dsdp function

Solve semidefinite programm with DSDP

Interface to DSDP semidefinite programming library.

dsdp(A,b,C,K,OPTIONS=NULL)

Arguments

• A: An object of class "matrix" with $m$ rows defining the block diagonal constraint matrices $A_i$. Each constraint matrix $A_i$ is specified by a row of $A$ as explained in the Details section.

• b: A numeric vector of length $m$ containg the right hand side of the constraints.

• C: An object of class "matrix" with one row or a valid class from the class hierarchy in the "Matrix" package. It defines the objective coefficient matrix $C$ with the same structure of A as explained above.

• K: Describes the sizes of each block of the sdp problem. It is a list with the following elements:

  • "s":: A vector of integers listing the dimension of positive semidefinite cone blocks.
  • "l":: A scaler integer indicating the dimension of the linear nonnegative cone block.

• OPTIONS: A list of OPTIONS parameters passed to dsdp. It may contain any of the following fields:

• print:: = k to display output at each k iteration, else = 0 [default 10].

• logsummary:: = 1 print timing information if set to 1.

• save:: to set the filename to save solution file in SDPA format.

• outputstats:: = 1 to output full information about the solution statistics in STATS.

• gaptol:: tolerance for duality gap as a fraction of the value of the objective functions [default 1e-6].

• maxit:: maximum number of iterations allowed [default 1000].

Please refer to DSDP User Guide for additional OPTIONS parameters available.

Details

All problem matrices are assumed to be of block diagonal structure, the input matrix A must be specified as follows:

1. The coefficients for nonnegative cone block are put in the first K$l columns of A.
2. The coefficients for positive semidefinite cone blocks are put after nonnegative cone block in the the same order as those in K$s. The ith positive semidefinite cone block takes (K$s)[i] times (K$s)[[i]] columns, with each row defining a symmetric matrix of size (K$s)[[i]].

This function does not check for symmetry in the problem data.

Returns

Returns a list of three objects: - X: Optimal primal solution $X$. A vector containing blocks in the same structure as explained above.

• y: Optimal dual solution $y$. A vector of the same length as argument b

• STATS: A list with three to eight fields that describe the solution of the problem:

  • stype:: PDFeasible if the solutions to both (D) and (P) are feasible, Infeasible if (D) is infeasible, and Unbounded if (D) is unbounded.
  • dobj:: objective value of (D).
  • pobj:: objective value of (P).
  • r:: the multiple of the identity element added to $C-\mathcal{A}^{*}y$ in the final solution to make $S$ positive definite.
  • mu:: the final barrier parameter $\nu$.
  • pstep:: the final step length in (P)
  • dstep:: the final step length in (D)
  • pnorm:: the final value $\| P(\nu)\|$.

  The last five fields are optional, and only available when OPTIONS$outputstats is set to $1$.

References

• Steven J. Benson and Yinyu Ye:

  DSDP5 User Guide --- Software for Semidefinite Programming Technical Report ANL/MCS-TM-277, 2005

  https://www.mcs.anl.gov/hs/software/DSDP/DSDP5-Matlab-UserGuide.pdf

Examples

K=NULL
        K$s=c(2,3)
        K$l=2

        C=matrix(c(0,0,2,1,1,2,c(3,0,1,
                       0,2,0,
                       1,0,3)),1,15,byrow=TRUE)
        A=matrix(c(0,1,0,0,0,0,c(3,0,1,
                         0,4,0,
                         1,0,5),
                1,0,3,1,1,3,rep(0,9)), 2,15,byrow=TRUE)
        b <- c(1,2)
        
    OPTIONS=NULL
    OPTIONS$gaptol=0.000001
    OPTIONS$logsummary=0
    OPTIONS$outputstats=1
        
    result = dsdp(A,b,C,K,OPTIONS)