Optimal Experimental Design

Optimal experimental design is a theory at the interface of statistics and optimization, which studies how to allocate some statistical trials within a set of available design points. The goal is to allow for the best possible estimation of an unknown parameter \theta. In what follows, we assume the standard linear model with multiresponse experiments: a trial in the i^{\textrm{th}} design point gives a multidimensional observation that can be written as y_i = A_i^T \theta+\epsilon_i, where y_i is of dimension l_i, A_i is a m \times l_i- matrix, and the error vectors \epsilon_i are i.i.d. with a unit variance.

Several optimization criteria exist, leading to different SDP, SOCP and LP formulations. As such, optimal experimental design problens are natural examples for problems in conic optimization. For a review of the different formulations and more references, see [1].

The code below initializes the data used in all the examples of this page. It should be run prior to any of the codes presented in this page.

>>> import cvxopt as cvx
>>> import picos
>>> #---------------------------------#
>>> # First generate some data :      #
>>> #       _ a list of 8 matrices A  #
>>> #       _ a vector c              #
>>> #---------------------------------#
>>> A = [cvx.matrix([[1,0,0,0,0],
...                  [0,3,0,0,0],
...                  [0,0,1,0,0]]),
...      cvx.matrix([[0,0,2,0,0],
...                  [0,1,0,0,0],
...                  [0,0,0,1,0]]),
...      cvx.matrix([[0,0,0,2,0],
...                  [4,0,0,0,0],
...                  [0,0,1,0,0]]),
...      cvx.matrix([[1,0,0,0,0],
...                  [0,0,2,0,0],
...                  [0,0,0,0,4]]),
...      cvx.matrix([[1,0,2,0,0],
...                  [0,3,0,1,2],
...                  [0,0,1,2,0]]),
...      cvx.matrix([[0,1,1,1,0],
...                  [0,3,0,1,0],
...                  [0,0,2,2,0]]),
...      cvx.matrix([[1,2,0,0,0],
...                  [0,3,3,0,5],
...                  [1,0,0,2,0]]),
...      cvx.matrix([[1,0,3,0,1],
...                  [0,3,2,0,0],
...                  [1,0,0,2,0]])
... ]
>>> c = cvx.matrix([1,2,3,4,5])

Multi-response c-optimal design (SOCP)

We compute the c-optimal design (c=[1,2,3,4,5]) for the observation matrices A[i].T from the variable A defined above. The results below suggest that we should allocate 12.8% of the experimental effort on design point #5, and 87.2% on the design point #7.

Primal problem

The SOCP for multiresponse c-optimal design is:

\begin{center}
\begin{eqnarray*}
&\underset{\substack{\mu \in \mathbb{R}^s\\
                     \forall i \in [s],\ z_i \in \mathbb{R}^{l_i}}}{\mbox{minimize}}
                   & \sum_{i=1}^s \mu_i\\
&\mbox{subject to} & \sum_{i=1}^s A_i z_i = c\\
&                  & \forall i \in [s],\ \Vert z_i \Vert_2 \leq \mu_i,
\end{eqnarray*}
\end{center}

>>> # create the problem, variables and params
>>> c_primal_SOCP = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> cc = picos.Constant('c', c)
>>> z  = [picos.RealVariable('z[{0}]'.format(i), AA[i].size[1])
...        for i in range(s)]
>>> mu = picos.RealVariable('mu', s)
>>> # define the constraints and objective function
>>> cones = c_primal_SOCP.add_list_of_constraints([abs(z[i]) <= mu[i] for i in range(s)])
>>> lin   = c_primal_SOCP.add_constraint(picos.sum([AA[i] * z[i] for i in range(s)]) == cc)
>>> c_primal_SOCP.set_objective('min', (1|mu) )
>>> print(c_primal_SOCP)
Second Order Cone Program
  minimize ∑(mu)
  over
    3×1 real variable z[i] ∀ i ∈ [0…7]
    8×1 real variable mu
  subject to
    ‖z[i]‖ ≤ mu[i] ∀ i ∈ [0…7]
    ∑(A[i]·z[i] : i ∈ [0…7]) = c
>>> #solve the problem and retrieve the optimal weights of the optimal design.
>>> solution = c_primal_SOCP.solve(solver='cvxopt')
>>> mu = mu.value
>>> w = mu / sum(mu) #normalize mu to get the optimal weights

The optimal design is:

>>> print(w)
[...]
[...]
[...]
[...]
[ 1.28e-01]
[...]
[ 8.72e-01]
[...]

The [...] above indicate a numerical zero entry (i.e., which can be something like 2.84e-10). We use the ellipsis ... instead for clarity and compatibility with doctest.

Dual problem

This is only to check that we obtain the same solution with the dual problem, and to provide one additional example in this tutorial:

\begin{center}
\begin{eqnarray*}
&\underset{u \in \mathbb{R}^m}{\mbox{maximize}}
                   & c^T u\\
&\mbox{subject to} & \forall i \in [s],\ \Vert A_i^T u \Vert_2 \leq 1
\end{eqnarray*}
\end{center}

>>> # create the problem, variables and params
>>> c_dual_SOCP = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> cc = picos.Constant('c',c)
>>> u  = picos.RealVariable('u',c.size)
>>> # define the constraints and objective function
>>> cones = c_dual_SOCP.add_list_of_constraints(
...         [abs(AA[i].T*u)<=1 for i in range(s)])
>>> c_dual_SOCP.set_objective('max', (cc|u) )
>>> print(c_dual_SOCP)#
Second Order Cone Program
  maximize ⟨c, u⟩
  over
    5×1 real variable u
  subject to
    ‖A[i]ᵀ·u‖ ≤ 1 ∀ i ∈ [0…7]
>>> #solve the problem and retrieve the weights of the optimal design
>>> solution = c_dual_SOCP.solve(solver='cvxopt')
>>> mu = [cons.dual[0] for cons in cones] #Lagrangian duals of the SOC constraints
>>> mu = cvx.matrix(mu)
>>> w=mu/sum(mu) #normalize mu to get the optimal weights

The optimal design is:

>>> print(w)
[...]
[...]
[...]
[...]
[ 1.28e-01]
[...]
[ 8.72e-01]
[...]

Single-response c-optimal design (LP)

When the observation matrices are row vectors (single-response framework), the SOCP above reduces to a simple LP, because the variables z_i are scalar. We solve below the LP for the case where there are 11 available design points, corresponding to the columns of the matrices A[4], A[5], A[6], and A[7][:,:-1] defined in the preambule.

The optimal design allocates 3.37% to point #5 (2nd column of A[5]), 27.9% to point #7 (1st column of A[6]), 11.8% to point #8 (2nd column of A[6]), 27.6% to point #9 (3rd column of A[6]), and 29.3% to point #11 (2nd column of A[7]).

>>> # create the problem, variables and params
>>> c_primal_LP = picos.Problem()
>>> A1 = [cvx.sparse(a[:,i],tc='d') for i in range(3) for a in A[4:]] #12 column vectors
>>> A1 = A1[:-1] # remove the last design point (it is the same as the last-but-one)
>>> s = len(A1)
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A1)] # each AA[i].T is a 1 x 5 observation matrix
>>> cc = picos.Constant('c', c)
>>> z = [picos.RealVariable('z[{0}]'.format(i), 1) for i in range(s)]
>>> mu = picos.RealVariable('mu', s)
>>> #define the constraints and objective function
>>> abs_con = c_primal_LP.add_list_of_constraints([abs(z[i]) <= mu[i] for i in range(s)])
>>> lin_con = c_primal_LP.add_constraint(picos.sum([AA[i]*z[i] for i in range(s)]) == cc)
>>> c_primal_LP.set_objective('min', (1|mu))

Note that there are no cone constraints, because the constraints of the form |z_i| \leq \mu_i are handled as two inequalities when z_i is scalar, so the problem is a LP indeed:

>>> print(c_primal_LP)
Linear Program
  minimize ∑(mu)
  over
    1×1 real variable z[i] ∀ i ∈ [0…10]
    11×1 real variable mu
  subject to
    |z[i]| ≤ mu[i] ∀ i ∈ [0…10]
    ∑(A[i]·z[i] : i ∈ [0…10]) = c
>>> #solve the problem and retrieve the weights of the optimal design
>>> solution = c_primal_LP.solve(solver='cvxopt')
>>> mu = mu.value
>>> w = mu / sum(mu) #normalize mu to get the optimal weights

The optimal design is:

>>> print(w)
[...]
[...]
[...]
[...]
[ 3.37e-02]
[...]
[ 2.79e-01]
[ 1.18e-01]
[ 2.76e-01]
[...]
[ 2.93e-01]

SDP formulation of c-optimal design

We give below the SDP for c-optimality, in primal and dual form. You can observe that we obtain the same results as with the SOCP presented earlier: 12.8% on design point #5, and 87.2% on design point #7.

Primal problem

The SDP formulation of the c-optimal design problem is:

\begin{center}
\begin{eqnarray*}
&\underset{\mu \in \mathbb{R}^s}{\mbox{minimize}}
                   & \sum_{i=1}^s \mu_i\\
&\mbox{subject to} & \sum_{i=1}^s \mu_i A_i A_i^T \succeq c c^T,\\
&                  & \mu \geq 0.
\end{eqnarray*}
\end{center}

>>> # create the problem, variables and params
>>> c_primal_SDP = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> cc = picos.Constant('c', c)
>>> mu = picos.RealVariable('mu',s)
>>> # define the constraints and objective function
>>> lmi = c_primal_SDP.add_constraint(
...         picos.sum([mu[i] * AA[i] * AA[i].T for i in range(s)]) >> cc*cc.T)
>>> lin_cons = c_primal_SDP.add_constraint(mu >= 0)
>>> c_primal_SDP.set_objective('min', (1|mu) )
>>> print(c_primal_SDP)
Semidefinite Program
  minimize ∑(mu)
  over
    8×1 real variable mu
  subject to
    ∑(mu[i]·A[i]·A[i]ᵀ : i ∈ [0…7]) ≽ c·cᵀ
    mu ≥ 0
>>> #solve the problem and retrieve the weights of the optimal design
>>> solution = c_primal_SDP.solve(solver='cvxopt')
>>> w = mu.value
>>> w = w / sum(w) #normalize mu to get the optimal weights

The optimal design is:

>>> print(w)
[...]
[...]
[...]
[...]
[ 1.28e-01]
[...]
[ 8.72e-01]
[...]

Dual problem

This is only to check that we obtain the same solution with the dual problem, and to provide one additional example in this tutorial:

\begin{center}
\begin{eqnarray*}
&\underset{X \in \mathbb{R}^{m \times m}}{\mbox{maximize}}
                   &  c^T X c\\
&\mbox{subject to} & \forall i \in [s],\ \langle A_i A_i^T,\ X \rangle \leq 1,\\
&                  &  X \succeq 0.
\end{eqnarray*}
\end{center}

>>> #create the problem, variables and params
>>> c_dual_SDP = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> cc = picos.Constant('c', c)
>>> m  = c.size[0]
>>> X  = picos.SymmetricVariable('X',(m,m))
>>> #define the constraints and objective function
>>> lin_cons = c_dual_SDP.add_list_of_constraints(
...                  [(AA[i]*AA[i].T | X ) <= 1 for i in range(s)])
>>> psd = c_dual_SDP.add_constraint(X>>0)
>>> c_dual_SDP.set_objective('max', cc.T*X*cc)
>>> print(c_dual_SDP)
Semidefinite Program
  maximize cᵀ·X·c
  over
    5×5 symmetric variable X
  subject to
    ⟨A[i]·A[i]ᵀ, X⟩ ≤ 1 ∀ i ∈ [0…7]
    X ≽ 0
>>> # solve the problem and retrieve the weights of the optimal design
>>> solution = c_dual_SDP.solve(solver='cvxopt')
>>> mu = [cons.dual for cons in lin_cons] #Lagrangian duals of the linear constraints
>>> mu = cvx.matrix(mu)
>>> w = mu / sum(mu) #normalize mu to get the optimal weights

The optimal design is:

>>> print(w)
[...]
[...]
[...]
[...]
[ 1.28e-01]
[...]
[ 8.72e-01]
[...]

And the optimal positive semidefinite matrix X is:

>>> print(X)
[ 5.92e-03  8.98e-03  2.82e-03 -3.48e-02 -1.43e-02]
[ 8.98e-03  1.36e-02  4.27e-03 -5.28e-02 -2.17e-02]
[ 2.82e-03  4.27e-03  1.34e-03 -1.66e-02 -6.79e-03]
[-3.48e-02 -5.28e-02 -1.66e-02  2.05e-01  8.39e-02]
[-1.43e-02 -2.17e-02 -6.79e-03  8.39e-02  3.44e-02]

A-optimality (SOCP)

We compute the A-optimal design for the observation matrices A[i].T defined in the preambule. The optimal design allocates 24.9% on design point #3, 14.2% on point #4, 8.51% on point #5, 12.1% on point #6, 13.2% on point #7, and 27.0% on point #8.

Primal problem

The SOCP for the A-optimal design problem is:

\begin{center}
\begin{eqnarray*}
&\underset{\substack{\mu \in \mathbb{R}^s\\
                     \forall i \in [s],\ Z_i \in \mathbb{R}^{l_i \times m}}}{\mbox{minimize}}
                   & \sum_{i=1}^s \mu_i\\
&\mbox{subject to} & \sum_{i=1}^s A_i Z_i = I\\
&                  & \forall i \in [s],\ \Vert Z_i \Vert_F \leq \mu_i,
\end{eqnarray*}
\end{center}

>>> # create the problem, variables and params
>>> A_primal_SOCP = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> Z = [picos.RealVariable('Z[{0}]'.format(i), AA[i].T.size) for i in range(s)]
>>> mu = picos.RealVariable('mu', s)
>>> #define the constraints and objective function
>>> cone_cons = A_primal_SOCP.add_list_of_constraints(
...                     [abs(Z[i]) <= mu[i] for i in range(s)])
>>> lin_cons = A_primal_SOCP.add_constraint(
...                      picos.sum([AA[i] * Z[i] for i in range(s)]) == 'I')
>>> A_primal_SOCP.set_objective('min', (1|mu) )
>>> print(A_primal_SOCP)
Second Order Cone Program
  minimize ∑(mu)
  over
    3×5 real variable Z[i] ∀ i ∈ [0…7]
    8×1 real variable mu
  subject to
    ‖Z[i]‖ ≤ mu[i] ∀ i ∈ [0…7]
    ∑(A[i]·Z[i] : i ∈ [0…7]) = I
>>> # solve the problem and retrieve the weights of the optimal design
>>> solution = A_primal_SOCP.solve(solver='cvxopt')
>>> w = mu.value
>>> w = w / sum(w) #normalize mu to get the optimal weights

The optimal design is:

>>> print(w)
[...]
[...]
[ 2.49e-01]
[ 1.42e-01]
[ 8.51e-02]
[ 1.21e-01]
[ 1.32e-01]
[ 2.70e-01]

Dual problem

This is only to check that we obtain the same solution with the dual problem, and to provide one additional example in this tutorial:

\begin{center}
\begin{eqnarray*}
&\underset{U \in \mathbb{R}^{m \times m}}{\mbox{maximize}}
                   &  \mbox{trace}\ U\\
&\mbox{subject to} & \forall i \in [s],\ \Vert A_i^T U \Vert_2 \leq 1
\end{eqnarray*}
\end{center}

>>> #create the problem, variables and params
>>> D_SOCPual_A=picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> m  = AA[0].size[0]
>>> U  = picos.RealVariable('U',(m,m))
>>> #define the constraints and objective function
>>> cone_cons = D_SOCPual_A.add_list_of_constraints(
...       [abs(AA[i].T*U) <= 1 for i in range(s)])
>>> D_SOCPual_A.set_objective('max', 'I'|U)
>>> print(D_SOCPual_A)
Second Order Cone Program
  maximize tr(U)
  over
    5×5 real variable U
  subject to
    ‖A[i]ᵀ·U‖ ≤ 1 ∀ i ∈ [0…7]
>>> # solve the problem and retrieve the weights of the optimal design
>>> solution = D_SOCPual_A.solve(solver='cvxopt')
>>> mu = [cons.dual[0] for cons in cone_cons] # Lagrangian duals of the SOC constraints
>>> mu = cvx.matrix(mu)
>>> w = mu / sum(mu) # normalize mu to get the optimal weights

The optimal design is:

>>> print(w)
[...]
[...]
[ 2.49e-01]
[ 1.42e-01]
[ 8.51e-02]
[ 1.21e-01]
[ 1.32e-01]
[ 2.70e-01]

A-optimality with multiple constraints (SOCP)

A-optimal designs can also be computed by SOCP when the vector of weights \mathbf{w} is subject to several linear constraints. To give an example, we compute the A-optimal design for the observation matrices given in the preambule, when the weights must satisfy: \sum_{i=0}^3 w_i \leq 0.5 and \sum_{i=4}^7 w_i \leq 0.5. This problem has the following SOCP formulation:

\begin{center}
\begin{eqnarray*}
&\underset{\substack{\mathbf{w} \in \mathbb{R}^s\\
                     \mu \in \mathbb{R}^s\\
                     \forall i \in [s],\ Z_i \in \mathbb{R}^{l_i \times m}}}{\mbox{minimize}}
                   & \sum_{i=1}^s \mu_i\\
&\mbox{subject to} & \sum_{i=1}^s A_i Z_i = I\\
&                  & \sum_{i=0}^3 w_i \leq 0.5\\
&                  & \sum_{i=4}^7 w_i \leq 0.5\\
&                  & \forall i \in [s],\ \Vert Z_i \Vert_F^2 \leq \mu_i w_i,
\end{eqnarray*}
\end{center}

The optimal solution allocates 29.7% and 20.3% to the design points #3 and #4, and respectively 6.54%, 11.9%, 9.02% and 22.5% to the design points #5 to #8:

>>> # create the problem, variables and params
>>> A_multiconstraints = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> mu = picos.RealVariable('mu',s)
>>> w  = picos.RealVariable('w',s)
>>> Z  = [picos.RealVariable('Z[{0}]'.format(i), AA[i].T.size)
...                          for i in range(s)]
>>> # define the constraints and objective function
>>> lin_cons0 = A_multiconstraints.add_constraint(
...         picos.sum([AA[i] * Z[i] for i in range(s)]) == 'I')
>>> lin_cons1 = A_multiconstraints.add_constraint( (1|w[:4]) <= 0.5)
>>> lin_cons2 = A_multiconstraints.add_constraint( (1|w[4:]) <= 0.5)
>>> cone_cons = A_multiconstraints.add_list_of_constraints(
...       [ abs(Z[i]) **2 <= mu[i] * w[i] for i in range(s)])
>>> A_multiconstraints.set_objective('min', (1|mu) )
>>> print(A_multiconstraints)
Quadratically Constrained Program
  minimize ∑(mu)
  over
    3×5 real variable Z[i] ∀ i ∈ [0…7]
    8×1 real variable mu, w
  subject to
    ∑(A[i]·Z[i] : i ∈ [0…7]) = I
    ∑(w[:4]) ≤ 0.5
    ∑(w[4:]) ≤ 0.5
    ‖Z[i]‖² ≤ mu[i]·w[i] ∀ i ∈ [0…7]
>>> # solve the problem and retrieve the weights of the optimal design
>>> solution = A_multiconstraints.solve(solver='cvxopt')
>>> w = w.value
>>> w = w / sum(w) # normalize w to get the optimal weights

The optimal design is:

>>> print(w)
[...]
[...]
[ 2.97e-01]
[ 2.03e-01]
[ 6.54e-02]
[ 1.19e-01]
[ 9.02e-02]
[ 2.25e-01]

Exact A-optimal design (MISOCP)

In the exact version of A-optimality, a number N \in \mathbb{N} of trials is given, and the goal is to find the optimal number of times n_i \in \mathbb{N} that a trial on design point #i should be performed, with \sum_i n_i =N.

The SOCP formulation of A-optimality for constrained designs also accept integer constraints, which results in a MISOCP for exact A-optimality:

\begin{center}
\begin{eqnarray*}
&\underset{\substack{\mathbf{t} \in \mathbb{R}^s\\
                     \mathbf{n} \in \mathbb{N}^s\\
                     \forall i \in [s],\ Z_i \in \mathbb{R}^{l_i \times m}}}{\mbox{minimize}}
                   & \sum_{i=1}^s t_i\\
&\mbox{subject to} & \sum_{i=1}^s A_i Z_i = I\\
&                  & \forall i \in [s],\ \Vert Z_i \Vert_F^2 \leq n_i t_i,\\
&                  & \sum_{i=1}^s n_i = N.
\end{eqnarray*}
\end{center}

The exact optimal design is \mathbf{n}=[0,0,5,3,2,2,3,5]:

>>> # create the problem, variables and params
>>> A_exact = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> m  = AA[0].size[0]
>>> N  = picos.Constant('N', 20) # number of trials allowed
>>> I = picos.Constant('I', cvx.spmatrix([1]*m,range(m),range(m),(m,m))) #identity matrix
>>> Z = [picos.RealVariable('Z[{0}]'.format(i), AA[i].T.size) for i in range(s)]
>>> n = picos.IntegerVariable('n', s)
>>> t = picos.RealVariable('t', s)
>>> # define the constraints and objective function
>>> cone_cons = A_exact.add_list_of_constraints(
...         [ abs(Z[i])**2 <= n[i] * t[i] for i in range(s)])
>>> lin_cons = A_exact.add_constraint(
...          picos.sum([AA[i]*Z[i] for i in range(s)]) == I)
>>> wgt_cons = A_exact.add_constraint( (1|n) <= N )
>>> A_exact.set_objective('min',1|t)
>>> print(A_exact)
Mixed-Integer Quadratically Constrained Program
  minimize ∑(t)
  over
    8×1 integer variable n
    3×5 real variable Z[i] ∀ i ∈ [0…7]
    8×1 real variable t
  subject to
    ‖Z[i]‖² ≤ n[i]·t[i] ∀ i ∈ [0…7]
    ∑(A[i]·Z[i] : i ∈ [0…7]) = I
    ∑(n) ≤ N
>>> #solve the problem and display the optimal design
>>> solution = A_exact.solve()
>>> print(n)
[...]
[...]
[ 5.00e+00]
[ 3.00e+00]
[ 2.00e+00]
[ 2.00e+00]
[ 3.00e+00]
[ 5.00e+00]

Note

The above output is not validated as we lack an appropriate solver on the build server.

Approximate and exact D-optimal design ((MI)SOCP)

The D-optimal design problem has a SOCP formulation involving a geometric mean in the objective function:

\begin{center}
\begin{eqnarray*}
&\underset{\substack{\mathbf{L} \in \mathbb{R}^{m \times m}\\
                     \mathbf{w} \in \mathbb{R}^s\\
                     \forall i \in [s],\ V_i \in \mathbb{R}^{l_i \times m}}}{\mbox{maximize}}
                   & \left(\prod_{i=1}^m L_{i,i}\right)^{1/m}\\
&\mbox{subject to} & \sum_{i=1}^s A_i V_i = L,\\
&                  & L\ \mbox{lower triangular},\\
&                  & \Vert V_i \Vert_F \leq \sqrt{m}\ w_i,\\
&                  & \sum_{i=1}^s w_i \leq 1.
\end{eqnarray*}
\end{center}

By introducing a new variable t such that t \leq \left(\prod_{i=1}^m L_{i,i}\right)^{1/m}, we can pass this problem to PICOS with the function geomean, which reformulates the geometric mean inequality as a set of equivalent second order cone constraints. The example below allocates respectively 22.7%, 3.38%, 1.65%, 5.44%, 31.8% and 35.1% to the design points #3 to #8.

>>> #create the problem, variables and params
>>> D_SOCP = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> m  = AA[0].size[0]
>>> mm = picos.Constant('m', m)
>>> L = picos.RealVariable('L', (m,m))
>>> V = [picos.RealVariable('V['+str(i)+']', AA[i].T.size) for i in range(s)]
>>> w = picos.RealVariable('w',s)
>>> # additional variable to handle the geometric mean in the objective function
>>> t = picos.RealVariable('t',1)
>>> # define the constraints and objective function
>>> lin_cons = D_SOCP.add_constraint(picos.sum([AA[i]*V[i] for i in range(s)]) == L)
>>> # L is lower triangular
>>> lowtri_cons = D_SOCP.add_list_of_constraints( [L[i,j] == 0
...                for i in range(m)
...                for j in range(i+1,m)])
>>> cone_cons = D_SOCP.add_list_of_constraints([abs(V[i]) <= (mm**0.5)*w[i]
...                                                 for i in range(s)])
>>> wgt_cons = D_SOCP.add_constraint(1|w <= 1)
>>> geomean_cons = D_SOCP.add_constraint(t <= picos.geomean(picos.maindiag(L)))
>>> D_SOCP.set_objective('max',t)
>>> #solve the problem and display the optimal design
>>> print(D_SOCP)
Optimization Problem
  maximize t
  over
    1×1 real variable t
    3×5 real variable V[i] ∀ i ∈ [0…7]
    5×5 real variable L
    8×1 real variable w
  subject to
    L = ∑(A[i]·V[i] : i ∈ [0…7])
    L[i,j] = 0 ∀ (i,j) ∈ zip([0,0,…,2,3],[1,2,…,4,4])
    ‖V[i]‖ ≤ m^(1/2)·w[i] ∀ i ∈ [0…7]
    ∑(w) ≤ 1
    geomean(maindiag(L)) ≥ t
>>> solution = D_SOCP.solve(solver='cvxopt')
>>> print(w)
[...]
[...]
[ 2.27e-01]
[ 3.38e-02]
[ 1.65e-02]
[ 5.44e-02]
[ 3.18e-01]
[ 3.51e-01]

As for the A-optimal problem, there is an alternative SOCP formulation of D-optimality [2], in which integer constraints may be added. This allows us to formulate the exact D-optimal problem as a MISOCP. For N=20, we obtain the following N-exact D-optimal design: \mathbf{n}=[0,0,5,1,0,1,6,7]:

>>> # create the problem, variables and params
>>> D_exact = picos.Problem()
>>> L = picos.RealVariable('L',(m,m))
>>> V = [picos.RealVariable('V['+str(i)+']',AA[i].T.size) for i in range(s)]
>>> T = picos.RealVariable('T', (s,m))
>>> n = picos.IntegerVariable('n', s)
>>> N = picos.Constant('N', 20)
>>> # additional variable to handle the geomean inequality
>>> t = picos.RealVariable('t',1)
>>> # define the constraints and objective function
>>> lin_cons = D_exact.add_constraint(
...         picos.sum([AA[i]*V[i] for i in range(s)]) == L)
>>> # L is lower triangular
>>> lowtri_cons = D_exact.add_list_of_constraints( [L[i,j] == 0
...                                  for i in range(m)
...                                  for j in range(i+1,m)])
>>> cone_cons = D_exact.add_list_of_constraints([ abs(V[i][:,k])**2 <= n[i]/N*T[i,k]
...                 for i in range(s) for k in range(m)])
>>> lin_cons2 = D_exact.add_list_of_constraints([(1|T[:,k]) <= 1
...                       for k in range(m)])
>>> wgt_cons = D_exact.add_constraint(1|n <= N)
>>> geomean_cons = D_exact.add_constraint(t <= picos.geomean( picos.maindiag(L)))
>>> D_exact.set_objective('max',t)
>>> print(D_exact)
Mixed-Integer Optimization Problem
  maximize t
  over
    8×1 integer variable n
    1×1 real variable t
    3×5 real variable V[i] ∀ i ∈ [0…7]
    5×5 real variable L
    8×5 real variable T
  subject to
    L = ∑(A[i]·V[i] : i ∈ [0…7])
    L[i,j] = 0 ∀ (i,j) ∈ zip([0,0,…,2,3],[1,2,…,4,4])
    ‖V[i][:,j]‖² ≤ n[i]/N·T[i,j] ∀ (i,j) ∈
      zip([0,0,…,7,7],[0,1,…,3,4])
    ∑(T[:,i]) ≤ 1 ∀ i ∈ [0…4]
    ∑(n) ≤ N
    geomean(maindiag(L)) ≥ t
>>> #solve the problem and display the optimal design
>>> solution = D_exact.solve()
>>> print(n)
[...]
[...]
[ 5.00e+00]
[ 1.00e+00]
[...]
[ 1.00e+00]
[ 6.00e+00]
[ 7.00e+00]

Note

The above output is not validated as we lack an appropriate solver on the build server.

Former MAXDET formulation of the D-optimal design (SDP)

A so-called MAXDET Programming formulation of the D-optimal design has been known since the late 90’s [3], and can be reformulated as a SDP thanks to the detrootn function. The following code finds the same design as the SOCP approach presented above.

>>> # problem, variables and parameters
>>> D_MAXDET = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> m  = AA[0].size[0]
>>> w = picos.RealVariable('w', s, lower=0)
>>> t = picos.RealVariable('t', 1)
>>> # constraint and objective
>>> wgt_cons = D_MAXDET.add_constraint(1|w <= 1)
>>> Mw = picos.sum([w[i] * AA[i] * AA[i].T for i in range(s)])
>>> detrootn_cons = D_MAXDET.add_constraint(t <= picos.DetRootN(Mw))
>>> D_MAXDET.set_objective('max', t)
>>> print(D_MAXDET)
Optimization Problem
  maximize t
  over
    1×1 real variable t
    8×1 real variable w (bounded below)
  subject to
    ∑(w) ≤ 1
    det(∑(w[i]·A[i]·A[i]ᵀ : i ∈ [0…7]))^(1/5) ≥ t
>>> #solve and display
>>> solution = D_MAXDET.solve(solver='cvxopt')
>>> print(w)
[ ...]
[ ...]
[ 2.27e-01]
[ 3.38e-02]
[ 1.65e-02]
[ 5.44e-02]
[ 3.18e-01]
[ 3.51e-01]

General Phi_p optimal design (SDP)

The A- and D-optimal design problems presented above can be obtained as special cases of the general Kiefer \Phi_p- optimal design problem, where p is a real in (-\infty,1] :

\begin{center}
\begin{eqnarray*}
&\underset{w \in \mathbb{R}^s}{\mbox{maximize}}
                   &\quad \left( \frac{1}{m} \operatorname{trace}\ \big(\sum_{i=1}^s w_i A_i A_i^T \big)^p \right)^{1/p} \\
&\textrm{subject to} &\quad w\geq0,\ \sum_{i=1}^s w_i \leq 1.
\end{eqnarray*}

\end{center}

These problems are easy to enter in PICOS, thanks to the tracepow function, that automatically replaces inequalities involving trace of matrix powers as a set of equivalent linear matrix inequalities (SDP) (cf. [4] ). Below are two examples with p=0.2 and p=-3, allocating respectively (20.6%, 0.0%, 0.0%, 0.92%, 40.8%, 37.7%), and (24.8%, 16.6%, 10.8%, 14.1%, 7.84%, 26.0%) of the trials to the design points 3 to 8.

>>> #problems, variables and parameters
>>> P0dot2_SDP  = picos.Problem()
>>> Pminus3_SDP = picos.Problem()
>>> AA = [picos.Constant('A[{0}]'.format(i), Ai)
...       for i, Ai in enumerate(A)] # each AA[i].T is a 3 x 5 observation matrix
>>> s  = len(AA)
>>> m  = AA[0].size[0]
>>> w = picos.RealVariable('w', s, lower=0)
>>> t = picos.RealVariable('t', 1)
>>> # constraint and objective
>>> wgt02_cons = P0dot2_SDP.add_constraint(1|w <= 1)
>>> wgtm3_cons = Pminus3_SDP.add_constraint(1|w <= 1)
>>> Mw = picos.sum([w[i]*AA[i]*AA[i].T for i in range(s)])
>>> tracep02_cons = P0dot2_SDP.add_constraint(t <= picos.PowerTrace(Mw, 0.2))
>>> P0dot2_SDP.set_objective('max', t)
>>> tracepm3_cons = Pminus3_SDP.add_constraint(t >= picos.PowerTrace(Mw, -3))
>>> Pminus3_SDP.set_objective('min', t)
>>> # p=0.2
>>> print(P0dot2_SDP)
Optimization Problem
  maximize t
  over
    1×1 real variable t
    8×1 real variable w (bounded below)
  subject to
    ∑(w) ≤ 1
    tr(∑(w[i]·A[i]·A[i]ᵀ : i ∈ [0…7])^(1/5)) ≥ t
>>> #solve and display
>>> solution = P0dot2_SDP.solve(solver='cvxopt')
>>> print(w)
[ ...]
[ ...]
[ 2.06e-01]
[ ...]
[ ...]
[ 9.20e-03]
[ 4.08e-01]
[ 3.77e-01]
>>> # p=-3
>>> print(Pminus3_SDP)
Optimization Problem
  minimize t
  over
    1×1 real variable t
    8×1 real variable w (bounded below)
  subject to
    ∑(w) ≤ 1
    tr(∑(w[i]·A[i]·A[i]ᵀ : i ∈ [0…7])^(-3)) ≤ t
>>> solution = Pminus3_SDP.solve(solver='cvxopt')
>>> print(w)
[ ...]
[ ...]
[ 2.48e-01]
[ 1.66e-01]
[ 1.08e-01]
[ 1.41e-01]
[ 7.83e-02]
[ 2.60e-01]

References

  1. Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming”, G. Sagnol, Journal of Statistical Planning and Inference, 141(5), p. 1684-1708, 2011.

  2. Computing exact D-optimal designs by mixed integer second order cone programming”, G. Sagnol and R. Harman, Submitted: arXiv:1307.4953.

  3. Determinant maximization with linear matrix inequality constraints”, L. Vandenberghe, S. Boyd and S.P. Wu, SIAM journal on matrix analysis and applications, 19(2), 499-533, 1998.

  4. On the semidefinite representations of real functions applied to symmetric matrices”, G. Sagnol, Linear Algebra and its Applications, 439(10), p. 2829-2843, 2013.