Abstract
The paper helps to understand the essence of stochastic population-based searches that solve ill-conditioned global optimization problems. This condition manifests itself by presence of lowlands, i.e., connected subsets of minimizers of positive measure, and inability to regularize the problem. We show a convenient way to analyze such search strategies as dynamic systems that transform the sampling measure. We can draw informative conclusions for a class of strategies with a focusing heuristic. For this class we can evaluate the amount of information about the problem that can be gathered and suggest ways to verify stopping conditions. Next, we show the Hierarchic Memetic Strategy coupled with Multi-Winner Evolutionary Algorithm (HMS/MWEA) that follow the ideas from the first part of the paper. We introduce a complex, ergodic Markov chain of their dynamics and prove an asymptotic guarantee of success. Finally, we present numerical solutions to ill-conditioned problems: two benchmarks and a real-life engineering one, which show the strategy in action. The paper recalls and synthesizes some results already published by authors, drawing new qualitative conclusions. The totally new parts are Markov chain models of the HMS structure of demes and of the MWEA component, as well as the theorem of their ergodicity.
Highlights
1.1 Ill-conditioned global optimization problemsMany problems in machine learning, optimal control, medical diagnostics, optimal design, geophysics, etc. are formulated as global optimization ones
We show the Hierarchic Memetic Strategy coupled with Multi-Winner Evolutionary Algorithm (HMS/MWEA) that follow the ideas from the first part of the paper
The possible solution is to use a cascade of stochastic searches, in which the upper ones are designated to global search, while the lowest ones deliver the sample concentrated in the basins of attraction of lowlands or minimum manifolds
Summary
Many problems in machine learning, optimal control, medical diagnostics, optimal design, geophysics, etc. are formulated as global optimization ones. They are frequently irreversibly ill-conditioned and possess many. Illconditioning of IPs, mentioned above, is caused mainly due to unavailability of complete and accurate measurements, e.g. insufficient set of data used for Artificial Neural Network (ANN) learning or pointwise measurement of the electric field called ‘‘logging curve’’ for investigation of oil and gas resources (see [28, 31] and [9]). We may refer to the representative examples of engineering ill-conditioned IPs: regression solved by Deep Neural Networks (DNNs) [18], ambiguity in lens design [23], calibration of conceptual rainfall-runoff models [12], investigation of oil and gas resources [56], and diagnosis of tumor tissue [37]
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