6.6 Semidefinite Optimization

Semidefinite optimization is a generalization of conic optimization, allowing the use of matrix variables belonging to the convex cone of positive semidefinite matrices

\[\PSD^r = \left\lbrace X \in \Symm^r: z^T X z \geq 0, \quad \forall z \in \real^r \right\rbrace,\]

where \(\Symm^r\) is the set of \(r \times r\) real-valued symmetric matrices.

MOSEK can solve semidefinite optimization problems of the form

\[\begin{split}\begin{array}{lccccll} \mbox{minimize} & & & \sum_{j=0}^{n-1} c_j x_j + \sum_{j=0}^{p-1} \left\langle \barC_j, \barX_j \right\rangle + c^f & & &\\ \mbox{subject to} & l_i^c & \leq & \sum_{j=0}^{n-1} a_{ij} x_j + \sum_{j=0}^{p-1} \left\langle \barA_{ij}, \barX_j \right\rangle & \leq & u_i^c, & i = 0, \ldots, m-1,\\ & l_j^x & \leq & x_j & \leq & u_j^x, & j = 0, \ldots, n-1,\\ & & & x \in \K, \barX_j \in \PSD^{r_j}, & & & j = 0, \ldots, p-1 \end{array}\end{split}\]

where the problem has \(p\) symmetric positive semidefinite variables \(\barX_j\in \PSD^{r_j}\) of dimension \(r_j\) with symmetric coefficient matrices \(\barC_j\in \Symm^{r_j}\) and \(\barA_{i,j}\in \Symm^{r_j}\). We use standard notation for the matrix inner product, i.e., for \(A,B\in \real^{m\times n}\) we have

\[\left\langle A,B \right\rangle := \sum_{i=0}^{m-1} \sum_{j=0}^{n-1} A_{ij} B_{ij}.\]

We demonstrate the setup of semidefinite variables and the matrices \(\barC\), \(\barA\) on the following examples:

6.6.1 Example SDO1

We consider the simple optimization problem with semidefinite and conic quadratic constraints:

(6.10)\[\begin{split}\begin{array} {llcc} \mbox{minimize} & \left\langle \left[ \begin{array} {ccc} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{array} \right], \barX \right\rangle + x_0 & & \\ \mbox{subject to} & \left\langle \left[ \begin{array} {ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right], \barX \right\rangle + x_0 & = & 1, \\ & \left\langle \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right], \barX \right\rangle + x_1 + x_2 & = & 1/2, \\ & x_0 \geq \sqrt{{x_1}^2 + {x_2}^2}, & \barX \succeq 0, & \end{array}\end{split}\]

The problem description contains a 3-dimensional symmetric semidefinite variable which can be written explicitly as:

\[\begin{split}\barX = \left[ \begin{array} {ccc} \barX_{00} & \barX_{10} & \barX_{20} \\ \barX_{10} & \barX_{11} & \barX_{21} \\ \barX_{20} & \barX_{21} & \barX_{22} \end{array} \right] \in \PSD^3,\end{split}\]

and a conic quadratic variable \((x_0, x_1, x_2) \in \Q^3\). The objective is to minimize

\[2(\barX_{00} + \barX_{10} + \barX_{11} + \barX_{21} + \barX_{22}) + x_0,\]

subject to the two linear constraints

\[\begin{split}\begin{array}{ccc} \barX_{00} + \barX_{11} + \barX_{22} + x_0 & = & 1, \\ \barX_{00} + \barX_{11} + \barX_{22} + 2(\barX_{10} + \barX_{20} + \barX_{21}) + x_1 + x_2 & = & 1/2. \end{array}\end{split}\]

Setting up the linear and conic part

The linear and conic parts (constraints, variables, objective, cones) are set up using the methods described in the relevant tutorials; Sec. 6.1 (Linear Optimization), Sec. 6.3 (Conic Quadratic Optimization), Sec. 6.5 (Conic Exponential Optimization), Sec. 6.4 (Power Cone Optimization). Here we only discuss the aspects directly involving semidefinite variables.

Appending semidefinite variables

The dimensions of semidefinite variables are passed in prob$bardim.

Coefficients of semidefinite terms.

Every term of the form \((\barA_{i,j})_{k,l}(\barX_j)_{k,l}\) is determined by four indices \((i,j,k,l)\) and a coefficient value \(v=(\barA_{i,j})_{k,l}\). Here \(i\) is the number of the constraint in which the term appears, \(j\) is the index of the semidefinite variable it involves and \((k,l)\) is the position in that variable. This data is passed in the structure prob$barA. Note that only the lower triangular part should be specified explicitly, that is one always has \(k\geq l\).

Semidefinite terms \((\barC_j)_{k,l}(\barX_j)_{k,l}\) of the objective are specified in the same way in prob$barc but only include \((j,k,l)\) and \(v\).

Source code

Listing 6.6 R implementation of model (6.10). Click here to download.
library("Rmosek")

getbarvarMatrix <- function(barvar, bardim)
{
    N <- as.integer(bardim)
    new("dspMatrix", x=barvar, uplo="L", Dim=c(N,N))
}

sdo1 <- function()
{
    # Specify the non-matrix variable part of the problem.
    prob <- list(sense="min")
    prob$c     <- c(1, 0, 0)
    prob$A <- sparseMatrix(i=c(1, 2, 2),
                           j=c(1, 2, 3),
                           x=c(1, 1, 1), dims=c(2, 3))
    prob$bc    <- rbind(blc=c(1, 0.5), 
                        buc=c(1, 0.5))
    prob$bx    <- rbind(blx=rep(-Inf,3), 
                        bux=rep( Inf,3))
    prob$cones <- cbind(list("QUAD", c(1, 2, 3)))

    # Specify semidefinite matrix variables (one 3x3 block)
    prob$bardim <- c(3)

    # Block triplet format specifying the lower triangular part 
    # of the symmetric coefficient matrix 'barc':
    prob$barc$j <- c(1, 1, 1, 1, 1)
    prob$barc$k <- c(1, 2, 3, 2, 3)
    prob$barc$l <- c(1, 2, 3, 1, 2)
    prob$barc$v <- c(2, 2, 2, 1, 1)

    # Block triplet format specifying the lower triangular part 
    # of the symmetric coefficient matrix 'barA':
    prob$barA$i <- c(1, 1, 1, 2, 2, 2, 2, 2, 2)
    prob$barA$j <- c(1, 1, 1, 1, 1, 1, 1, 1, 1)
    prob$barA$k <- c(1, 2, 3, 1, 2, 3, 2, 3, 3)
    prob$barA$l <- c(1, 2, 3, 1, 2, 3, 1, 1, 2)
    prob$barA$v <- c(1, 1, 1, 1, 1, 1, 1, 1, 1)

    # Solve the problem
    r <- mosek(prob)

    # Print matrix variable and return the solution
    stopifnot(identical(r$response$code, 0))
    print( list(barx=getbarvarMatrix(r$sol$itr$barx[[1]], prob$bardim[1])) )
    r$sol
}

sdo1()

The numerical values of \(\barX_j\) are returned in the list r$sol$itr$barx; the \(j\)-th element of the list is the lower triangular part of each \(\barX_j\) stacked column-by-column into a numeric vector. Similarly, the dual semidefinite variables \(\barS_j\) are recovered through r$sol$itr$bars.

6.6.2 Example SDO2

We now demonstrate how to define more than one semidefinite variable using the following problem with two matrix variables and two types of constraints:

(6.11)\[\begin{split}\begin{array}{lrll} \mbox{minimize} & \langle C_1,\barX_1\rangle + \langle C_2,\barX_2\rangle & & \\ \mbox{subject to} & \langle A_1,\barX_1\rangle + \langle A_2,\barX_2\rangle & = & b, \\ & (\barX_2)_{01} & \leq & k, \\ & \barX_1, \barX_2 & \succeq & 0. \end{array}\end{split}\]

In our example \(\dim(\barX_1)=3\), \(\dim(\barX_2)=4\), \(b=23\), \(k=-3\) and

\[\begin{split}C_1= \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 6 \end{array}\right], A_1= \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 2 \end{array}\right],\end{split}\]
\[\begin{split}C_2= \left[\begin{array}{cccc} 1 & -3 & 0 & 0\\ -3 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right], A_2= \left[\begin{array}{cccc} 0 & 1 & 0 & 0\\ 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -3 \\ \end{array}\right],\end{split}\]

are constant symmetric matrices.

Note that this problem does not contain any scalar variables, but they could be added in the same fashion as in Sec. 6.6.1 (Example SDO1).

For explanations of data structures used in the example see Sec. 6.6.1 (Example SDO1). Note that the field bardim is used to specify that we have two semidefinite variables of dimensions 3 and 4.

The code representing the above problem is shown below.

Listing 6.7 Implementation of model (6.11). Click here to download.
library("Rmosek")

getbarvarMatrix <- function(barvar, bardim)
{
    N <- as.integer(bardim)
    new("dspMatrix", x=barvar, uplo="L", Dim=c(N,N))
}

sdo2 <- function()
{
    # Sample data in sparse lower-triangular triplet form
    C1_k <- c(1, 3);
    C1_l <- c(1, 3);
    C1_v <- c(1, 6);
    A1_k <- c(1, 3, 3);
    A1_l <- c(1, 1, 3);
    A1_v <- c(1, 1, 2);
    C2_k <- c(1, 2, 2, 3);
    C2_l <- c(1, 1, 2, 3);
    C2_v <- c(1, -3, 2, 1);
    A2_k <- c(2, 2, 4);
    A2_l <- c(1, 2, 4);
    A2_v <- c(1, -1, -3);
    b <- 23.0;
    k <- -3.0;
 
    # Specify the dimensions of the problem
    prob <- list(sense="min");
    # Two constraints
    prob$A <- Matrix(nrow=2, ncol=0);                   # 2 constraints
    prob$c <- numeric(0);
    prob$bx <- rbind(blc=numeric(0), 
                     buc=numeric(0));

    # Dimensions of semidefinite matrix variables 
    prob$bardim <- c(3, 4);
    # Constraint bounds    
    prob$bc <- rbind(blc=c(b, -Inf), 
                     buc=c(b, k));

    # Block triplet format specifying the lower triangular part 
    # of the symmetric coefficient matrix 'barc':
    prob$barc$j <- c(rep(1, length(C1_v)), 
                     rep(2, length(C2_v)));         # Which PSD variable (j)
    prob$barc$k <- c(C1_k, C2_k);                   # Entries: (k,l)->v
    prob$barc$l <- c(C1_l, C2_l);
    prob$barc$v <- c(C1_v, C2_v);

    # Block triplet format specifying the lower triangular part 
    # of the symmetric coefficient matrix 'barA':
    prob$barA$i <- c(rep(1, length(A1_v)+length(A2_v)), 2);   # Which constraint (i)
    prob$barA$j <- c(rep(1, length(A1_v)),
                     rep(2, length(A2_v)),
                     2);                                      # Which PSD variable (j)
    prob$barA$k <- c(A1_k, A2_k, 2);                          # Entries: (k,l)->v
    prob$barA$l <- c(A1_l, A2_l, 1);
    prob$barA$v <- c(A1_v, A2_v, 0.5);

    # Solve the problem
    r <- mosek(prob);

    # Print matrix variable and return the solution
    stopifnot(identical(r$response$code, 0));
    print( list(X1=getbarvarMatrix(r$sol$itr$barx[[1]], prob$bardim[1])) );
    print( list(X2=getbarvarMatrix(r$sol$itr$barx[[2]], prob$bardim[2])) );

}

sdo2();

The numerical values of \(\barX_j\) are returned in the list r$sol$itr$barx; the \(j\)-th element of the list is the lower triangular part of each \(\barX_j\) stacked column-by-column into a numeric vector. Similarly, the dual semidefinite variables \(\barS_j\) are recovered through r$sol$itr$bars.