Grey box optimization

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The breadth of the optimization field is wide. At one extremity, there are optimization problems for which the structure is well-known and exploitable: linear programming, or convex optimization with explicit algebraic formulations for example. This is not the target class of problems of my research. At the opposite extremity, blackbox optimization (BBO) refers to situations in which the structure of the objective function and of the constraints are unknown and cannot be exploited during the optimization process. These situations frequently arise when the functions defining the problem are computed through a time-consuming simulation.  These are the problems at the center of this research project. One of the most useful tools for solving optimization problems is the derivative. Indeed, the gradient vector is the steepest ascent direction and can be followed in a maximization context. Derivative-free optimization (DFO) refers to the situation where the derivatives of the functions are unavailable, or potentially difficult to estimate. A subtle distinction between DFO and BBO is that in the latter there is no reason to believe that derivatives even exist, and in the former they might exist but their expression is not available. Prior to the 1990's, only punctual developments were made in the DFO and BBO fields. But since then, there has been a steady increase in the interest devoted to these research areas. These are among the most rapidly expanding areas of nonlinear optimization research. This may be explained in part by the fact that computers are now able to simulate complex engineering processes in reasonable time, and by the successful utilization of algorithms on real engineering problems in industrial environments. The research project described in this proposal builds on my prior NSERC-funded work on direct search algorithms for DFO and BBO problems. The objective of this project is to explore the grey zone in the wide optimization field: problems for which part of the structure, but not all, is available. For example, previous work has focused on situations where the objective function is the sum of the squares of blackbox functions, and where crude information with respect to the monotonicity of some constraints with respect to certain variables was explicitly known. The present project will study other types of grey box optimization problems in which high-level qualitative or quantitative information about the constraints, the nature of the problem and the surrogates are available. All developments in the projects outlined in this proposal will be tested on real engineering test problems and will be analyzed using tools from nonsmooth calculus for a rigorous convergence analysis. The outcome of this research is useful to our industrial collaborators in Canada.  We plan to continue to apply our work in areas such as hydrological sciences, pharmaceutical and bioinformatic industry, alloy design, metamaterial design and aeronautics.