Adaptive survey designs to minimize survey mode effects – a case study on the Dutch Labor Force Survey
3. An algorithm for solving the multi-mode optimization problemAdaptive survey designs to minimize survey mode effects – a case study on the Dutch Labor Force Survey
3. An algorithm for solving the multi-mode optimization problem
In the previous section, we introduced the quality and
cost functions and constructed a multi-mode optimization problem. The
subpopulation comparability constraint, i.e., the upper limit to the maximum
absolute difference between group method effects, makes the problem nonconvex
and hard to solve. As a consequence, when trying to solve the multi-mode
optimization problem, most general-purpose nonlinear solvers cannot do better
than a local optimum. Therefore, the choice of starting points in the solvers
plays an important role. As such, we propose a two-step approach. In the first
step, we solve a linear programming problem (LP) that addresses the linear
constraints (2.1), (2.5), (2.6) and (2.9)
(2.10). In the second step, we use the optimal
solution obtained in step 1 as a starting point for a local search algorithm to
solve the nonconvex nonlinear problem (NNLP).
We reformulate the optimization problem to make it
computationally more tractable. Since
we can rewrite the objective function via an
and impose that
has to be nonnegative. The constraints
themselves do not change, they are simply replaced. The multi-mode optimization
problem is given in (3.2).
We can derive the LP by removing the non-linear
constraints on the comparability of method effects across subpopulations and by
replacing the non-linear objective function by one of the linear constraints.
We choose for minimization of costs as the LP objective. The resulting LP
problem formulation is given by
To solve the linear problem, we use the simplex method
available in R in package
Our proposed two-step algorithm thus handles (3.1)
in the first step. Denote by
the optimal solution obtained in the LP. In
the second step,
is submitted to a nonlinear optimization
algorithm as a starting point in order to solve (3.2). For this step, we use
nonlinear algorithms available in NLOPT (see Johnson 2013), an open-source library for nonlinear optimization
that can be called from R through the
package. The NNLP second step of the algorithm
is performed only if the minimal required budget found in the LP first step is
smaller than or equal to the available budget
If the minimal budget is larger, then there is
no feasible solution to the optimization problem.
Given that the performance of these algorithms is
problem-dependent, we choose to combine two local search algorithms in order to
increase the convergence speed. Global optimization algorithms are available in
the NLOPT library but their performance for our problem was significantly worse
than the selected local optimization algorithms. The two selected local search
algorithms are COBYLA (Constrained Optimization by Linear Approximations),
introduced by Powell (1998)
(see Roy 2007 for an
and the Augmented Lagrangian Algorithm
(AUGLAG), described in Conn, Gould and
Toint (1991) and Birgin and Martinez (2008). The COBYLA method builds
successive linear approximations of the objective function and constraints via
a simplex of
dimensions), and optimizes these
approximations in a trust region at each step. The AUGLAG method combines the
objective function and the nonlinear constraints into a single function, i.e.,
the objective plus a penalty for any violated constraint. The resulting
function is then passed to another optimization algorithm as an unconstrained
problem. If the constraints are violated by the solution of this sub-problem,
then the size of the penalties is increased and the process is repeated.
Eventually, the process must converge to the desired solution, if that exists.
As local optimizer for the AUGLAG method we choose MMA
(Method of Moving Asymptotes, introduced in Svanberg 2002), based on its
performance for our numerical experiments. The strategy behind MMA is as
follows. At each point
MMA forms a local approximation, that is both
convex and separable, using the gradient of
and the constraint functions, plus a quadratic
penalty term to make the approximations conservative, e.g., upper bounds for
the exact functions. Optimizing the approximation leads to a new candidate
If the constraints are met, then the process
continues from the new point
otherwise, the penalty term is increased and
the process is repeated.
The reason for using two local search algorithms is that
AUGLAG performs better in finding the neighborhood of the global optimum but
COBYLA provides a greater accuracy in locating the optimum. Therefore, the LP
optimal solution is first submitted to AUGLAG and after a number of iterations,
when the improvement in the objective value is below a specified threshold, the
current solution of AUGLAG is submitted to COBYLA for increased accuracy. For
our case study, given the precision requirements of the obtained statistics in
the survey (0.5%), the results are considered accurate enough if the obtained
objective value is within
away from the global optimum. Any further
accuracy gains are completely blurred by the sampling variation and accuracy of
the input parameters themselves. The computational times can run up to a few
hours. Since the optimization problem does not need to be solved during data
collection, this will, however, not pose practical problems.
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