Couenne::exprQuad Class Reference

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

#include <CouenneExprQuad.hpp>

Inheritance diagram for Couenne::exprQuad:
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List of all members.

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_

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.



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
 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
 eigenvalues and eigenvectors
int getnQTerms ()
 eigenvalues and eigenvectors
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
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 *, enum auxSign=expression::AUX_EQ)
 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

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 44 of file CouenneExprQuad.hpp.


Member Typedef Documentation

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

matrix

Definition at line 49 of file CouenneExprQuad.hpp.

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

Definition at line 50 of file CouenneExprQuad.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.

Definition at line 36 of file exprQuad.cpp.

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

Copy constructor.

copy constructor

Definition at line 135 of file exprQuad.cpp.


Member Function Documentation

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

eigenvalues and eigenvectors

Definition at line 94 of file CouenneExprQuad.hpp.

int Couenne::exprQuad::getnQTerms (  )  [inline]

eigenvalues and eigenvectors

Definition at line 97 of file CouenneExprQuad.hpp.

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

cloning method

Reimplemented from Couenne::exprGroup.

Definition at line 101 of file CouenneExprQuad.hpp.

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

Print expression to an iostream.

I/O.

Reimplemented from Couenne::exprGroup.

Definition at line 178 of file exprQuad.cpp.

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

Function for the evaluation of the expression.

Compute sum of linear and nonlinear terms.

Reimplemented from Couenne::exprGroup.

Definition at line 293 of file CouenneExprQuad.hpp.

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

return l-2 norm of gradient at given point

Reimplemented from Couenne::exprGroup.

Definition at line 530 of file exprQuad.cpp.

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

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

differentiation

Reimplemented from Couenne::exprGroup.

Definition at line 227 of file exprQuad.cpp.

expression * exprQuad::simplify (  )  [virtual]

Simplify expression.

Reimplemented from Couenne::exprGroup.

Definition at line 547 of file exprQuad.cpp.

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

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

Reimplemented from Couenne::exprGroup.

Definition at line 121 of file CouenneExprQuad.hpp.

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

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

Reimplemented from Couenne::exprGroup.

Definition at line 25 of file conv-exprQuad.cpp.

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

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

Reimplemented from Couenne::exprGroup.

Definition at line 37 of file conv-exprQuad.cpp.

void exprQuad::generateCuts ( expression w,
OsiCuts &  cs,
const CouenneCutGenerator cg,
t_chg_bounds chg = NULL,
int  wind = -1,
CouNumber  lb = -COUENNE_INFINITY,
CouNumber  ub = 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 Couenne::exprGroup.

Definition at line 44 of file conv-exprQuad.cpp.

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

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

Computes alpha coefficients for an alpha under- and overestimator of the quadratic term.

For the underestimator, dCoeffLo_ is computed such that

x^TQx + sum_i dCoeffLo_i (x_i - lb x_i)(ub x_i - x-i) is convex and underestimating (alpha_i is negative),

Regarding the overestimator, dCoeffUp_ are computed such that

x^TQx + sum_i dCoeffUp_i (x_i - lb x_i)(ub x_i - x-i) is concave and overestimating (alpha_i is positive).

If the method hasn't been called, dIndex_ will be NULL. If Q is positive semidefinite, then dCoeffLo_ will be NULL. If Q is negative semidefinite, then dCoeffUp_ will be NULL.

Return true if a convexification is there, false otherwise.

Definition at line 43 of file alphaConvexify.cpp.

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

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

Definition at line 22 of file quadCuts.cpp.

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

Compare two exprQuad.

compare quadratic terms

Reimplemented from Couenne::exprGroup.

Definition at line 303 of file exprQuad.cpp.

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

Code for comparisons.

Reimplemented from Couenne::exprGroup.

Definition at line 231 of file CouenneExprQuad.hpp.

int exprQuad::rank (  )  [virtual]

Used in rank-based branching variable choice.

used in rank-based branching variable choice

Reimplemented from Couenne::exprGroup.

Definition at line 349 of file exprQuad.cpp.

bool exprQuad::isInteger (  )  [virtual]

is this expression integer?

is this quadratic expression integer?

Reimplemented from Couenne::exprGroup.

Definition at line 403 of file exprQuad.cpp.

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 Couenne::exprGroup.

Definition at line 387 of file exprQuad.cpp.

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.

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

Definition at line 23 of file branchExprQuad.cpp.

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

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

update dependence set with index of this variable

Reimplemented from Couenne::exprGroup.

Definition at line 371 of file exprQuad.cpp.

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

replace variable x with new (aux) w

Reimplemented from Couenne::exprGroup.

Definition at line 447 of file exprQuad.cpp.

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

replace variable x with new (aux) w

Redirect variables to proper variable vector.

Reimplemented from Couenne::exprGroup.

Definition at line 478 of file exprQuad.cpp.

bool exprQuad::impliedBound ( int  wind,
CouNumber l,
CouNumber u,
t_chg_bounds chg,
enum auxSign  sign = expression::AUX_EQ 
) [virtual]

implied bound processing

implied bound processing for quadratic form upon change in lower- and/or upper bound of w, whose index is wind

Reimplemented from Couenne::exprSum.

Definition at line 37 of file impliedBounds-exprQuad.cpp.

CouNumber exprQuad::computeQBound ( int  sign  ) 

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

Definition at line 18 of file exprBQuad.cpp.

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

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

Definition at line 519 of file exprQuad.cpp.

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

return lower and upper bound of quadratic expression

Definition at line 30 of file compQuadFinBounds.cpp.

virtual bool Couenne::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 Couenne::expression.

Definition at line 286 of file CouenneExprQuad.hpp.


Member Data Documentation

sparseQ Couenne::exprQuad::matrix_ [mutable, protected]

Definition at line 61 of file CouenneExprQuad.hpp.

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

eigenvalues and eigenvectors

Definition at line 73 of file CouenneExprQuad.hpp.

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

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

Definition at line 76 of file CouenneExprQuad.hpp.

int Couenne::exprQuad::nqterms_ [protected]

number of non-zeroes in Q

Definition at line 79 of file CouenneExprQuad.hpp.


The documentation for this class was generated from the following files:

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