text: // This file is part of libigl, a simple c++ geometry processing library.
  // 
  // Copyright (C) 2013 Alec Jacobson <alecjacobson@gmail.com>
  // 
  // This Source Code Form is subject to the terms of the Mozilla Public License 
  // v. 2.0. If a copy of the MPL was not distributed with this file, You can 
  // obtain one at http://mozilla.org/MPL/2.0/.
  #ifndef IGL_ACTIVE_SET_H
  #define IGL_ACTIVE_SET_H
  
  #include "igl_inline.h"
  #include "SolverStatus.h"
  #include <Eigen/Core>
  #include <Eigen/Sparse>
  
  namespace igl
  {
    struct active_set_params;
    // Known Bugs: rows of [Aeq;Aieq] **must** be linearly independent. Should be
    // using QR decomposition otherwise:
    //   http://www.okstate.edu/sas/v8/sashtml/ormp/chap5/sect32.htm
    //
    // ACTIVE_SET Minimize quadratic energy 
    //
    // 0.5*Z'*A*Z + Z'*B + C with constraints
    //
    // that Z(known) = Y, optionally also subject to the constraints Aeq*Z = Beq,
    // and further optionally subject to the linear inequality constraints that
    // Aieq*Z <= Bieq and constant inequality constraints lx <= x <= ux
    //
    // Inputs:
    //   A  n by n matrix of quadratic coefficients
    //   B  n by 1 column of linear coefficients
    //   known  list of indices to known rows in Z
    //   Y  list of fixed values corresponding to known rows in Z
    //   Aeq  meq by n list of linear equality constraint coefficients
    //   Beq  meq by 1 list of linear equality constraint constant values
    //   Aieq  mieq by n list of linear inequality constraint coefficients
    //   Bieq  mieq by 1 list of linear inequality constraint constant values
    //   lx  n by 1 list of lower bounds [] implies -Inf
    //   ux  n by 1 list of upper bounds [] implies Inf
    //   params  struct of additional parameters (see below)
    //   Z  if not empty, is taken to be an n by 1 list of initial guess values
    //     (see output)
    // Outputs:
    //   Z  n by 1 list of solution values
    // Returns true on success, false on error
    //
    // Benchmark: For a harmonic solve on a mesh with 325K facets, matlab 2.2
    // secs, igl/min_quad_with_fixed.h 7.1 secs
    //
    template <
      typename AT, 
      typename DerivedB,
      typename Derivedknown, 
      typename DerivedY,
      typename AeqT,
      typename DerivedBeq,
      typename AieqT,
      typename DerivedBieq,
      typename Derivedlx,
      typename Derivedux,
      typename DerivedZ
      >
    IGL_INLINE igl::SolverStatus active_set(
      const Eigen::SparseMatrix<AT>& A,
      const Eigen::PlainObjectBase<DerivedB> & B,
      const Eigen::PlainObjectBase<Derivedknown> & known,
      const Eigen::PlainObjectBase<DerivedY> & Y,
      const Eigen::SparseMatrix<AeqT>& Aeq,
      const Eigen::PlainObjectBase<DerivedBeq> & Beq,
      const Eigen::SparseMatrix<AieqT>& Aieq,
      const Eigen::PlainObjectBase<DerivedBieq> & Bieq,
      const Eigen::PlainObjectBase<Derivedlx> & lx,
      const Eigen::PlainObjectBase<Derivedux> & ux,
      const igl::active_set_params & params,
      Eigen::PlainObjectBase<DerivedZ> & Z
      );
  };
  
  #include "EPS.h"
  struct igl::active_set_params
  {
    // Input parameters for active_set:
    //   Auu_pd  whether Auu is positive definite {false}
    //   max_iter  Maximum number of iterations (0 = Infinity, {100})
    //   inactive_threshold  Threshold on Lagrange multiplier values to determine
    //     whether to keep constraints active {EPS}
    //   constraint_threshold  Threshold on whether constraints are violated (0
    //     is perfect) {EPS}
    //   solution_diff_threshold  Threshold on the squared norm of the difference
    //     between two consecutive solutions {EPS}
    bool Auu_pd;
    int max_iter;
    double inactive_threshold;
    double constraint_threshold;
    double solution_diff_threshold;
    active_set_params():
      Auu_pd(false),
      max_iter(100),
      inactive_threshold(igl::DOUBLE_EPS),
      constraint_threshold(igl::DOUBLE_EPS),
      solution_diff_threshold(igl::DOUBLE_EPS)
      {};
  };
  
  #ifndef IGL_STATIC_LIBRARY
  #  include "active_set.cpp"
  #endif
  
  #endif
  