(508b) Constrained Grey-Box Global Optimization of High Dimensional Problems

Authors: 
Boukouvala, F., Texas A&M University
Floudas, C. A., Princeton University

The use of deterministic global optimization methods based on analytical Cfunctions is prohibitive in a large portion of engineering applications which have one or multiple of the following characteristics: (a) high computational cost which implies that the calculation of derivatives is not viable, (b) numerical noise which leads to unreliable derivative information, and (c) lack of closed-form model equations which implies lack of derivative information [1-2]. Optimization without derivatives is becoming necessary in many recent applications dealing with detailed multiscale expensive simulations (i.e., finite element method models, computational fluid dynamic models and non-linear partial differential equation systems). In this work we are interested inconstrained grey-box systems, which are defined as optimization formulations which contain a set of explicitly known equations (constraints and/or objective function) and a set of explicitly unknown equations (constraints and/or objective).  In prior work, we have shown that well-designed sampling coupled with iterative fitting of surrogate models coupled with deterministic global optimization has a competitive performance in locating global solutions at a low sampling expense [3-4]. However, treatment of a large set of general constraints (number of constraints > 50) in high dimensional problems (number of dimensions > 100) still remains a challenge in the field of constrained grey-box optimization due to the computational expense associated with collecting samples in high-dimensional spaces, solving large parameter estimation problems and finally  optimizing the formulated grey-box formulations globally. Existing approaches in the literature treat such problems as pure “black-boxes”, disregarding any information in the form of constraints or problem structure and limiting their communication and integration with existing deterministic optimization software. Moreover, existing methods have been tested on high dimensional box-constrained problems using local optimization concepts- which are highly dependent on the existence of a good initial point [5]. 

            The ARGONAUT framework (Algorithms for Global Optimization of constraiNed Grey-Box compUtational sysTems), is comprised of several key components which are enhanced in this work to handle higher dimensional problems and problems with a large number of constraints. The main components of the algorithm are: (a) Optimized sampling in constrained regions defined by any known information of the problem, (b) Bounds tightening when this is possible based on known information, (c) Selection of optimal surrogate functions for each individual unknown model from a larger pool of possible regression and interpolation techniques (i.e., linear, quadratic, signomial, kriging and radial-basis functions), (d) Exploration of model sparsity, (e) Sequential treatment of variables based on  variable importance and (f) Parallelization of sampling, parameter estimation, model validation and global optimization procedures for computational savings. The proposed framework is highly intertwined with the deterministic global optimization solver ANTIGONE [6] which is necessary for optimizing the constrained non-linear grey-box formulations providing a diverse set of local and global solutions which are good candidates for further sampling. We provide thorough analysis of results on a large set of test problems, and propose different strategies based on the size of the problem as well as the computational expense of the simulation model. Finally, the performance of the developed algorithms are compared to existing software for constrained optimization without derivatives.

References:

1. Conn, A.R., Scheinberg, K. Vicente, L.N.:Introduction to Derivative-free Optimization. MPS-SIAM Series on Optimization, vol. 8, SIAM, Philadelphia (2009).

2. Boukouvala, F. , Misener, R., Floudas, C.A.: Global Optimization Advances in Mixed-Integer Nonlinear Programming, MINLP and Constrained Derivative-free Optimization, CDFO. European Journal of Operational Research. Under Review (2015).

3. Hasan, M.M.F., Boukouvala, F., First, E.L., Floudas, C.A.: Nationwide, Regional, and Statewide CO2 Capture, Utilization, and Sequestration Supply Chain Network Optimization. Industrial Engineering & Chemistry Research 53(18), 7489-7506 (2014).

4. Boukouvala, F., Hasan, M.M.F., Floudas, C.A.:  Global Optimization of General Constrained Grey-Box Models: New Method and its Application to Constrained PDEs. Submitted to Journal of Global Optimization (2015).

5. Adjengue, L., Audet, C., Yahia, I.B.: A variance-based method to rank input variables of the Mesh Adaptive Direct Search algorithm. Optim Letters 8, 1599- 1610 (2014).

6. Misener, R., Floudas, C.A.: ANTIGONE: Algorithms for Continuous/ Integer Global Optimization of Nonlinear Equations, Journal of Global Optimization  59(2-3), 503-526 (2014).