Preparing the data

  The problem is defined by the following data:

a structure with three fields:

a matrix with m rows and columns. The kth row holds the symmetric block diagonal matrix stored as a vector

a matrix with m rows and columns. The kth row holds the block vector

a matrix with m rows and columns. The kth row holds the vector

the vector b defining the dual objective function. Its length m is equal to the number of primal constraints

a structure with three fields, defining the primal objective function:

the block diagonal matrix

the block vector

the vector

a structure with three fields defining the block sizes:

a vector whose length is the number of blocks in the block diagonal matrix and with entries set to the matrix block sizes . Note that these numbers must be all greater or equal to two, as any blocks with size one should be incorporated in the vector

a vector whose length is the number of blocks in the block vector and with entries set to the vector block sizes . Note that these numbers must be all greater or equal to two, as each vector block has a special first component and at least one other component

the scalar (the length of )

Important: If one or more of the three parts of the SQLP is not present, the corresponding field of may be set to 0, or to the Matlab empty matrix [ ], or left unspecified. In any of these cases, the corresponding fields of and , if set, will be ignored.

Important: In order to be able to use the mex files, all block diagonal matrices must be stored in Matlab's sparse format, unless they contain only one block.

In particular, must be stored in sparse format, unless it has only one block. If it has only one block, it may be stored in either sparse or full format. The vectors , and are stored in full format. The matrices and may be stored in either sparse or full format. There are four different ways to set up the matrix , whose rows represent :

1. Constructing directly using the function svec.m, which converts block diagonal matrices to vector representation. The function smat.m restores the symmetric matrix from such a vector. These routines are invoked by

   

   The block diagonal matrix M passed to svec.m must be stored in sparse format if it has more than one block. By default, the blocks are assumed to be dense. If the user wishes to take advantage of the sparsity in the blocks of the matrix M, then a third, optional parameter sparseblks can be passed to svec.m. When , svec.m treats the blocks as sparse, and returns a sparse vector.

   These routines preserve the inner product, i.e. if and for block diagonal matrices M, N, then

   

2. Using the routine makeA.m, which calls svec.m, to construct from given predefined matrices. This is invoked by

   

   where blk.s is as above and Amat is a one-dimensional cell array. The kth component of the cell array Amat is the matrix . Cell arrays are indexed using braces . For example, to set assign a matrix value to Amat{3}; this must be in sparse format if it has more than one block. The individual blocks of the block diagonal data matrices are treated by default as being dense. If they are sparse, and the user wishes to take advantage of this sparsity, a third optional parameter spblocks may be passed to makeA.m, and this parameter should be set to the value 1. In this case the matrix will be stored using the sparse matrix storage option.

3. Using the routine import.m to load all the data ( , , and ) from a plain ASCII file. This is invoked by

   

   The ASCII data in the file must be stored in a special compact format that is described in Appendix A. The file name must have a period and an extension (anything other than the standard Matlab extensions mat, mex etc.) following the period. The user-specified extension is important as Matlab treats file names without an extension as mat files. The routine export.m implements the reverse operation, saving a problem's data in an ASCII file, in a format recognized by import.m.

4. Loading a mat file defining the data , , , , saved previously by Matlab's save command, using Matlab's load command. This option may be used to load all the examples (from control theory and truss topology design) benchmarked in Appendix C, for which mat files are available from the SDPpack home page.