exprQuad Class Reference

class exprQuad, with constant, linear and quadratic terms More...

#include <exprQuad.hpp>

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List of all members.

Convexification data structures

These are filled by alphaConvexify, which implements the alpha-convexification method described in the LaGO paper by Nowak and Vigerske -- see also Adjiman and Floudas.

 exprQuad (CouNumber c0, std::vector< std::pair< exprVar *, CouNumber > > &lcoeff, std::vector< quadElem > &qcoeff, expression **al=NULL, int n=0)
 Constructor.
 exprQuad (const exprQuad &src, Domain *d=NULL)
 Copy constructor.
sparseQgetQ () const
 Constructor.
int getnQTerms ()
 Constructor.
virtual expressionclone (Domain *d=NULL) const
 cloning method
virtual void print (std::ostream &=std::cout, bool=false) const
 Print expression to an iostream.
virtual CouNumber operator() ()
 Function for the evaluation of the expression.
CouNumber gradientNorm (const double *x)
 return l-2 norm of gradient at given point
virtual expressiondifferentiate (int index)
 Compute derivative of this expression with respect to variable whose index is passed as argument.
virtual expressionsimplify ()
 Simplify expression.
virtual int Linearity ()
 Get a measure of "how linear" the expression is.
virtual void getBounds (expression *&, expression *&)
 Get lower and upper bound of an expression (if any).
virtual void getBounds (CouNumber &, CouNumber &)
 Get lower and upper bound of an expression (if any).
virtual void generateCuts (expression *w, OsiCuts &cs, const CouenneCutGenerator *cg, t_chg_bounds *=NULL, int=-1, CouNumber=-COUENNE_INFINITY, CouNumber=COUENNE_INFINITY)
 Generate cuts for the quadratic expression, which are supporting hyperplanes of the concave upper envelope and the convex lower envelope.
virtual bool alphaConvexify (const CouenneProblem *)
 Compute data for $\alpha$-convexification of a quadratic form (fills in dCoeff_ and dIndex_ for the convex underestimator).
void quadCuts (expression *w, OsiCuts &cs, const CouenneCutGenerator *cg)
 method exprQuad::quadCuts Based on the information (dIndex_, dCoeffLo_, dCoeffUp_) created/modified by alphaConvexify(), create convexification cuts for this expression.
virtual int compare (exprQuad &)
 Compare two exprQuad.
virtual enum expr_type code ()
 Code for comparisons.
virtual int rank ()
 Used in rank-based branching variable choice.
virtual bool isInteger ()
 is this expression integer?
virtual int DepList (std::set< int > &deplist, enum dig_type type=ORIG_ONLY)
 fill in the set with all indices of variables appearing in the expression
virtual CouNumber selectBranch (const CouenneObject *obj, const OsiBranchingInformation *info, expression *&var, double *&brpts, double *&brDist, int &way)
 Set up branching object by evaluating many branching points for each expression's arguments.
virtual void fillDepSet (std::set< DepNode *, compNode > *dep, DepGraph *g)
 Fill dependence set of the expression associated with this auxiliary variable.
virtual void replace (exprVar *x, exprVar *w)
 replace variable x with new (aux) w
virtual void realign (const CouenneProblem *p)
 replace variable x with new (aux) w
virtual bool impliedBound (int, CouNumber *, CouNumber *, t_chg_bounds *)
 implied bound processing
CouNumber computeQBound (int sign)
 method to compute the bound based on sign: -1 for lower, +1 for upper
virtual void closestFeasible (expression *varind, expression *vardep, CouNumber &left, CouNumber &right) const
 compute $y^{lv}$ and $y^{uv}$ for Violation Transfer algorithm
void computeQuadFiniteBound (CouNumber &qMin, CouNumber &qMax, CouNumber *l, CouNumber *u, int &indInfLo, int &indInfUp)
 return lower and upper bound of quadratic expression
virtual bool isCuttable (CouenneProblem *problem, int index) const
 can this expression be further linearized or are we on its concave ("bad") side
std::vector< std::pair< CouNumber,
std::vector< std::pair< exprVar *,
CouNumber > > > > 
eigen_
 eigenvalues and eigenvectors
std::map< exprVar *, std::pair<
CouNumber, CouNumber > > 
bounds_
 current bounds (checked before re-computing eigenvalues/vectors)
int nqterms_
 number of non-zeroes in Q

Public Types

typedef std::vector< std::pair<
exprVar *, CouNumber > > 
sparseQcol
 matrix
typedef std::vector< std::pair<
exprVar *, sparseQcol > > 
sparseQ

Protected Attributes

Q matrix storage
Sparse implementation: given expression of the form $\sum_{i \in N, j \in N} q_{ij} x_i x_j$, qindexI_ and qindexJ_ contain respectively entries $i$ and $j$ for which $q_{ij}$ is nonzero in $q_{ij} x_i x_j$:

sparseQ matrix_

Detailed Description

class exprQuad, with constant, linear and quadratic terms

It represents an expression of the form $a_0 + \sum_{i\in I} b_i x_i + x^T Q x + \sum_{i \in J} h_i (x)$, with $a_0 + \sum_{i\in I} b_i x_i$ an affine term, $x^T Q x$ a quadratic term, and a nonlinear sum $\sum_{i \in J} h_i (x)$. Standardization checks possible quadratic or linear terms in the latter and includes them in the former parts.

If $h_i(x)$ is a product of two nonlinear, nonquadratic functions $h'(x)h''(x)$, two auxiliary variables $w'=f'(x)$ and $w''=h''(x)$ are created and the product $w'w''$ is included in the quadratic part of the exprQuad. If $h(x)$ nonquadratic, nonlinear function, an auxiliary variable $w=h(x)$ is created and included in the linear part.

Definition at line 42 of file exprQuad.hpp.


Member Typedef Documentation

typedef std::vector<std::pair <exprVar *, CouNumber> > exprQuad::sparseQcol

matrix

Definition at line 47 of file exprQuad.hpp.

typedef std::vector<std::pair <exprVar *, sparseQcol> > exprQuad::sparseQ

Definition at line 48 of file exprQuad.hpp.


Constructor & Destructor Documentation

exprQuad::exprQuad ( CouNumber  c0,
std::vector< std::pair< exprVar *, CouNumber > > &  lcoeff,
std::vector< quadElem > &  qcoeff,
expression **  al = NULL,
int  n = 0 
)

Constructor.

Referenced by clone().

exprQuad::exprQuad ( const exprQuad src,
Domain d = NULL 
)

Copy constructor.


Member Function Documentation

sparseQ& exprQuad::getQ (  )  const [inline]

Constructor.

Definition at line 92 of file exprQuad.hpp.

References matrix_.

int exprQuad::getnQTerms (  )  [inline]

Constructor.

Definition at line 95 of file exprQuad.hpp.

References nqterms_.

virtual expression* exprQuad::clone ( Domain d = NULL  )  const [inline, virtual]

cloning method

Reimplemented from exprGroup.

Definition at line 99 of file exprQuad.hpp.

References exprQuad().

virtual void exprQuad::print ( std::ostream &  = std::cout,
bool  = false 
) const [virtual]

Print expression to an iostream.

Reimplemented from exprGroup.

CouNumber exprQuad::operator() (  )  [inline, virtual]

Function for the evaluation of the expression.

Reimplemented from exprGroup.

Definition at line 291 of file exprQuad.hpp.

References expression::Index(), matrix_, and exprGroup::operator()().

CouNumber exprQuad::gradientNorm ( const double *  x  )  [virtual]

return l-2 norm of gradient at given point

Reimplemented from exprGroup.

virtual expression* exprQuad::differentiate ( int  index  )  [virtual]

Compute derivative of this expression with respect to variable whose index is passed as argument.

Reimplemented from exprGroup.

virtual expression* exprQuad::simplify (  )  [virtual]

Simplify expression.

Reimplemented from exprGroup.

virtual int exprQuad::Linearity (  )  [inline, virtual]

Get a measure of "how linear" the expression is.

Reimplemented from exprGroup.

Definition at line 119 of file exprQuad.hpp.

References exprGroup::c0_, CONSTANT, COUENNE_EPS, exprGroup::lcoeff_, LINEAR, exprSum::Linearity(), matrix_, QUADRATIC, and ZERO.

virtual void exprQuad::getBounds ( expression *&  ,
expression *&   
) [virtual]

Get lower and upper bound of an expression (if any).

Reimplemented from exprGroup.

virtual void exprQuad::getBounds ( CouNumber ,
CouNumber  
) [virtual]

Get lower and upper bound of an expression (if any).

Reimplemented from exprGroup.

virtual void exprQuad::generateCuts ( expression w,
OsiCuts &  cs,
const CouenneCutGenerator cg,
t_chg_bounds = NULL,
int  = -1,
CouNumber  = -COUENNE_INFINITY,
CouNumber  = COUENNE_INFINITY 
) [virtual]

Generate cuts for the quadratic expression, which are supporting hyperplanes of the concave upper envelope and the convex lower envelope.

Reimplemented from exprGroup.

virtual bool exprQuad::alphaConvexify ( const CouenneProblem  )  [virtual]

Compute data for $\alpha$-convexification of a quadratic form (fills in dCoeff_ and dIndex_ for the convex underestimator).

void exprQuad::quadCuts ( expression w,
OsiCuts &  cs,
const CouenneCutGenerator cg 
)

method exprQuad::quadCuts Based on the information (dIndex_, dCoeffLo_, dCoeffUp_) created/modified by alphaConvexify(), create convexification cuts for this expression.

The original constraint is :

\[ \eta = a_0 + a^T x + x^T Q x \]

where $ \eta $ is the auxiliary corresponding to this expression and $ w_j $ are the auxiliaries corresponding to the other non-linear terms contained in the expression.

The under-estimator of $ x^T Q x$ is given by

\[ x^T Q x + \sum \lambda_{\min,i} (x_i - l_i ) (u_i - x_i ) \]

and its over-estimator is given by

\[ x^T Q x + \sum \lambda_{\max, i} (x_i - l_i ) (u_i - x_i ) \]

(where $ \lambda_{\min, i} = \frac{\lambda_{\min}}{w_i^2} $, $ \lambda_{\max, i} = \frac{\lambda_{\max}}{w_i^2} $, and $w_i = u_i - l_i$), where $\lambda_{\max}$ ($\lambda_{\max}$) is the minimum (maximum) eigenvalue of the matrix $A={\rm Diag}({\bf u} - {\bf l}) Q {\rm Diag}({\bf u} - {\bf l})$, obtained by pre- and post-multiplying $ Q $ by the diagonal matrix whose $i$-th element is $u_i - l_i$.

Let $ \tilde a_0(\lambda)$, $ \tilde a(\lambda) $ and $ \tilde Q (\lambda) $ be

\[ \tilde a_0(\lambda) = a_0 - \sum_{i = 1}^n \lambda_i l_i u_i \]

\[ \tilde a(\lambda) = a + \left[ \begin{array}{c} \lambda_1 (u_1 + l_1) \\ \vdots \\ \lambda_n (u_n + l_n) \end{array} \right], \]

\[ \tilde Q(\lambda) = Q - \left( \begin{array}{ccc} {\lambda_1} & & 0 \\ & \ddots & \\ 0 & & \lambda_n \end{array} \right). \]

The convex relaxation of the initial constraint is then given by the two constraints

\[ \eta \geq \tilde a_0(\lambda_{\min}) + \tilde a(\lambda_{\min})^T x + x^T \tilde Q(\lambda_{\min}) x \]

\[ \eta \leq \tilde a_0(\lambda_{\max}) + \tilde a(\lambda_{\max})^T x + x^T \tilde Q(\lambda_{\max}) x \]

The cut is computed as follow. Let $ (x^*, \eta^*) $ be the solution at hand. The two outer-approximation cuts are:

\[ \eta \geq \tilde a_0(\lambda_{\min}) + \tilde a(\lambda_{\min})^T x + {x^*}^T \tilde Q(\lambda_{\min}) (2x - x^*) \]

and

\[ \eta \leq \tilde a_0(\lambda_{\max}) + \tilde a(\lambda_{\max})^T x + {x^*}^T \tilde Q(\lambda_{\max}) (2x - x^*); \]

grouping coefficients, we get:

\[ {x^*}^T \tilde Q(\lambda_{\min}) x^* - \tilde a_0(\lambda_{\min}) \geq (\tilde a(\lambda_{\min}) + 2 \tilde Q(\lambda_{\min} ) x^*)^T x - \eta \]

and

\[ {x^*}^T \tilde Q(\lambda_{\max}) x^* - \tilde a_0(\lambda_{\max}) \leq (\tilde a(\lambda_{\max}) + 2 \tilde Q (\lambda_{\max}) x^* )^T x - \eta \]

virtual int exprQuad::compare ( exprQuad  )  [virtual]

Compare two exprQuad.

virtual enum expr_type exprQuad::code (  )  [inline, virtual]

Code for comparisons.

Reimplemented from exprGroup.

Definition at line 229 of file exprQuad.hpp.

References COU_EXPRQUAD.

virtual int exprQuad::rank (  )  [virtual]

Used in rank-based branching variable choice.

Reimplemented from exprGroup.

virtual bool exprQuad::isInteger (  )  [virtual]

is this expression integer?

Reimplemented from exprGroup.

virtual int exprQuad::DepList ( std::set< int > &  deplist,
enum dig_type  type = ORIG_ONLY 
) [virtual]

fill in the set with all indices of variables appearing in the expression

Reimplemented from exprGroup.

virtual CouNumber exprQuad::selectBranch ( const CouenneObject obj,
const OsiBranchingInformation *  info,
expression *&  var,
double *&  brpts,
double *&  brDist,
int &  way 
) [virtual]

Set up branching object by evaluating many branching points for each expression's arguments.

Reimplemented from expression.

virtual void exprQuad::fillDepSet ( std::set< DepNode *, compNode > *  dep,
DepGraph g 
) [virtual]

Fill dependence set of the expression associated with this auxiliary variable.

Reimplemented from exprGroup.

virtual void exprQuad::replace ( exprVar x,
exprVar w 
) [virtual]

replace variable x with new (aux) w

Reimplemented from exprGroup.

virtual void exprQuad::realign ( const CouenneProblem p  )  [virtual]

replace variable x with new (aux) w

Reimplemented from exprGroup.

virtual bool exprQuad::impliedBound ( int  ,
CouNumber ,
CouNumber ,
t_chg_bounds  
) [virtual]

implied bound processing

Reimplemented from exprSum.

CouNumber exprQuad::computeQBound ( int  sign  ) 

method to compute the bound based on sign: -1 for lower, +1 for upper

virtual void exprQuad::closestFeasible ( expression varind,
expression vardep,
CouNumber left,
CouNumber right 
) const [virtual]

compute $y^{lv}$ and $y^{uv}$ for Violation Transfer algorithm

Reimplemented from expression.

void exprQuad::computeQuadFiniteBound ( CouNumber qMin,
CouNumber qMax,
CouNumber l,
CouNumber u,
int &  indInfLo,
int &  indInfUp 
) [protected]

return lower and upper bound of quadratic expression

virtual bool exprQuad::isCuttable ( CouenneProblem problem,
int  index 
) const [inline, protected, virtual]

can this expression be further linearized or are we on its concave ("bad") side

Reimplemented from expression.

Definition at line 284 of file exprQuad.hpp.


Member Data Documentation

sparseQ exprQuad::matrix_ [mutable, protected]

Definition at line 59 of file exprQuad.hpp.

Referenced by getQ(), Linearity(), and operator()().

std::vector<std::pair <CouNumber, std::vector <std::pair <exprVar *, CouNumber> > > > exprQuad::eigen_ [mutable, protected]

eigenvalues and eigenvectors

Definition at line 71 of file exprQuad.hpp.

std::map<exprVar *, std::pair <CouNumber, CouNumber> > exprQuad::bounds_ [protected]

current bounds (checked before re-computing eigenvalues/vectors)

Definition at line 74 of file exprQuad.hpp.

int exprQuad::nqterms_ [protected]

number of non-zeroes in Q

Definition at line 77 of file exprQuad.hpp.

Referenced by getnQTerms().


The documentation for this class was generated from the following file:
Generated on Sun Feb 14 03:04:53 2010 for Couenne by  doxygen 1.4.7