Publications

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We present SUSPECT, an open source toolkit that symbolically analyzes mixed-integer nonlinear optimization problems formulated using the Python algebraic modeling library Pyomo. We present the data structures and algorithms used to implement SUSPECT. SUSPECT works on a directed acyclic graph representation of the optimization problem to perform: bounds tightening, bound propagation, monotonicity detection, and convexity detection. We show how the tree-walking rules in SUSPECT balance the need for lightweight computation with effective special structure detection. SUSPECT can be used as a standalone tool or as a Python library to be integrated in other tools or solvers. We highlight the easy extensibility of SUSPECT with several recent convexity detection tricks from the literature. We also report experimental results on the MINLPLib 2 dataset.

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There is a need for efficient optimization strategies to efficiently solve large-scale, nonlinear optimization problems. Many problem classes, including design under uncertainty are inherently structured and can be accelerated with decomposition approaches. This paper describes a second-order multiplier update for the alternating direction method of multipliers (ADMM) to solve nonlinear stochastic programming problems. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for the method of multipliers (MM). We interpret ADMM as a decomposable version of MM and propose modifications to the multiplier update of the standard ADMM scheme based on improvements observed in MM. The modifications to the ADMM algorithm seek to accelerate solutions of optimization problems for design under uncertainty and the numerical effectiveness of the approaches is demonstrated on a set of ten stochastic programming problems. Practical strategies for improving computational performance are discussed along with comparisons between the algorithms. We observe that the second-order update achieves convergence in fewer unconstrained minimizations for MM on general nonlinear problems. In the case of ADMM, the second-order update reduces significantly the number of subproblem solves for convex quadratic programs (QPs).

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Algebraic modeling languages (AMLs) have drastically simplified the implementation of algebraic optimization problems. However, there are still many classes of optimization problems that are not easily represented in most AMLs. These classes of problems are typically reformulated before implementation, which requires significant effort and time from the modeler and obscures the original problem structure or context. In this work we demonstrate how the Pyomo AML can be used to represent complex optimization problems using high-level modeling constructs. We focus on the operation of dynamic systems under uncertainty and demonstrate the combination of Pyomo extensions for dynamic optimization and stochastic programming. We use a dynamic semibatch reactor model and a large-scale bubbling fluidized bed adsorber model as test cases.

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We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differential equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.