(461e) A Predictor-Corrector Algorithm for Projection-Based Derivative-Free Optimization
In this work, we present an algorithm to address bound-constrained and multi-dimensional black-box problems based on projection onto a one-dimensional space. Specifically, we consider a problem of minimizing G(t), which can be interpreted as the projection of the original objective function f(xi), i = 1, â¦, n taken in the t-space such that t is linear combination of xi. Additionally, we also provide useful properties of the univariate function G(t) and show that the minimum of G(t) also corresponds to the minimum of f(x). This enables us to solve the original optimization problem in two steps. The first step involves identifying G(t) and is referred as the inner loop . In this work, the inner loop is applied at discrete values of t(t1, t2, ..., tk). Once the solution at the previous t(tk-1)Â is known, sensitivity theorem  is utilized to obtain close solution for the inner loop at next t(tk). The second step is optimizing the univariate function G(t) to find the solution of the original problem. We also propose an alternate projection by normalizing t. We compare the performance of the two projections with existing DFO solvers. The step length, Î±, where Î± = tk - tk-1, is another critical parameter in determining the number of evaluations taken by the algorithm. We have considered and compared two strategies related to the step length. The first strategy uses a fixed or uniform step length. The second strategy uses an adaptive approach where the value of Î±Â is revised as the algorithm proceeds, based on the number of evaluations taken in the previous inner loop.
We have tested the algorithm to solve several high-dimensional test problems. For instance, we are able to find the global solution of a 50-dimensional test problem (emfl_eps) in 40,747 evaluations. We also obtain the solutions of explin2, expquad and qrtquad (each of which are 120-dimensional problems) using 85521, 62760 and 89484 function evaluations, respectively. The algorithm is applied to a large set of test problems taken from  and compared to existing model-based DFO solvers. The proposed approach performs superior relative to ORBIT  and SNOBFIT .
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