Publications

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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.

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In this work, we describe new capabilities for the Pyomo.GDP modeling environment, moving beyond classical reformulation approaches to include non-standard reformulations and a new logic-based solver, GDPopt. Generalized Disjunctive Programs (GDPs) address optimization problems involving both discrete and continuous decision variables. For difficult problems, advanced reformulations such as the disjunctive “basic step” to intersect multiple disjunctions or the use of procedural reformulations may be necessary. Complex nonlinear GDP models may also be tackled using logic-based outer approximation. These expanded capabilities highlight the flexibility that Pyomo.GDP offers modelers in applying novel strategies to solve difficult optimization problems.

Optimization problems under uncertainty involve making decisions without the full knowledge of the impact the decisions will have and before all the facts relevant to those decisions are known. These problems are common, for example, in process synthesis and design, planning and scheduling, supply chain management, and generation and distribution of electric power. The sources of uncertainty in optimization problems fall into two broad categories: endogenous and exogenous. Exogenous uncertain parameters are realized at a known stage (e.g., time period or decision point) in the problem irrespective of the values of the decision variables. For example, demand is generally considered to be independent of any capacity expansion decisions in process industries, and hence, is regarded as an exogenous uncertain parameter. In contrast, decisions impact endogenous uncertain parameters. The impact can either be in the resolution or in the distribution of the uncertain parameter. The realized values of a Type-I endogenous uncertain parameter are affected by the decisions. An example of this type of uncertainty would be facility protection problem where the likelihood of a facility failing to deliver goods or services after a disruptive event depends on the level of resources allocated as protection to that facility. On the other hand, only the realization times of Type-II endogenous uncertain parameters are affected by decisions. For example, in a clinical trial planning problem, whether a clinical trial is successful or not is only realized after the clinical trial has been completed, and whether the clinical trial is successful or not is not impacted by when the clinical trial is started. There are numerous approaches to modelling and solving optimization problems with exogenous and/or endogenous uncertainty, including (adjustable) robust optimization, (approximate) dynamic programming, model predictive control, and stochastic programming. Stochastic programming is a particularly attractive approach, as there is a straightforward translation from the deterministic model to the stochastic equivalent. The challenge with stochastic programming arises through the rapid, sometimes exponential, growth in the program size as we sample the uncertainty space or increase the number of recourse stages. In this talk, we will give an overview of our research activities developing practical stochastic programming approaches to problems with exogeneous and/or endogenous uncertainty. We will highlight several examples from power systems planning and operations, process modelling, synthesis and design optimization, artificial lift infrastructure planning for shale gas production, and clinical trial planning. We will begin by discussing the straightforward case of exogenous uncertainty. In this situation, the stochastic program can be expressed completely by a deterministic model, a scenario tree, and the scenario-specific parameterizations of the deterministic model. Beginning with the deterministic model, modelers create instances of the deterministic model for each scenario using the scenario-specific data. Coupling the scenario models occurs through the addition of nonanticipativity constraints, equating the stage decision variables across all scenarios that pass through the same stage node in the scenario tree. Modelling tools like PySP (Watson, 2012) greatly simplify the process of composing large stochastic programs by beginning either with an abstract representation of the deterministic model written in Pyomo (Hart, et al., 2017) and scenario data, or a function that will return the deterministic Pyomo model for a specific scenario. PySP automatically can create the extensive form (deterministic equivalent) model from a general representation of the scenario tree. The challenge with large scale stochastic programs with exogenous uncertainty arises through managing the growth of the problem size. Fortunately, there are several well-known approaches to decomposing the problem, both stage-wise (e.g., Benders’ decomposition) and scenario-based (e.g., Lagrangian relaxation or Progressive Hedging), enabling the direct solution of stochastic programs with hundreds or thousands of scenarios. We will then discuss developments in modelling and solving stochastic programs with endogenous uncertainty. These problems are significantly more challenging to both pose and to solve, due to the exponential growth in scenarios required to cover the decision-dependent uncertainties relative to the number of stages in the problem. In this situation, standardized frameworks for expressing stochastic programs do not exist, requiring a modeler to explicitly generate the representations and nonanticipativity constraints. Further, the size of the resulting scenario space (frequently exceeding millions of scenarios) precludes the direct solution of the resulting program. In this case, numerous decomposition algorithms and heuristics have been developed (e.g., Lagrangean decomposition-based algorithms (Tarhan, et al. 2013) or Knapsack-based decomposition Algorithms (Christian and Cremaschi, 2015)).

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.

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