (   )  const [pure virtual]
 

(Q_X().ordered_by() == BY_ROW)

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const MatrixSymOpNonsing& ConstrainedOptPack::QPSchurPack::QP::Ko (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const DVectorSlice ConstrainedOptPack::QPSchurPack::QP::fo (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual Constraints& ConstrainedOptPack::QPSchurPack::QP::constraints (   )  [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

virtual const Constraints& ConstrainedOptPack::QPSchurPack::QP::constraints (   )  const [pure virtual]
 

Implemented in ConstrainedOptPack::QPSchurPack::QPInitFixedFreeStd.

void ConstrainedOptPack::QPSchurPack::QP::dump_qp (  std::ostream &  out  )  [virtual]
 

Dump the definition of the QP to a stream.

This function is only to be used for debugging small problems.

Definition at line 5009 of file ConstrainedOptPack_QPSchur.cpp.

The documentation for this class was generated from the following files:
Generated on Thu Sep 18 12:34:18 2008 for ConstrainedOptPack: C++ Tools for Constrained (and Unconstrained) Optimization by doxygen 1.3.9.1