Sequential quadratic programming with indefinite Hessian approximations for nonlinear optimum experimental design for parameter estimation in differential–algebraic equations
In this thesis we develop algorithms for the numerical solution of problems from nonlinear
optimum experimental design (OED) for parameter estimation in differential–algebraic
equations. These OED problems can be formulated as special types of path- and control-
constrained optimal control (OC) problems. The objective is to minimize a functional on
the covariance matrix of the model parameters that is given by first-order sensitivities of the
model equations. Additionally, the objective is nonlinearly coupled in time, which make
OED problems a challenging class of OC problems. For their numerical solution, we propose
a direct multiple shooting parameterization to obtain a structured nonlinear programming
problem (NLP). An augmented system of nominal and variational states for the model
sensitivities is parameterized on multiple shooting intervals and the objective is decoupled
by means of additional variables and constraints. In the resulting NLP, we identify several
structures that allow to evaluate derivatives at greatly reduced costs compared to a standard
OC formulation.
For the solution of the block-structured NLPs, we develop a new sequential quadratic
programming (SQP) method. Therein, partitioned quasi-Newton updates are used to approximate the block-diagonal Hessian of the Lagrangian. We analyze a model problem with
indefinite, block-diagonal Hessian and prove that positive definite approximations of the
individual blocks prevent superlinear convergence. For an OED model problem, we show
that more and more negative eigenvalues appear in the Hessian as the multiple shooting grid
is refined and confirm the detrimental impact of positive definite Hessian approximations.
Hence, we propose indefinite SR1 updates to guarantee fast local convergence. We develop
a filter line search globalization strategy that accepts indefinite Hessians based on a new
criterion derived from the proof of global convergence. BFGS updates with a scaling strategy to prevent large eigenvalues are used as fallback if the SR1 update does not promote
convergence. For the solution of the arising sparse and nonconvex quadratic subproblems, a
parametric active set method with inertia control within a Schur complement approach is
developed. It employs a symmetric, indefinite LBL T -factorization for the large, sparse KKT
matrix and maintains and updates QR-factors of a small and dense Schur complement.
The new methods are complemented by two C++ implementations: muse transforms an
OED or OC problem instance to a structured NLP by means of direct multiple shooting.
A special feature is that fully independent grids for controls, states, path constraints, and
measurements are maintained. This provides higher flexibility to adapt the NLP formulation
to the characteristics of the problem at hand and facilitates comparison of different formulations in the light of the lifted Newton method. The software package blockSQP is an
implementation of the new SQP method that uses a newly developed variant of the quadratic
programming solver qpOASES. Numerical results are presented for a benchmark collection of
OED and OC problems that show how SR1 approximations improve local convergence over
BFGS. The new method is then applied to two challenging OED applications from chemical
engineering. Its performance compares favorably to an available existing implementation