Global Optimization of Constrained Grey-Box Computational Problems

World Congress on Global Optimization 2015 in Gainesville, Florida

February 24, 2015

C.A. Floudas F. Boukouvala

Grey-box global optimization refers to problems for which derivatives of the objective function and/or constraints of the original model are not directly employed for obtaining the global optimum. In typical applications of grey-box optimization, derivative information is either: (1) available but deceptive; (2) prohibitively expensive; or (3) partially to completely unavailable. Optimization without derivatives has been characterized as one of the most “challenging and open problems in science and engineering, which has a vast number of potential practical applications” (Conn et al., 2009). For example, derivative-free methods enable the optimization of costly simulation models developed to represent industrial processes with a high level of detail, coupling multiple physical, chemical, and mechanical phenomena across different scales. In addition, the independency of grey-box methods from derivatives allows the optimization of problems with embedded numerical noise, discontinuities and multiple local optima (Conn et al., 2009; Forrester et al., 2008; Martelli and Amaldi, 2014). Constrained grey-box methods have a vast pool of application areas ranging from expensive finite-element or partial-differential equation systems and flowsheet optimization to mechanical engineering design, molecular design, material screening, geosciences, supply chain optimization and pharmaceutical product development, to name a few (Boukouvala et al., 2014). Despite the increasing interest in derivative-free optimization, there is scarcity of global optimization approaches for multidimensional general constrained grey-box problems. Specifically, existing theoretical and algorithmic developments in grey-box optimization employ local optimization concepts and are predominantly developed for box-constrained problems or constrained problems with explicitly known constraints (Rios and Sahinidis, 2013). In this work, the problem of constrained grey-box optimization is formulated as a compilation of deterministic global optimization sub-problems stemming from sampling selection, parameter estimation and global optimization of surrogate formulations. We present a novel AlgoRithm for Global Optimization of coNstrAined grey-box compUTational problems (ARGONAUT), which is developed to solve constrained greybox optimization problems with a large number of input variables and constraints. The algorithm can address box constraints, known inequality and equality constraints, and unknown inequality and equality constraints. The objective function of the grey-box problem, as well as the set of unknown constraints are approximated by surrogate functions. The algorithm involves variable selection techniques, which aim to exploit the sparsity of the model with regards to the objective function and the set of unknown constraints. Subsequently, surrogate function selection for the objective and each of the unknown constraints is performed using parameter estimation and validation concepts, which are formulated as non-linear optimization problems solved to global optimality using deterministic global optimization solver ANTIGONE (Misener and Floudas, 2014). ARGONAUT also includes a rigorous formulation for the selection of samples used for parameter estimation, using a Mixed Integer Linear formulation developed to select an optimal subset of samples based on both their input space locations as well as their objective and constraint function values (Li and Floudas, 2014). The proposed algorithm is an iterative framework, which identifies new sampling locations during each iteration based on the global optimization of the grey-box formulation which is comprised of all of the known constraints and surrogate approximations of the unknown objective and constraints. Finally, a domain refinement procedure is embedded within ARGONAUT, which allows for the enhanced exploration of promising subspaces of the input domain. The capacity of ARGONAUT is shown through a large set of problems from known standard libraries for constrained global optimization. The performance of ARGONAUT is also compared with commercially available derivative-free optimization software.