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
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,
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
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. 15.8 (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:
Sec. 6.7.1 (Example SDO1): A problem with one semidefinite variable and linear and conic constraints.
Sec. 6.7.2 (Example SDO2): A problem with two semidefinite variables with a linear constraint and bound.
Sec. 6.7.3 (Example SDO_LMI: Linear matrix inequalities and the vectorized semidefinite domain): A problem with linear matrix inequalities and the vectorized semidefinite domain.
6.7.1 Example SDO1¶
We consider the simple optimization problem with semidefinite and conic quadratic constraints:
The problem description contains a 3-dimensional symmetric semidefinite variable which can be written explicitly as:
and an affine conic constraint (ACC) \((x_0, x_1, x_2) \in \Q^3\). The objective is to minimize
subject to the two linear constraints
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 appendbarvars.
# Append matrix variables of sizes in 'BARVARDIM'.
# The variables will initially be fixed at zero.
appendbarvars(task,barvardim)
Appending coefficient matrices
Coefficient matrices \(\barC_j\) and \(\barA_{ij}\) are constructed as weighted combinations of sparse symmetric matrices previously appended with the function appendsparsesymmat.
symc = appendsparsesymmat(task,barvardim[1],
barci,
barcj,
barcval)
syma0 = appendsparsesymmat(task,barvardim[1],
barai[1],
baraj[1],
baraval[1])
syma1 = appendsparsesymmat(task,barvardim[1],
barai[2],
baraj[2],
baraval[2])
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 appendsparsesymmat, we can combine them to set up the objective matrix coefficient \(\barC_j\) using 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.
putbarcj(task,1, [symc], [1.0])
Similarly, a constraint matrix coefficient \(\barA_{ij}\) is set up by the function putbaraij.
putbaraij(task,1, 1, [syma0], [1.0])
putbaraij(task,2, 1, [syma1], [1.0])
Retrieving the solution
After the problem is solved, we read the solution using getbarxj:
xx = getxx(task,MSK_SOL_ITR)
barx = getbarxj(task,MSK_SOL_ITR, 1)
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 Mosek
using Printf, SparseArrays
printstream(msg::String) = print(msg)
# Bound keys for constraints
bkc = [ MSK_BK_FX
MSK_BK_FX]
# Bound values for constraints
blc = [1.0, 0.5]
buc = [1.0, 0.5]
A = sparse( [1,2,2],[1,2,3],[1.0, 1.0, 1.0])
conesub = [1, 2, 3]
barci = [1, 2, 2, 3, 3]
barcj = [1, 1, 2, 2, 3]
barcval = [2.0, 1.0, 2.0, 1.0, 2.0]
barai = Any[ [1, 2, 3],
[1, 2, 3, 2, 3, 3] ]
baraj = Any[ [1, 2, 3],
[1, 1, 1, 2, 2, 3] ]
baraval = Any[ [1.0, 1.0, 1.0],
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0] ]
numvar = 3
numcon = length(bkc)
barvardim = [3]
# Create a task object and attach log stream printer
maketask() do task
# Use remote server: putoptserverhost(task,"http://solve.mosek.com:30080")
putstreamfunc(task,MSK_STREAM_LOG,printstream)
# Append 'numvar' variables.
# The variables will initially be fixed at zero (x=0).
appendvars(task,numvar)
# Append 'numcon' empty constraints.
# The constraints will initially have no bounds.
appendcons(task,numcon)
# Append matrix variables of sizes in 'BARVARDIM'.
# The variables will initially be fixed at zero.
appendbarvars(task,barvardim)
# Set the linear term c_0 in the objective.
putcj(task, 1, 1.0)
# Set the bounds on variable j
# blx[j] <= x_j <= bux[j]
putvarboundsliceconst(task,1,numvar+1,
MSK_BK_FR,
-Inf,
+Inf)
# Set the bounds on constraints.
# blc[i] <= constraint_i <= buc[i]
putconboundslice(task,1,numcon+1, bkc,blc,buc)
# Append the conic quadratic cone
let afei = getnumafe(task)+1,
dom = appendquadraticconedomain(task,3)
appendafes(task,3)
putafefentrylist(task,[1,2,3],
[1,2,3],
[1.0,1.0,1.0])
appendaccseq(task,dom,afei,nothing)
end
# Input row i of A
putacolslice(task,1,numvar+1,
A.colptr[1:numvar], A.colptr[2:numvar+1],
A.rowval,A.nzval)
symc = appendsparsesymmat(task,barvardim[1],
barci,
barcj,
barcval)
syma0 = appendsparsesymmat(task,barvardim[1],
barai[1],
baraj[1],
baraval[1])
syma1 = appendsparsesymmat(task,barvardim[1],
barai[2],
baraj[2],
baraval[2])
putbarcj(task,1, [symc], [1.0])
putbaraij(task,1, 1, [syma0], [1.0])
putbaraij(task,2, 1, [syma1], [1.0])
# Input the objective sense (minimize/maximize)
putobjsense(task,MSK_OBJECTIVE_SENSE_MINIMIZE)
# Solve the problem and print summary
optimize(task)
writedata(task,"sdo1.ptf")
solutionsummary(task,MSK_STREAM_MSG)
# Get status information about the solution
prosta = getprosta(task,MSK_SOL_ITR)
solsta = getsolsta(task,MSK_SOL_ITR)
if solsta == MSK_SOL_STA_OPTIMAL
# Output a solution
xx = getxx(task,MSK_SOL_ITR)
barx = getbarxj(task,MSK_SOL_ITR, 1)
@printf("Optimal solution: \n xx = %s\n barx = %s\n", xx',barx')
elseif solsta == MSK_SOL_STA_DUAL_INFEAS_CER
println("Primal or dual infeasibility.\n")
elseif solsta == MSK_SOL_STA_PRIM_INFEAS_CER
println("Primal or dual infeasibility.\n")
elseif solsta == MSK_SOL_STA_UNKNOWN
println("Unknown solution status")
else
println("Other solution status")
end
end
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:
In our example \(\dim(\barX_1)=3\), \(\dim(\barX_2)=4\), \(b=23\), \(k=-3\) and
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 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 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
# Input data
let numcon = 2, # Number of constraints.
numbarvar = 2,
dimbarvar = Int32[3, 4], # Dimension of semidefinite variables
# Objective coefficients concatenated
Cj = Int32[ 1, 1, 2, 2, 2, 2 ], # Which symmetric variable (j)
Ck = Int32[ 1, 3, 1, 2, 2, 3 ], # Which entry (k,l)->v
Cl = Int32[ 1, 3, 1, 1, 2, 3 ],
Cv = [ 1.0, 6.0, 1.0, -3.0, 2.0, 1.0 ],
# Equality constraints coefficients concatenated
Ai = Int32[ 1, 1, 1, 1, 1, 1 ], # Which constraint (i = 0)
Aj = Int32[ 1, 1, 1, 2, 2, 2 ], # Which symmetric variable (j)
Ak = Int32[ 1, 3, 3, 2, 2, 4 ], # Which entry (k,l)->v
Al = Int32[ 1, 1, 3, 1, 2, 4 ],
Av = [ 1.0, 1.0, 2.0, 1.0, -1.0, -3.0 ],
# The second constraint - one-term inequality
A2i = Int32[ 2 ], # Which constraint (i = 1)
A2j = Int32[ 2 ], # Which symmetric variable (j = 1)
A2k = Int32[ 2 ], # Which entry A(1,0) = A(0,1) = 0.5
A2l = Int32[ 1 ],
A2v = [ 0.5 ],
bkc = [ MSK_BK_FX,
MSK_BK_UP ],
blc = [ 23.0, 0.0 ],
buc = [ 23.0, -3.0 ]
maketask() do task
# Use remote server: putoptserverhost(task,"http://solve.mosek.com:30080")
putstreamfunc(task,MSK_STREAM_LOG,msg -> print(msg))
# Append numcon empty constraints.
# The constraints will initially have no bounds.
appendcons(task,numcon)
# Append numbarvar semidefinite variables.
appendbarvars(task,dimbarvar)
# Set objective (6 nonzeros).
putbarcblocktriplet(task,Cj, Ck, Cl, Cv)
# Set the equality constraint (6 nonzeros).
putbarablocktriplet(task,Ai, Aj, Ak, Al, Av)
# Set the inequality constraint (1 nonzero).
putbarablocktriplet(task,A2i, A2j, A2k, A2l, A2v)
# Set constraint bounds
putconboundslice(task,1, 3, bkc, blc, buc)
# Run optimizer
optimize(task)
solutionsummary(task,MSK_STREAM_MSG)
solsta = getsolsta(task,MSK_SOL_ITR)
if solsta == MSK_SOL_STA_OPTIMAL
# Retrieve the soution for all symmetric variables
println("Solution (lower triangular part vectorized):")
for i in 1:numbarvar
barx = getbarxj(task,MSK_SOL_ITR, i)
println("X$i: $barx")
end
elseif solsta == MSK_SOL_STA_DUAL_INFEAS_CER || if solsta == MSK_SOL_STA_PRIM_INFEAS_CER
println("Primal or dual infeasibility certificate found.")
elseif solsta == MSK_SOL_STA_UNKNOWN
println("The status of the solution could not be determined.")
else
println("Other solution status.")
end
end
end
end
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)
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 MSK_DOMAIN_SVEC_PSD_CONE. This allows us to express the constraints in (6.25) as the affine conic constraints shown in (6.26).
Vectorization of the LMI is performed as explained in Sec. 15.8 (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 putafebarfblocktriplet.
# barF block triplets
putafebarfblocktriplet(task,barf_i, barf_j, barf_k, barf_l, barf_v)
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
appendacc(task,appendrplusdomain(task,1), [1], nothing)
# Append SVEC_PSD domain and the corresponding ACC
appendacc(task,appendsvecpsdconedomain(task,3), [2,3,4], nothing)
Source code
using Mosek
let numafe = 4, # Number of affine expressions.
numvar = 2, # Number of scalar variables
dimbarvar = [2], # Dimension of semidefinite cone
lenbarvar = [2 * (2 + 1) / 2], # Number of scalar SD variables
barc_j = [1, 1],
barc_k = [1, 2],
barc_l = [1, 2],
barc_v = [1.0, 1.0],
afeidx = [1, 1, 2, 3, 3, 4],
varidx = [1, 2, 2, 1, 2, 1],
f_val = Float64[-1, -1, 3, sqrt(2.0), sqrt(2.0), 3],
g = Float64[0, -1, 0, -1],
barf_i = [1, 1],
barf_j = [1, 1],
barf_k = [1, 2],
barf_l = [1, 1],
barf_v = [0.0, 1.0]
maketask() do task
# Use remote server: putoptserverhost(task,"http://solve.mosek.com:30080")
# Append 'NUMAFE' empty affine expressions.
appendafes(task,numafe)
# Append 'NUMVAR' variables.
# The variables will initially be fixed at zero (x=0).
appendvars(task,numvar)
# Append 'NUMBARVAR' semidefinite variables.
appendbarvars(task,dimbarvar)
# Optionally add a constant term to the objective.
putcfix(task,1.0)
# Set the linear term c_j in the objective.
putcj(task,1, 1.0)
putcj(task,2, 1.0)
for j in 1:numvar
putvarbound(task,j, MSK_BK_FR, -0.0, 0.0)
end
# Set the linear term barc_j in the objective.
putbarcblocktriplet(task, barc_j, barc_k, barc_l, barc_v)
# Set up the affine conic constraints
# Construct the affine expressions
# F matrix
putafefentrylist(task,afeidx, varidx, f_val)
# g vector
putafegslice(task,1, 5, g)
# barF block triplets
putafebarfblocktriplet(task,barf_i, barf_j, barf_k, barf_l, barf_v)
# Append R+ domain and the corresponding ACC
appendacc(task,appendrplusdomain(task,1), [1], nothing)
# Append SVEC_PSD domain and the corresponding ACC
appendacc(task,appendsvecpsdconedomain(task,3), [2,3,4], nothing)
# Run optimizer
optimize(task)
# Print a summary containing information
# about the solution for debugging purposes
solutionsummary(task,MSK_STREAM_MSG)
solsta = getsolsta(task,MSK_SOL_ITR)
if solsta == MSK_SOL_STA_OPTIMAL
xx = getxx(task,MSK_SOL_ITR)
barx = getbarxj(task,MSK_SOL_ITR,1); # Request the interior solution.
println("Optimal primal solution, x = $xx, barx = $barx")
elseif solsta == MSK_SOL_STA_PRIM_INFEAS_CER || solsta == MSK_SOL_STA_DUAL_INFEAS_CER
println("Primal or dual infeasibility certificate found.")
elseif solsta == MSK_SOL_STA_UNKNOWN
println("The status of the solution could not be determined.")
else
println("Other solution status.")
end
end
end
