Symbolic Bucket Elimination for Piecewise Continuous Constrained Optimization

Abstract

Bucket elimination and its approximation extensions have proved to be effective techniques for discrete optimization. This paper addresses the extension of bucket elimination to continuous constrained optimization by leveraging the recent innovation of the extended algebraic decision diagram (XADD). XADDs support symbolic arithmetic and optimization operations on piecewise linear or univariate quadratic functions that permit the solution of continuous constrained optimization problems with a symbolic form of bucket elimination. The proposed framework is an efficient alternative for solving optimization problems with low tree-width constraint graphs without using a big-M formulation for piecewise, indicator, or conditional constraints. We apply this framework to difficult constrained optimization problems including XOR’s of linear constraints and temporal constraint satisfaction problems with “repulsive” preferences, and show that this new approach significantly outperforms Gurobi. Our framework also enables symbolic parametric optimization where closed-form solutions cannot be computed with tools like Gurobi, where we demonstrate a final novel application to parametric optimization of learned Relu-based deep neural networks.