Abstract
We present a new derivative-free optimization algorithm based on the sparse grid numerical integration. The algorithm applies to a smooth nonlinear objective function where calculating its gradient is impossible and evaluating its value is also very expensive. The new algorithm has: 1) a unique starting point strategy; 2) an effective global search heuristic; and 3) consistent local convergence. These are achieved through a uniform use of sparse grid numerical integration. Numerical experiment result indicates that the algorithm is accurate and efficient, and benchmarks favourably against several state-of-art derivative free algorithms.
Highlights
Derivative-based methods can be very efficient and have been widely used in solving optimization problems
We present a new derivative-free optimization algorithm based on the sparse grid numerical integration
These results cover important aspects of a nonlinear derivative free algorithm, including our choice of the searching direction; starting point strategy; and a new closed-form formula which we developed to calculate the number of function calls in the sparse grid numerical integration at each iteration
Summary
Derivative-based methods can be very efficient and have been widely used in solving optimization problems. Many scientific and engineering optimization problems fall into this category [1]. In the helicopter rotor blade design problem [2], the objective function can only be evaluated by very expensive simulation. Similar problems include the nonlinear optimization parameter tuning problem [3], medical image registration [4], dynamic pricing [5], and community groundwater problem [6]. Derivative-free methods must be used in these situations. For these problems, the derivative information is not available, the function evaluation could be inaccurate or noisy most of times as well. An algorithm needs the capability to generate robust searching directions without using derivative and without overly relying on individual function evaluations
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