Semidefinite Blocks (Basics)


Detailed Description

Routines that create the SDPCone object, set the data, and get the solution for problems.

#include dsdp5.h 

In DSDP Standard Form, a semidefinite program is given by the pair of problems

\[ \begin{array}{llllllllll} (P) \ \ \ & \mbox{minimize} & {\displaystyle \sum_{j=0}^{n_b-1} C_j \bullet X_j } &\mbox{subject to}& {\displaystyle \sum_{j=0}^{n_b-1} A_{i,j} \bullet X_{j} = b_i } ,& i=1,\ldots, m, & & X_j \succeq 0, \\ \end{array} \]

\[ \begin{array}{lllllllll} (D) \ \ \ & \mbox{maximize} & {\displaystyle \sum_{i=1}^m b_i \ y_i } &\mbox{subject to}&{\displaystyle \sum_{i=1}^m A_{i,j}y_i + S_{j} } = C_{j}, & j=0, \ldots, n_b-1, & S_j \succeq 0 \\ \end{array} \]

where the data $A_{i,j}$ and $C_j$ are symmetric matrices of the same dimension and the inner product of two $n \times n$ matrices $C=(c_{k,l})$ and $X=(x_{k,l})$ is defined by $ C \bullet X := trace (C^T X) = \sum_{k,l}c_{k,l} x_{k,l} $.

Blocks are labelled from 0 to nblocks, where nblocks is the total number of blocks in the SDPCone object.

Variables y are numbered 1 through m. Variable 0 designates the C matrices, which are also denoted $ A_{0,j}$.

By default, this cone represents symmetric $ n \times n$ matrices in packed symmetric format. This format uses an array of length $ n(n+1)/2 $ and orders the the elements of the matrix as follows:

\[ \begin{array}{llllllll} [ a_{1,1} & a_{2,1} & a_{2,2} & a_{3,1} & a_{3,2} & a_{3,3} & \ldots & a_{n,n} ] \\ \end{array}. \]

Return values:
0 if successful


Functions

int SDPConeGetXArray (SDPCone sdpcone, int blockj, double *xx[], int *nn)
 After applying the solver, set a pointer to the array in the object with the solution X.
int SDPConeSetADenseVecMat (SDPCone sdpcone, int blockj, int vari, int n, double alpha, double val[], int nnz)
 Set a matrix $A_{i,j}$ in a dense format.
int SDPConeSetASparseVecMat (SDPCone sdpcone, int blockj, int vari, int n, double alpha, int ishift, const int ind[], const double val[], int nnz)
 Set data matrix $A_{i,j}$ in a sparse format.
int SDPConeViewDataMatrix (SDPCone sdpcone, int blockj, int vari)
 Print a data matrix to the screen.
int SDPConeViewX (SDPCone sdpcone, int blockj, int n, double x[], int nn)
 Print a dense array X to the screen.


Function Documentation

int SDPConeGetXArray ( SDPCone  sdpcone,
int  blockj,
double *  xx[],
int *  nn 
)

After applying the solver, set a pointer to the array in the object with the solution X.

Parameters:
sdpcone semidefinite cone object
blockj block number
xx address of an array for dense matrices
*nn the length of the array
See also:
DSDPSolve()

DSDPComputeX()

SDPConeViewX()

DSDP dsdp;
SDPCone sdpcone;
double *xx;
int nn;

DSDPSolve(dsdp);
DSDPComputeX(dsdp);
SDPConeGetXArray(sdpcone,0,&xx,&nn);
SDPConeViewX(sdpcone,0,xx,nn);
SDPConeRestoreXArray(sdpcone,0,&xx,&nn);

DSDP uses a single dense array to add data matrices, compute the matrix X, and take the inner product of X with the data matrices. Therefore, the ordering of elements in this array must also be used in the data matrices.

Definition at line 328 of file dsdpadddata.c.

Referenced by DSDPPrintData(), DSDPPrintSolution(), LovaszTheta(), MaxCut(), and StableSet().

int SDPConeSetADenseVecMat ( SDPCone  sdpcone,
int  blockj,
int  vari,
int  n,
double  alpha,
double  val[],
int  nnz 
)

Set a matrix $A_{i,j}$ in a dense format.

Parameters:
sdpcone SDP cone
blockj block number j from 0 to nblocks
vari variable i from 0 through m
n dimension of the matrix
alpha multiple of the data (usually 1.0)
val array of elements in the matrix
nnz length of the array
For example, the matrix
img208.gif
can be inserted into the cone in packed symmetric format as follows
double val[]={3,2,0,0,6,0};
SDPConeSetDenseAVecMat(sdpcone,j,i,3,1.0,val,6);

Note:
DSDP will use the val array in many routines, but it will not modify or delete it.
See also:
SDPConeViewDataMatrix()

SDPConeSetASparseVecMat()

Definition at line 265 of file dsdpadddatamat.c.

Referenced by mexFunction(), and ReadSDPAFile().

int SDPConeSetASparseVecMat ( SDPCone  sdpcone,
int  blockj,
int  vari,
int  n,
double  alpha,
int  ishift,
const int  ind[],
const double  val[],
int  nnz 
)

Set data matrix $A_{i,j}$ in a sparse format.

Parameters:
sdpcone SDP cone
blockj block number j from 0 to nblocks
vari variable i from 0 through m
n dimension of the matrix
ishift index of $ a_{1,1} $ (usually 0)
alpha multiple of the data (usually 1)
val array of elements in the matrix
ind array of indices representing the location of the elements
nnz length of the previous two arrays
For example, the matrix
img208.gif
can be inserted into the cone in packed symmetric format in several ways.

Using the ordering of the packed symmetric format, we can index each element of the matrix with an integer between 0 and n(n+1)/2-1 (inclusive). If the first element in the val array is $a_{1,1}$, the first element in the ind array should be 0 . If the second element in the val array is $a_{2,1}$, then the second element in ind array should be 1 (the second element in dense representation). If the third element in the val array is $a_{3,2}$, then the third element in ind array should be 4. Explicitly,

double val1[]={3,2,6};
int ind1[]={0,1,4};
SDPConeSetASparseVecMat(sdpcone,j,i,3,1.0,0,ind1,val1,3);
If we index the elements from 1 through n(n+1)/2, we can use
double val2[]={3,2,6};
int ind2[]={1,2,5};
SDPConeSetASparseVecMat(sdpcone,j,i,3,1.0,1,ind2,val2,3);
The ordering of the elements in the array can be changed and zero elements can be included.
double val3[]={6,3,0,2};
int ind3[]={4,0,2,1};
SDPConeSetASparseVecMat(sdpcone,j,i,3,1.0,0,ind3,val3,4);
The elements can also be scaled.
double val1[]={6,4,12};
int ind1[]={0,1,4};
SDPConeSetASparseVecMat(sdpcone,j,i,3,0.5,0,ind1,val1,3);

Note:
DSDP will use the two arrays throughout in many routines, but it will not modify or delete them until finished with DSDP.
See also:
DSDPCreateSDPCone()

DSDPSetDualObjective()

SDPConeViewDataMatrix()

Definition at line 152 of file dsdpadddatamat.c.

Referenced by MaxCut(), mexFunction(), MinColoring(), ReadSDPAFile(), and SetStableSetData().

int SDPConeViewDataMatrix ( SDPCone  sdpcone,
int  blockj,
int  vari 
)

Print a data matrix to the screen.

Parameters:
sdpcone semidefinite cone object
blockj block number
vari variable number from 0 through m
See also:
SDPConeView()

Definition at line 205 of file dsdpadddata.c.

Referenced by MaxCut(), ReadSDPAFile(), and SetStableSetData().

int SDPConeViewX ( SDPCone  sdpcone,
int  blockj,
int  n,
double  x[],
int  nn 
)

Print a dense array X to the screen.

Parameters:
sdpcone semidefinite cone
blockj block number
n the dimension of the block
x dense matrix array.
nn length of the array x
See also:
SDPConeGetXArray()

Definition at line 223 of file sdpcone.c.


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