Rdsdp::dsdp {Rdsdp}

Solve semidefinite programm with DSDP

Description

Interface to DSDP semidefinite programming library.

Usage

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.

Value

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)