Symbolic regression methods simultaneously determine the model functional form and the regression parameter values by generating expression trees. Symbolic regression can capture the complexity of ...real‐world phenomena but the use of deterministic optimization for symbolic regression has been limited due to the complexity of the search space of existing formulations. We present a novel deterministic mixed‐integer nonlinear programming formulation for symbolic regression that incorporates derivative constraints through auxiliary expression trees. By applying the chain rule to mathematical operations, binary expression trees are capable of representing the calculation of first and second derivatives. We apply this formulation to illustrative examples using derivative information to show increased model discrimination capability. In addition, we perform a case study of a thermodynamic equation of state to gain insight on valid functional forms with thermodynamics‐based constraints on the first and second derivatives.
In this paper, an open-source solver for mixed-integer nonlinear programming (MINLP) problems is presented. The Supporting Hyperplane Optimization Toolkit (SHOT) combines a dual strategy based on ...polyhedral outer approximations (POA) with primal heuristics. The POA is achieved by expressing the nonlinear feasible set of the MINLP problem with linearizations obtained with the extended supporting hyperplane (ESH) and extended cutting plane (ECP) algorithms. The dual strategy can be tightly integrated with the mixed-integer programming (MIP) subsolver in a so-called single-tree manner,
i.e.
, only a single MIP optimization problem is solved, where the polyhedral linearizations are added as lazy constraints through callbacks in the MIP solver. This enables the MIP solver to reuse the branching tree in each iteration, in contrast to most other POA-based methods. SHOT is available as a COIN-OR open-source project, and it utilizes a flexible task-based structure making it easy to extend and modify. It is currently available in GAMS, and can be utilized in AMPL, Pyomo and JuMP as well through its ASL interface. The main functionality and solution strategies implemented in SHOT are described in this paper, and their impact on the performance are illustrated through numerical benchmarks on 406 convex MINLP problems from the MINLPLib problem library. Many of the features introduced in SHOT can be utilized in other POA-based solvers as well. To show the overall effectiveness of SHOT, it is also compared to other state-of-the-art solvers on the same benchmark set.
This work presents a review of the main deterministic mixed‐integer nonlinear programming (MINLP) solution methods for problems with convex and nonconvex functions. An overview for deriving MINLP ...formulations through generalized disjunctive programming (GDP), which is an alternative higher‐level representation of MINLP problems, is also presented. A review of solution methods for GDP problems is provided. Some relevant applications of MINLP and GDP in process systems engineering are described in this work.
The optimal design, planning, and scheduling of chemical processes requires the use of mixed‐integer nonlinear programming (MINLP) or generalized disjunctive programming (GDP) models to optimize discrete and continuous variables. Here, applications of MINLP and GDP are described, and a review of deterministic methods to solve these problems is provided.
A sustainable water infrastructure cannot be made possible without improving its energy efficiency in terms of the consumption value and the type of supplying resources. One solution to reduce the ...external energy requirement of the water systems is to decrease the energy consumption, while the internal energy recovery capacity is increased. Such is achieved by the present work through (1) optimizing the water demand and tanks’ and pumps’ scheduling and operations and (2) installing pumps-as-turbines (PATs) where there is excess pressure head in the water. In this paper, two optimization models are developed. The first optimization problem is designed to find the optimal number and location of the PATs among the eligible positions across a water system. The optimal solution will be incorporated in the second optimization problem, i.e., an economic dispatch problem, where the cost of energy generation of an integrated water-energy system is minimized while the water and electrical load demands, as well as the hydraulic integrity of the system are satisfied. The energy system consists of conventional power generation units aggregated of natural gas and diesel based generators, solar photovoltaic units, wind generators, and battery energy storage systems. The proposed models are verified with case studies, and the numerical results show more than 30% reduction in the daily electricity consumption of water systems and $79 in daily operating cost of energy generation in water-energy systems when 3 PATs were installed at optimal locations.
•Two optimization models using mixed integer nonlinear programming are proposed.•First optimization model finds optimal place and number of pumps-as-turbines (PATs).•Next optimization model takes in optimal PATs to minimize cost of energy generation.•Effect of demand response on energy savings of water-energy system is studied.•Effect of optimal PATs on resource allocation of water-energy system is studied.
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