6.7 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 stated in the primal form,

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

where the problem has $$p$$ symmetric positive semidefinite variables $$\barX_j\in \PSD^{r_j}$$ of dimension $$r_j$$. The symmetric coefficient matrices $$\barC_j\in \Symm^{r_j}$$ and $$\barA_{i,j}\in \Symm^{r_j}$$ are used to specify PSD terms in the linear objective and the linear constraints, respectively. The symmetric coefficient matrices $$\barF_{i,j}\in \Symm^{r_j}$$ are used to specify PSD terms in the affine conic constraints. Note that $$q$$ ((6.22)) is the total dimension of all the cones, i.e. $$q=\text{dim}(\K_1 \times \ldots \times \K_k)$$, given there are $$k$$ ACCs. We use standard notation for the matrix inner product, i.e., for $$A,B\in \real^{m\times n}$$ we have

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

In addition to the primal form presented above, semidefinite problems can be expressed in their dual form. Constraints in this form are usually called linear matrix inequalities (LMIs). LMIs can be easily specified in MOSEK using the vectorized positive semidefinite cone which is defined as:

• Vectorized semidefinite domain:

$\PSD^{d,\mathrm{vec}} = \left\{(x_1,\ldots,x_{d(d+1)/2})\in \real^n~:~ \mathrm{sMat}(x)\in\PSD^d\right\},$

where $$n=d(d+1)/2$$ and,

$\begin{split}\mathrm{sMat}(x) = \left[\begin{array}{cccc}x_1 & x_2/\sqrt{2} & \cdots & x_{d}/\sqrt{2} \\ x_2/\sqrt{2} & x_{d+1} & \cdots & x_{2d-1}/\sqrt{2} \\ \cdots & \cdots & \cdots & \cdots \\ x_{d}/\sqrt{2} & x_{2d-1}/\sqrt{2} & \cdots & x_{d(d+1)/2}\end{array}\right],\end{split}$

or equivalently

$\PSD^{d,\mathrm{vec}} = \left\{\mathrm{sVec}(X)~:~X\in\PSD^d\right\},$

where

$\mathrm{sVec}(X) = (X_{11},\sqrt{2}X_{21},\ldots,\sqrt{2}X_{d1},X_{22},\sqrt{2}X_{32},\ldots,X_{dd}).$

In other words, the domain consists of vectorizations of the lower-triangular part of a positive semidefinite matrix, with the non-diagonal elements additionally rescaled. LMIs can be expressed by restricting appropriate affine expressions to this cone type.

For other types of cones supported by MOSEK, see Sec. 15.11 (Supported domains) and the other tutorials in this chapter. Different cone types can appear together in one optimization problem.

We demonstrate the setup of semidefinite variables and their coefficient matrices in the following examples:

6.7.1 Example SDO1¶

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

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

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 an affine conic constraint (ACC) $$(x_0, x_1, x_2) \in \Q^3$$. 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, ACC) are set up using the methods described in the relevant tutorials; Sec. 6.1 (Linear Optimization), Sec. 6.2 (From Linear to Conic Optimization). Here we only discuss the aspects directly involving semidefinite variables.

Appending semidefinite variables

First, we need to declare the number of semidefinite variables in the problem, similarly to the number of linear variables and constraints. This is done with the function Task.appendbarvars.

Appending coefficient matrices

Coefficient matrices $$\barC_j$$ and $$\barA_{ij}$$ are constructed as weighted combinations of sparse symmetric matrices previously appended with the function Task.appendsparsesymmat.

barc_i,
barc_j,
barc_v);

The arguments specify the dimension of the symmetric matrix, followed by its description in the sparse triplet format. Only lower-triangular entries should be included. The function produces a unique index of the matrix just entered in the collection of all coefficient matrices defined by the user.

After one or more symmetric matrices have been created using Task.appendsparsesymmat, we can combine them to set up the objective matrix coefficient $$\barC_j$$ using Task.putbarcj, which forms a linear combination of one or more symmetric matrices. In this example we form the objective matrix directly, i.e. as a weighted combination of a single symmetric matrix.

Similarly, a constraint matrix coefficient $$\barA_{ij}$$ is set up by the function Task.putbaraij.

Retrieving the solution

double[] barx = task.getbarxj(mosek.soltype.itr, 0);   /* Request the interior solution. */

The function returns the half-vectorization of $$\barX_j$$ (the lower triangular part stacked as a column vector), where the semidefinite variable index $$j$$ is passed as an argument.

Source code

using System;

namespace mosek.example
{
public class sdo1
{
public static void Main(string[] args)
{
int    numcon    = 2;  /* Number of constraints.              */
int    numvar    = 3;  /* Number of conic quadratic variables */
int[]  dimbarvar = { 3 };         /* Dimensions of semidefinite cones */
int[]  lenbarvar = { 3 * (3 + 1) / 2 }; /* Number of scalar SD variables  */

mosek.boundkey[] bkc = { mosek.boundkey.fx, mosek.boundkey.fx };
double[]     blc     = { 1.0, 0.5 };
double[]     buc     = { 1.0, 0.5 };

int[]        barc_i  = { 0,   1,   1,   2,   2 },
barc_j  = { 0,   0,   1,   1,   2 };
double[]     barc_v  = { 2.0, 1.0, 2.0, 1.0, 2.0 };

int[][]      asub    = { new int[] {0},   new int[] {1, 2}}; /* column subscripts of A */
double[][]   aval    = { new double[] {1.0}, new double[] {1.0, 1.0}};

int[][]      bara_i  = { new int[] {0,   1,   2},   new int[] {0,   1 ,  2,   1,   2,   2 } },
bara_j  = { new int[] {0,   1,   2},   new int[] {0,   0 ,  0,   1,   1,   2 } };
double[][]   bara_v  = { new double[] {1.0, 1.0, 1.0}, new double[] {1.0, 1.0, 1.0, 1.0, 1.0, 1.0}};

using (mosek.Env env = new mosek.Env())
{
{
// Directs the log task stream to the user specified
// method msgclass.streamCB
/* Append 'NUMCON' empty constraints.
The constraints will initially have no bounds. */

/* Append 'NUMVAR' variables.
The variables will initially be fixed at zero (x=0). */

/* Append 'NUMBARVAR' semidefinite variables. */

/* Optionally add a constant term to the objective. */

/* Set the linear term c_j in the objective.*/

for (int j = 0; j < numvar; ++j)

/* Set the linear term barc_j in the objective.*/
{
long[] idx = new long[1];
double[] falpha = { 1.0 };
barc_i,
barc_j,
barc_v);
}

/* Set the bounds on constraints.
for i=1, ...,numcon : blc[i] <= constraint i <= buc[i] */

for (int i = 0; i < numcon; ++i)
bkc[i],      /* Bound key.*/
blc[i],      /* Numerical value of lower bound.*/
buc[i]);     /* Numerical value of upper bound.*/

/* Input A row by row */
for (int i = 0; i < numcon; ++i)
asub[i],
aval[i]);

/* Append the conic quadratic constraint */
// Diagonal F matrix
task.putafefentrylist(new long[]{0,1,2}, new int[]{0,1,2}, new double[]{1.0,1.0,1.0});

/* Add the first row of barA */
{
long[] idx = new long[1];
double[] falpha = {1.0};
bara_i[0],
bara_j[0],
bara_v[0],
out idx[0]);

}

{
long[] idx = new long[1];
double[] falpha = {1.0};
/* Add the second row of barA */
bara_i[1],
bara_j[1],
bara_v[1],
out idx[0]);

}

/* Run optimizer */

/* Print a summary containing information
about the solution for debugging purposes*/

switch (solsta)
{
case mosek.solsta.optimal:
double[] barx = task.getbarxj(mosek.soltype.itr, 0);   /* Request the interior solution. */
Console.WriteLine("Optimal primal solution");
for (int i = 0; i < numvar; ++i)
Console.WriteLine("x[{0}]   : {1}", i, xx[i]);

for (int i = 0; i < lenbarvar[0]; ++i)
Console.WriteLine("barx[{0}]: {1}", i, barx[i]);
break;
case mosek.solsta.dual_infeas_cer:
case mosek.solsta.prim_infeas_cer:
Console.WriteLine("Primal or dual infeasibility certificate found.");
break;
case mosek.solsta.unknown:
Console.WriteLine("The status of the solution could not be determined.");
break;
default:
Console.WriteLine("Other solution status.");
break;
}
}
}
}
}

class msgclass : mosek.Stream
{
string prefix;
public msgclass (string prfx)
{
prefix = prfx;
}

public override void streamCB (string msg)
{
Console.Write ("{0}{1}", prefix, msg);
}
}
}

6.7.2 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:

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

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. 6.7.1 (Example SDO1).

Other than in Sec. 6.7.1 (Example SDO1) we don’t append coefficient matrices separately but we directly input all nonzeros in each constraint and all nonzeros in the objective at once. 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 call to Task.putbarablocktriplet. 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 Task.putbarcblocktriplet but only include $$(j,k,l)$$ and $$v$$.

For explanations of other data structures used in the example see Sec. 6.7.1 (Example SDO1).

The code representing the above problem is shown below.

using (mosek.Env env = new mosek.Env())
{
{
// Directs the log task stream to the user specified

/* Append numcon empty constraints.
The constraints will initially have no bounds. */

/* Append numbarvar semidefinite variables. */

/* Set objective (6 nonzeros).*/

/* Set the equality constraint (6 nonzeros).*/

/* Set the inequality constraint (1 nonzero).*/

/* Set constraint bounds */

/* Run optimizer */

switch (solsta) {
case mosek.solsta.optimal:

/* Retrieve the soution for all symmetric variables */
Console.WriteLine("Solution (lower triangular part vectorized):");
for(int i = 0; i < numbarvar; i++) {
int dim = dimbarvar[i] * (dimbarvar[i] + 1) / 2;
double[] barx = new double[dim];

Console.Write("X" + (i+1) + ": ");
for (int j = 0; j < dim; ++j)
Console.Write(barx[j] + " ");
Console.WriteLine();
}

break;
case mosek.solsta.dual_infeas_cer:
case mosek.solsta.prim_infeas_cer:
Console.WriteLine("Primal or dual infeasibility certificate found.");
break;
case mosek.solsta.unknown:
Console.WriteLine("The status of the solution could not be determined.");
break;
default:
Console.WriteLine("Other solution status.");
break;
}
}
}

6.7.3 Example SDO_LMI: Linear matrix inequalities and the vectorized semidefinite domain¶

The standard form of a semidefinite problem is usually either based on semidefinite variables (primal form) or on linear matrix inequalities (dual form). However, MOSEK allows mixing of these two forms, as shown in (6.25)

(6.25)$\begin{split}\begin{array} {llcc} \mbox{minimize} & \left\langle \left[ \begin{array} {cc} 1 & 0 \\ 0 & 1 \end{array} \right], \barX \right\rangle + x_0 + x_1 + 1 & & \\ \mbox{subject to} & \left\langle \left[ \begin{array} {cc} 0 & 1 \\ 1 & 0 \end{array} \right], \barX \right\rangle - x_0 - x_1 & \in & \real_{\geq 0}^1, \\ & x_0 \left[ \begin{array}{cc} 0 & 1 \\ 1 & 3 \end{array} \right] + x_1 \left[ \begin{array}{cc} 3 & 1 \\ 1 & 0 \end{array} \right] - \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] & \succeq & 0, \\ & \barX \succeq 0. & & \end{array}\end{split}$

The first affine expression is restricted to a linear domain and could also be modelled as a linear constraint (instead of an ACC). The lower triangular part of the linear matrix inequality (second constraint) can be vectorized and restricted to the domaintype.svec_psd_cone. This allows us to express the constraints in (6.25) as the affine conic constraints shown in (6.26).

(6.26)$\begin{split}\begin{array}{ccccccc} \left\langle\left[\begin{array}{cc}0&1\\1&0\end{array}\right],\barX \right\rangle & + & \left[\begin{array}{cc}-1&-1\end{array}\right] x & + & \left[\begin{array}{c}0\end{array}\right] & \in & \real_{\geq 0}^1, \\ & & \left[\begin{array}{cc}0&3\\ \sqrt{2}&\sqrt{2}\\3&0\end{array}\right] x & + & \left[\begin{array}{c}-1\\0\\-1\end{array}\right] & \in & \PSD^{3,\mathrm{vec}} \end{array}\end{split}$

Vectorization of the LMI is performed as explained in Sec. 15.11 (Supported domains).

Setting up the linear part

The linear parts (objective, constraints, variables) and the semidefinite terms in the linear expressions are defined exactly as shown in the previous examples.

Setting up the affine conic constraints with semidefinite terms

To define the affine conic constraints, we first set up the affine expressions. The $$F$$ matrix and the $$g$$ vector are defined as usual. Additionally, we specify the coefficients for the semidefinite variables. The semidefinite coefficients shown in (6.26) are setup using the function Task.putafebarfblocktriplet.

/* barF block triplets */

These affine expressions are then included in their corresponding domains to construct the affine conic constraints. Lastly, the ACCs are appended to the task.

/* Append R+ domain and the corresponding ACC */
/* Append SVEC_PSD domain and the corresponding ACC */

Source code

using System;

namespace mosek.example
{
public class sdo_lmi
{
public static void Main(string[] args)
{
int    numafe    = 4;  /* Number of affine expressions.              */
int    numvar    = 2;  /* Number of scalar variables */
int[]  dimbarvar = { 2 };         /* Dimension of the semidefinite variable */
int[]  lenbarvar = { 2 * (2 + 1) / 2 }; /* Number of scalar SD variables  */

int[]     barc_j  = { 0, 0 },
barc_k  = { 0, 1 },
barc_l  = { 0, 1 };
double[]  barc_v  = { 1.0, 1.0 };

long[]    afeidx = {0, 0, 1, 2, 2, 3};
int[]     varidx = {0, 1, 1, 0, 1, 0};
double[]  f_val  = {-1, -1, 3, Math.Sqrt(2), Math.Sqrt(2), 3},
g  = {0, -1, 0, -1};

long[]    barf_i = { 0, 0 };
int[]     barf_j = { 0, 0 },
barf_k = { 0, 1 },
barf_l = { 0, 0 };
double[]  barf_v = { 0.0, 1.0 };

using (mosek.Env env = new mosek.Env())
{
{
// Directs the log task stream to the user specified
// method msgclass.streamCB
/* Append 'NUMAFE' empty affine expressions.*/

/* Append 'NUMVAR' variables.
The variables will initially be fixed at zero (x=0). */

/* Append 'NUMBARVAR' semidefinite variables. */

/* Optionally add a constant term to the objective. */
/* Set the linear term c_j in the objective.*/

for (int j = 0; j < numvar; ++j)

/* Set the linear term barc_j in the objective.*/

/* Set up the affine conic constraints */

/* Construct the affine expressions */
/* F matrix */
/* g vector */

/* barF block triplets */

/* Append R+ domain and the corresponding ACC */
/* Append SVEC_PSD domain and the corresponding ACC */

/* Run optimizer */

/* Print a summary containing information
about the solution for debugging purposes*/

switch (solsta)
{
case mosek.solsta.optimal:
double[] barx = task.getbarxj(mosek.soltype.itr, 0);   /* Request the interior solution. */
Console.WriteLine("Optimal primal solution");
for (int i = 0; i < numvar; ++i)
Console.WriteLine("x[{0}]   : {1}", i, xx[i]);

for (int i = 0; i < lenbarvar[0]; ++i)
Console.WriteLine("barx[{0}]: {1}", i, barx[i]);
break;
case mosek.solsta.dual_infeas_cer:
case mosek.solsta.prim_infeas_cer:
Console.WriteLine("Primal or dual infeasibility certificate found.");
break;
case mosek.solsta.unknown:
Console.WriteLine("The status of the solution could not be determined.");
break;
default:
Console.WriteLine("Other solution status.");
break;
}
}
}
}
}

class msgclass : mosek.Stream
{
string prefix;
public msgclass (string prfx)
{
prefix = prfx;
}

public override void streamCB (string msg)
{
Console.Write ("{0}{1}", prefix, msg);
}
}
}