Derivative-free and blackbox optimization for engineering problems

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This research project is the continuation of my prior work funded by NSERC on the development of the Mesh Adaptive Direct Search algorithm (MADS) for derivative-free and blackbox optimization problems.  In many engineering optimization problems, the objective and constraint functions are nonsmooth and evaluated through a computer simulation. The outputs of these codes are often of limited precision, contaminated with numerical noise, expensive to evaluate, and sometimes the code fails to return a value. ***There has been an important amount of work in developing models to improve the efficiency of direct search optimization methods.  Quadratic models, Gaussian processes, kriging interpolation as well as radial basis functions are such examples.  There has also been work on expanding the capabilities to integer, categorical and periodic variables, on efficient treatment of constraints, on bi- and multi objective problems.  In the present proposal we plan to pursue the development of MADS methods in many ways.***The current MADS algorithm generates trial points on a discretization of the space of variables called the mesh.  Several engineers that have applied MADS to practical problems complained that a consequence of the way that the mesh size parameter is updated leads to modifications in the values of the variables below a reasonable granularity.  We plan to alter the way that the mesh size parameter is updated, which is at the heart of the MADS algorithm, and the cornerstone of the nonsmooth convergence analysis.  ****Some constraints return a binary value indicating if it is satisfied or not, e.g., a flag indicating if the simulation failed or not.  Such a binary constraint is difficult to model using the above-mentioned modelling techniques.  We plan to import classification methods and data mining tools from the machine learning community to model them.  We proposed the BiMADS algorithm for biobjective blackbox constrained optimization.  For such problems, BiMADS returns a finite list of trade-off solutions, with respect to two conflicting objectives.  We plan to once again use classification tools to analyze the nature of the list of trade-off solutions, and to create representative solutions.***Surrogates are used by MADS as an inexpensive substitute of the true simulation.  MADS uses them by sampling the surrogate several times before deciding where to launch the true simulation.  Our numerical experiments systematically show that surrogates considerably help in finding better solutions.  We plan to devise a generic framework to handle surrogates with parametrizable precision, and to build additional models to dynamically estimate the difference between the true and surrogate functions.   ***All the projects outlined in this proposal will be rigorously analyzed using nonsmooth calculus.**