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Global optimization of nonlinear least-squares problems by branch-and-bound and optimality constraints. (English) Zbl 1258.93102
Summary: We study a simple, yet unconventional approach to the global optimization of unconstrained nonlinear least-squares problems. Non-convexity of the sum of least-squares objective in parameter estimation problems may often lead to the presence of multiple local minima. Here, we focus on the spatial branch-and-bound algorithm for global optimization and experiment with one of its implementations, BARON (see [N. V. Sahinidis, J. Glob. Optim. 8, No. 2, 201–205, (1996; Zbl 0856.90104)]), to solve parameter estimation problems. Through the explicit use of first-order optimality conditions, we are able to significantly expedite convergence to global optimality by strengthening the relaxation of the lower-bounding problem that forms a crucial part of the spatial branch-and-bound technique. We analyze the results obtained from 69 test cases taken from the statistics literature and discuss the successes and limitations of the proposed idea. In addition, we discuss software implementation for the automation of our strategy.

MSC:
93E10 Estimation and detection in stochastic control theory
93E24 Least squares and related methods for stochastic control systems
49M05 Numerical methods based on necessary conditions
65D10 Numerical smoothing, curve fitting
65K05 Numerical mathematical programming methods
90C26 Nonconvex programming, global optimization
90C46 Optimality conditions and duality in mathematical programming
Software:
BARON; GiNaC; INTOPT_90; nlmdl
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