EFFICIENT DESIGN OPTIMIZATION OVER MIXED-COMBINATORIAL SPACES ENABLED BY GRAPH-LEARNING

Nonlinear design optimization problems that involve a mixture of continuous variables and combinatorial variables or a finite set of combinations remain one of the most challenging classes of problems to solve. Search mechanics that apply to continuous variables, or even independent (separable) integer variables, do not directly apply to the combinatorial space. Existing solution approaches either make binary transformations leading to an explosion in design dimensions and inability to account for relations between combinations, make integer (or indexing) approximations leading to the imposition of artificial relations, or pursue problem-specific direct encoding approaches that do not generalize well. The combinatorial space can, however, be exactly represented as graphs where each valid combination is treated as a node. Building on this representation, this paper presents a new efficient two-step optimization algorithm to solve mixed-combinatorial non-linear programming (MCNLP) type of design problems. The first step involves constructing a graph neural network called GNN-ReCo that learns to recommend the best-suited combination given a candidate design vector including all the non-combinatorial (namely continuous and independent integer) variables. A list-wise loss function is key to training this GNN in a manner that scales and generalizes well across the global graph of combinations for a given problem, while training on subgraph snapshots. In step 2, a standard population-based optimizer that can search through continuous and integer spaces operates on the non-combinatorial variables, with GNN-ReCo embedded into the function evaluation part to automatically retrieve the best-suited combination for any candidate design produced by the optimizer. A well-known Particle Swarm Optimization (PSO) algorithm is used as the optimizer in the current implementation. Applied to a benchmark analytical problem with 10 continuous variables and 101 combinations with 10 integer-valued features each, the new GNN-ReCo-aided optimization algorithm shows a significant reduction in function evaluations required and a small improvement in accuracy compared to the baseline (direct implementation of PSO). The new algorithm is then also demonstrated on a more complex real-world problem - designing the physical configuration and control choices for an actively maneuverable tether-net system intended for capturing large space debris.