Trends and perspectives in deterministic MINLP optimization for integrated planning, scheduling, control, and design of chemical processes

4.1 MIP reformulations

In MIP reformulations, nonlinearities are reformulated as linear relationships between integer and continuous variables. MIP reformulations of MINLP problems may involve exact linearization, e.g., when linearly reformulating the multiplication of two binary variables, or the multiplication of binary and continuous variables. Similarly, another example emerges when continuous variables in nonlinear models are restricted to take discrete values thus resulting in an MIP formulation (Voudouris and Grossmann 1992). Exact linearization have been successfully applied to integrated scheduling and planning problems involving batch processing plants (Orçun et al. 2001; Tan and Khoshnevis 2004). In more general scenarios, exact linearization transformations may not be available and MIP input–output approaches that only provide an approximation of the original MINLP may be needed. These approaches rely on the approximation of nonlinearities as linear operators, leading to a formulation that can be readily solved with MIP methods, at the expense of having a model that is valid within a reduced space where the linearization was performed. This approach has been successfully applied to capital investment planning and scheduling of an oil refinery (Menezes et al. 2015).

4.2 Hybrid surrogate models

Hybrid surrogate models combining data-driven and first-principles models have also been considered to approximate one or some of the decision layers of the original integrated MINLP problem, e.g., using a data-driven surrogate model to approximate process dynamics in simultaneous design and control (Ricardez-Sandoval et al. 2009; Sachio et al. 2021; Trainor et al. 2013). Surrogate models aim to provide an accurate approximate model of some decision layers using sampled input–output data taken over a search region of interest, i.e., they aim to be valid over the whole optimization region of interest. The surrogate model itself can take the form of a linear model, a piecewise linear model, a polynomial model, a neural network, among others; consequently, the resulting hybrid surrogate model may remain as an MINLP that is often solved using general-purpose solvers. For instance, the integration of design and control through the explicit representation of the optimal controller using reinforcement learning has been proposed in the literature (Sachio et al. 2022). The approximation of process dynamics has also been considered for the integration of scheduling and robust model predictive control, through the use of two surrogate models in the form of artificial neural networks: one for the identification of the feasible space of operation, and the other for the approximation of the state and input variables of the dynamic system (Dias and Ierapetritou 2020).

Many hybrid surrogate models can also be regarded as MIP reformulations. For example, piecewise linear reformulations of data-driven low-order models (e.g., Hammerstein–Wiener models, and piecewise affine models) were proposed to represent nonlinear process dynamics in simultaneous scheduling and control problems, resulting in an MIP integrated problem (Simkoff and Baldea 2019). This strategy has been tested for instance on a continuous CSTR with cyclic production (Zhuge and Ierapetritou 2015), a continuous multiproduct polymerization reactor (Kelley et al. 2018a), and an air separation unit (Kelley et al. 2018b). A similar strategy was employed to simultaneously consider planning, scheduling, and dynamic optimization of batch processes, where dynamic models were replaced by linear surrogate models, resulting in an MIP formulation (Chu and You 2014a). Piecewise linear approximations that transform an MINLP into an MIP have also been considered for the planning and scheduling of flexible process networks under uncertainty (Yue and You 2013). Overall, hybrid surrogate methods may simplify the effort of model development and speed-up the solution of the optimization problem; however, they require training and validation of data sets that are assumed to cover/predict the entire operating range of the plant/model.

4.3 Continuous reformulations

In continuous reformulations, discrete decisions are expressed as continuous variables, and discrete requirements are enforced via additional constraints. Continuous reformulations have been developed to reformulate MINLPs as Nonlinear Programs (NLPs) to reduce the high computational costs often faced by MINLP and MIP techniques. Examples of reformulations include complementarity constraints, circle constraints, nonlinear complementarity problem functions, among others (Baumrucker et al. 2008; Burre et al. 2023; Stein et al. 2004). These techniques often require the iterative optimization of the resulting NLP with varying tolerance parameters that either enforce integer requirements as they go to zero or the consideration of integer requirements through objective function penalization. Despite this, they are usually faster than traditional MINLP solvers. Integrated decision-making MINLP applications include the integration of optimal scheduling and control of heat exchanger networks under fouling (Santamaria and Macchietto 2019), scheduling and NMPC of a multiproduct continuous reactor (Andrés-Martínez and Ricardez-Sandoval 2022a), simultaneous design and control of plug flow and CSTR reactor networks (Zhao and Marquardt 2017), design and control of extractive distillation column systems (Ramos et al. 2014).

When using NLP local solvers, the main drawback of continuous reformulations is that they inevitably lead to a nonconvex optimization problem that generally requires educated initial guesses to achieve convergence (Stein et al. 2004). Alternatively, when a global solver such as BARON or MAiNGO is used, case studies involving superstructure design have demonstrated that, from a computational performance point of view, an MINLP superstructure formulation is usually desirable over its continuous reformulation for problems with convex functions. However, when solving nonconvex MINLPs with the same global solvers, continuous reformulations may result in smaller problems and tighter relaxations (Burre et al. 2023). Despite these preliminary advantages, the performance of continuous reformulations using global solvers for integrated decision-making problems has not been reported in the literature, presumably due to the long computational times that may be required when compared to local NLP solvers.

4.4 Lagrangian relaxation

Lagrangian relaxation corresponds to the general idea of penalizing a subset of complicating constraints in the objective function of an optimization problem, i.e., complicating constraints are dualized. The nonnegative weights used to penalize these constraints are known as Lagrangian multipliers (Guignard 2003). An application of Lagrangian relaxation that has become relevant in the context of MINLP integrated decision-making in chemical engineering is known as Lagrangian decomposition (LD), which is a useful approach to exploit optimization problem structures where two (or more) subset of constraints interconnected through common variables appear in the problem formulation. For instance, in simultaneous scheduling and control, some constraints and variables are used to model scheduling decisions, and others are used to capture transient operation, i.e., dynamic path constraints. These two groups of constraints are often connected through common variables. In the case of two subsets of constraints, if common variables are duplicated, the linking constraints that connect these variables can be dualized resulting in a decomposable formulation where each subset of constraints considers independent variables (Guignard 2003).

Note that the value of Lagrangian multipliers used to penalize constraints is initially unknown. Hence, once the problem is decomposed, an adaptive approach is used to update the Lagrangian multipliers. This adaptive approach is incorporated within the main iterative process of LD, which is as follows: 1) the solution of the two independent problems is used to compute a lower bound to the problem and 2) a problem with fixed integer variables that returns an upper bound to the problem is computed. For example, chemical engineering applications in the field of planning, scheduling, and control of a multiproduct CSTR (Mora-Mariano et al. 2020), or the scheduling and control of polymerization reactors (Terrazas-Moreno et al. 2008a) have implemented the subgradient method developed by Fisher (1985) to iteratively find and update Lagrangian multipliers. Other relevant works in the field include the simultaneous supply chain design and planning in the electric motor industry (Yongheng et al. 2014), and the integration of refinery planning and crude-oil scheduling (Mouret et al. 2011; Yang et al. 2020).

In general, LD helps to reduce the computational cost of solving integrated decision-making problems, against the direct solution without decomposition techniques (Andrés-Martínez and Ricardez-Sandoval 2022b). Note that there are some challenges in the application of LD to MINLP integrated decision-making. For instance, an adequate strategy should be used to update Lagrangian multipliers, e.g., the algorithm may not converge if the subgradient step size is not defined or updated properly. Furthermore, problem-specific heuristics may be needed to solve the resulting MINLP subproblems efficiently (Mouret et al. 2011). In addition, and due to the presence of discrete variables and nonconvexities, the duality gap obtained at each iteration is only expected to become small enough to obtain near-optimal heuristic solutions, but it is not guaranteed to reach a value near zero, i.e., the lower bound may remain strictly lower than the true optimum of the problem (Mora-Mariano et al. 2020; Mouret et al. 2011).

4.5 Discrete steepest descent algorithm

Concepts from discrete convex analysis (Murota 2003) can be incorporated into novel decomposition-based algorithms particularly tailored to handle ordered discrete decisions, i.e., integer variables and ordered discrete structures of Boolean or binary variables. Variables of this type appear frequently in integrated decision-making MINLP problems, e.g., number of units in series/parallel in a reactor superstructure; selection of the most sensitive stage for distillation temperature control applications; number of times a task is executed during a production schedule. The key characteristic of algorithms based on discrete convex analysis is that they verify local optimality of ordered discrete decisions by performing objective function evaluations over a discrete neighborhood of a solution candidate, while the optimality of the remaining variables of the problem is determined by a general-purpose solver. As a result, these algorithms may yield locally optimal solutions that may not be found with traditional MINLP methods. An algorithm that follows these principles is the discrete steepest descent algorithm (D-SDA) (Liñán and Ricardez-Sandoval 2024a).

In the context of MINLP integrated decision making, the D-SDA has been applied for the simultaneous design and control of conventional and catalytic distillation columns (Liñán and Ricardez-Sandoval 2022a; Palma-Flores et al. 2023), and the simultaneous scheduling and dynamic optimization of network batch processes (Liñán and Ricardez-Sandoval 2024a). For these applications, D-SDA converged to local optima with objective function values that are better than those found with general-purpose MINLP solvers in shorter computational times. Note that the local optimality guarantees of D-SDA rely on the exploration of a neighborhood (i.e., an integrally convex neighborhood) whose size increases exponentially with the number of ordered discrete decisions. Integrally convex neighborhood explorations may lead to lengthy computational times; thus, using subsets of the integrally convex neighborhood helps to speed up convergence but may worsen the objective function value attained.

4.6 Multiparametric programming

Multiparametric programming (MP) is a tool used to systematically analyze the effect of varying parameters (e.g., uncertainties, disturbances, design parameters) in optimization problems (Pistikopoulos et al. 2007). Multiparametric optimization has played a significant role in the solution of optimization problems arising in plantwide integrated decision-making. This technique aims to find exact offline solutions to (or part of) the MIDO problems arising in integrated decision-making, by deriving explicit maps of optimal decisions, i.e., the parameter space is partitioned into subregions known as “critical regions,” whereas optimization variables are expressed as a piecewise function of the parameters. Since optimal decision maps are derived offline, these can be readily incorporated within online feedback decision loops, e.g., Explicit Model Predictive Control (MPC). Thus, one of the main features offered by multiparametric optimization is the possibility of replacing parts or the complete optimization problem by piecewise function evaluations (Pistikopoulos 2012).

Relevant MP applications include integrated design, scheduling ,and control, with applications in the operation of a multiproduct continuous and batch reactors (Burnak et al. 2019b; Burnak and Pistikopoulos 2020; Zhuge and Ierapetritou 2014), and the design and control of conventional and reactive distillation systems (Sakizlis et al. 2003; Tian et al. 2021). Different decomposition methodologies have been proposed depending on the application, e.g., embedding closed-loop MPC dynamics in the scheduling formulation via MP; deriving offline multiparametric expressions for both scheduling and MPC control interconnected through bridging models; considering design variables as parameters in a multiparametric MPC problem to enable design and control integration, among others (Pappas et al. 2021). Despite their successful applications, one of the main challenges in MP is to overcome its dimensionality limitations. For instance, determining critical regions generally involves the solution of multiple optimization subproblems with fixed parameters, which may be computationally intensive for general MIDO systems involving multiple states, e.g., increasing the number of decision variables in a multiparametric MPC formulation leads to an exponential increase in the number of critical regions (Burnak and Pistikopoulos 2020; Pappas et al. 2021).

4.7 Bilevel optimization

Bilevel optimization plays a key role in integrated decision-making problems where two decision layers consider different objectives. In a bilevel optimization formulation, a lower-level optimization problem is embedded within an upper-level optimization problem. One example is a lower-level model predictive controller (MPC) that aims to optimize set-point tracking and an upper-level scheduling layer that aims to minimize production costs. In contrast to sequential solution techniques where different decision layers are optimized independently, bilevel methods aim to enable the interaction between the decision layers involved. Chemical engineering applications include the simultaneous scheduling and dynamic optimization of batch plants and dynamic transitions in a polymer manufacturing process, where dynamic optimization problems are solved a priori to obtain a piecewise response function of the lower-level problem, which is latter embedded into the upper-level and solved as single-level optimization problems (Chu and You 2012, 2014b). The reformulation of a bilevel problem into a single level has also been proposed in the context of integration of planning and scheduling under demand uncertainty, through surrogate-based and data-driven techniques (Beykal et al. 2022). A bilevel decomposition approach has also been proposed for the simultaneous planning and scheduling of single-stage multiproduct continuous plants with parallel lines, with an upper-level planning and relaxed scheduling model and a lower-level simultaneous planning and scheduling model (Erdirik-Dogan and Grossmann 2008).

Another approach recently proposed in the literature integrates NMPC with process design, through the Karush–Kuhn–Tucker (KKT) transformation of the NMPC optimization layer. This leads into a highly nonconvex single-level problem that has the potential to be solved with conventional optimization solvers. In the context of MINLP, this strategy has been used to address the simultaneous design and control of distillation units (Palma-Flores et al. 2023). Note that the solution of a bilevel optimization problem reformulated as a single level problem brings additional challenges, e.g., the single-level formulation that results from using the KKT transformation violates multiple constraint qualifications at every feasible point, due to the KKT complementarity slackness constraints. Moreover, an optimal solution for the reformulated single-level problem may not represent an optimal solution for the original bilevel problem (Dempe et al. 2015).

Note that customized decomposition algorithms for single-level MINLP optimization are also referred in the literature as bilevel optimization approaches, e.g., Lara et al. (2019) deal with the MINLP design and planning of manufacturing networks through a customized decomposition that result in an MIP master problem and NLP subproblems.

4.8 Back-off algorithms

Back-off algorithms have been used to effectively handle uncertainties and disturbances in optimal process integration. When uncertainty is not considered in optimal decision-making, the solution typically lies in the boundary of the feasible space of operation (Patilas and Kookos 2021b). Consequently, uncertainty and slight disturbance variations may shift the operation of a process outside their feasible operating window, e.g., weeping may occur in a distillation column. Back-off algorithms iteratively move away from an initial nominal solution to a new optimal and feasible operating point in the presence of uncertainty/disturbances. A back-off algorithm serves as a more suitable technique to deal with MINLP optimization problems under uncertainty, when compared to traditional uncertainty handling strategies in optimization, e.g., the direct solution of the monolithic multiscenario MINLP formulation using general-purpose solvers (C. Li and Grossmann 2021).

The main challenge in this approach is to determine a systematic strategy to determine the amount of back-off needed to accommodate uncertainties and disturbances in the operation. In the context of MINLP integrated decision-making, the back-off approach has been applied to the simultaneous design and control of a CSTR, an ideal distillation column, and a reactive distillation process (Patilas and Kookos 2021b, 2021a). The amount of back-off was determined iteratively by finding the maximum violation of the vector of inequality constraints for a given time horizon and disturbance scenario, i.e., a worst-case approximation. Aiming to avoid conservative solutions due to the worst-case approximation, a stochastic back-off approach that models disturbance and uncertainty with stochastic random variables has been proposed and applied to the simultaneous scheduling and dynamic optimization of batch plants (Valdez-Navarro and Ricardez-Sandoval 2019), and the design, control, and scheduling of a multiproduct reactor (Koller et al. 2018). Note that further research is still needed to overcome the stochastic back-off computational challenges, e.g., due to its random nature, the stochastic back-off approach may yield different solutions every time the algorithm runs, and the need to run Monte Carlo simulations for back-off quantification remains the key computational bottleneck of this approach.

5.1 Branch and bound

Based on the BB principle, Ma and Li (2022) proposed the homotopy continuation-enhanced BB (HCBB) algorithm. The key novelty of this strategy is the introduction of a homotopy path between a node and its parent node that improves the convergence and efficiency of solving NLP subproblems through the branching tree. Thus, this feature exploits the potential similarity between the NLP subproblems solved in BB to generate improved variable initializations. As highlighted in that work, this reduces the variability in the local solution obtained for strongly nonconvex MINLPs when testing different initializations, compared to the SBB and DICOPT solvers. Consequently, this feature may benefit the optimization of integrated decision-making MINLP problems, whose NLP subproblems are in general nonconvex and challenging to converge.

Despite the benefits of HCBB, an issue not considered by this algorithm is the convergence to a feasible solution at the root node of the branching tree, i.e., HCBB assumes that good initial guesses are available at the root node. To circumvent this issue, Liu et al. (2022) proposed the feasible path-based BB (FPBB) algorithm, which brings improvements with respect to HCBB by incorporating a feasible path-based framework that partitions continuous variables into independent and dependent variables. By expressing dependent variables as implicit functions of independent variables, the equality constraints linking these variables can be ignored during the BB procedure and convergence is improved. The authors illustrate the benefits of FPBB with different case studies involving separation systems involving multiple interconnected conventional distillation units modeled through nonideal tray-by-tray models. Thus, FPBB is a promising strategy to incorporate highly nonlinear modeling features such those arising in separation systems into integrated decision-making MINLP problems. Accordingly, these algorithmic improvements may be helpful to address integrated decision-making problems involving intensified or heat integrated distillation sequences modeled with rigorous first-principles models, which have been rarely considered in the literature due to their complexity, e.g., see (Dias and Ierapetritou 2019).

5.2 Outer approximation

During the last years, several improvements to the standard OA technique have been recently proposed, e.g., level-based OA (LBOA) (Kronqvist et al. 2020), quadratic OA (QOA) (Kronqvist et al. 2020), proximal OA (POA) (De Mauri et al. 2020), decomposition-based OA (DECOA) (Muts et al. 2020), and regularized OA (ROA) (Bernal et al. 2022b). LBOA and QOA aim to accelerate OA by carefully choosing integer combinations in the master problem. In the case of LBOA, it differs from OA by considering two steps when generating trial integer combinations: the solution of an MIP, and the solution of an auxiliary Mixed Integer Quadratic Problem (MIQP) that introduces a regularization based on the Euclidean norm to each iteration to keep the next trial integer combination close to the previous one. QOA follows the same principle as LBOA, but it also incorporates second-order derivative information when choosing trial integer combinations. Test instances demonstrated the computational advantages of LBOA and QOA when solving nonlinear convex MINLPs (Kronqvist et al. 2020). More recently, ROA generalized the regularization idea introduced by LBOA to consider a user-specified regularization function (Bernal et al. 2022b). Due to the success of OA methods, LBOA, QOA, and ROA may benefit integrated decision-making by speeding-up the convergence in the solution. Specifically, the regularization ideas from the algorithms discussed above can be incorporated into the solution of integrated decision-making problems. For instance, subproblem convergence in decomposition methods (e.g., those based on GBD or OA) may be slow when the discrete variables generated by the master problem are not carefully selected as the iterations progress. This poses limitations in the context of integrated decision-making if there is a computational time limit that must be satisfied, e.g., in closed-loop scheduling and control, where decisions must be made in the order of magnitude of the controller sampling time in a bottom-up integration scheme (Caspari et al. 2020). To tackle this issue, regularization techniques may help to keep integer combinations close to an incumbent solution, resulting in subproblems that are similar to each other. This means that subproblems may have access to better initializations and converge faster, if subproblem variables are reinitialized from previous subproblems.

Another recent development is the feasibility pump implemented within OA solvers such as DICOPT, which facilitates the identification of feasible solutions in a preprocessing step (Bernal et al. 2020). POA improves the feasibility pump by integrating the feasibility pump in OA as it iterates. At each iteration, the discrete trial point selection by the OA’s master problem solely relies on improving the objective function of an MIP approximation of the problem; instead, POA balances objective function improvement and feasibility by generating guesses that are closer to the nonlinear feasible space. One disadvantage of POA is the need to tune parameters that define the trade-off between objective function values and constraint feasibility. However, a careful selection of tuning parameters in POA showed superior performance when compared to OA and the execution of the feasibility pump at preprocessing (De Mauri et al. 2020). These improvements are relevant in integrated decision-making problems, where identifying feasible solutions usually requires optimization expertise to develop custom feasibility algorithms depending on the application. For example, the initialization of the single-level MINLP problem in integrated design and NMPC control through bilevel decomposition requires educated initial guesses due to the nonconvexity of complementarity constraints (Palma-Flores et al. 2023). Another example includes the initialization of problems involving partial differential equations (PDE), resulting in mixed-integer PDE-constrained nonlinear optimization formulations that may be difficult to solve without customized initializations, e.g., when considering the scheduling and control of tubular reactors (Flores-Tlacuahuac and Grossmann 2010a), or oilfield production planning and control problems (Aristizabal et al. 2023). Thus, feasibility detection improvements may foster the application of MINLP by engineering practitioners that are not necessarily familiar with feasibility and variable initialization strategies.

The recently proposed DECOA algorithm presents an OA variant that relies on a block-reformulation of the original MINLP problem. This reformulation poses the original problem as multiples blocks of nonlinear constraints and variables interconnected through coupling constraints. Compared to OA, DECOA simplifies the solution of subproblems as multiple smaller subproblems that can be solved in parallel. Compared to OA, DECOA generally reduces the total number of MIP master problems that need to be solved (Muts et al. 2020). Note that integrated decision-making problems may have block-separable structures without the need of additional reformulations; e.g., Kelley et al. (2018a) have identified and exploited block-separable structures for simultaneous scheduling and dynamic optimization, while Li and Ierapetritou (2010) have identified this structure in the integration of planning and scheduling. Thus, DECOA may be a suitable strategy to address decision-making problems integrating multiple (e.g., more than three) decision layers by exploiting their block structure. Note that the OA techniques discussed herein impose the requirement that the MINLP must be convex to guarantee global optimality. In this regard, previous works have attempted to generate cuts using convex and concave McCormick relaxations, solve NLP subproblems to global optimality, and avoid cycling through no-good integer cuts, thus allowing OA to guarantee global optimality (Kesavan et al. 2004; Z. Peng et al. 2024).

5.3 Generalized Benders decomposition

Variants of GBD have also been proposed in recent years. Nonconvex sensitivity-based GBD (NSGBD) (J.-J. Lin et al. 2023) is a recent technique that offers convergence improvements compared to GBD. Traditional GBD techniques have been criticized in the past due to its convergence properties, e.g., for a nonconvex NLP problem, GBD may converge to a point that does not satisfy KKT conditions (Geoffrion 1972). NSGBD addresses this issue and guarantees convergence to a KKT point for nonconvex problems. To do that, NSGBD manipulates Benders cuts to guarantee that they do not cut-off the local optimum. NSGBD has been tested on the dynamic optimization of a fluid catalytic cracking unit, but it has not been tested yet in the field of integrated decision-making. GBD has been widely applied in simultaneous design and control problems (e.g., design and control of reactive distillation columns (Vasudevan et al. 2018) and simultaneous design and scheduling problems (e.g., design and scheduling of multipurpose batch plants (X. Lin and Floudas 2001); hence, NSGBD may bring the additional benefit of converging faster to a point that satisfies first order necessary optimality conditions for these problems.

A GBD variant that aims to solve nonconvex MINLP problems to global optimality is nonconvex nested Benders decomposition (NCNBD) (Füllner and Rebennack 2022). This method aims to address nonconvex multistage MINLP problems, where discrete and continuous variables as well as nonlinearities appear in any of the stages. NCNBD combines a variety of tools such as piecewise linear relaxations, regularization, binary approximations, and cutting planes. It relies on the dynamic refinement of the different approximations considered in the method, e.g., piecewise linear Benders cuts are constantly refined as iterations progress. NCNBD involves two iteration loops: in the inner loop, NCNBD optimizes an MIP that outer approximates the original multistage MINLP problem, whereas in the outer loop, NCNBD iteratively refines the outer approximation to improve its accuracy. NCNBD was tested with a variety of unit commitment problems, showing that it becomes more efficient than existing global solvers such as LINDO Global and Couenne as the number of stages increases. As the first decomposition method proposed for general multistage nonconvex MINLPs, NCNBD may be useful to incorporate multistage uncertainty in integrated decision-making MINLP problems, while providing global optimality guarantees. For instance, multistage uncertainty appears in all the operational decision layers (planning, scheduling, and control), which are decision processes executed during multiple sequential instants (i.e., stages) of time, and for which new information is revealed and unforeseen events may occur at any of those decision stages. An illustrative example in this context is the case of optimal generation planning and scheduling in hydro-based power systems over a multiyear planning horizon, which is divided into a series of coupling problems with different time scales including a long-term planning horizon with monthly steps and a shorter scheduling horizon with weekly time steps (Beltrán et al. 2021). Unforeseen events in this case include uncertainty in market prices and inflow energy, which can be modeled and considered by NCNBD as multistage uncertainty.

5.4 Column generation

Column generation is traditionally used for large-scale LP and has been embedded within BB in branch-and-price to handle MIP problems. Column generation has been recently extended to solve MINLP problems to global optimality. Allman and Zhang (2021) proposed a branch-and-price method for MINLPs (MINLP-BP), which combines column generation with the global optimization of MINLPs. MINLP-BP relies on the presence of complicating constraints in the formulation, i.e., when complicating constraints are removed, the resulting MINLP problem (or subproblems) can be solved more efficiently. As previously discussed in the Lagrangian decomposition section, integrated decision-making problems in chemical engineering usually follow this structure; however, MINLP-BP has not been used in this field. Nonetheless, MINLP-BP has shown to be useful when solving chemical engineering optimization problems such as the multiscenario synthesis of water networks and the multiscenario design of multiproduct batch plants (Allman and Zhang 2021). Overall, MINLP-BP showed improved performance when compared to BARON, e.g., from 25 problem instances tested, MINLP-BP solved 16 to global optimality, while BARON could not solve any of the instances to global optimality under 10,000 s. Note that this solution strategy exploits problems with complicating constraints; hence, it may be suitable to solve those integrated decision-making problems that have been tackled with LD (e.g., simultaneous refinery planning and crude-oil scheduling), with the additional benefit of providing global optimality guarantees.

Another column generation (CG)-based strategy for MINLPs (MINLP-CG) (Muts et al. 2023) combines CG with the Frank–Wolfe method to generate columns, thus avoiding the need to rely on a BB search. Similarly to DECOA, MINLP-CG also relies on a block-separable reformulation of the MINLP and considers these blocks as columns. Consequently, MINLP-CG is another solution technique that is worth exploring in the context of the integration of two and three decision layers, where each block would represent a decision layer. A key benefit of MINLP-CG is that it enables the implementation of parallel computation strategies to generate new columns, thanks to the block-separable structure of the MINLP. The superstructure-based synthesis of decentralized energy supply systems was used as a case study. When compared to BARON, MINLP-CG showed better dual and primal bounds for problems with thousands of variables, which are often found in energy systems optimization problems.

5.5 Logic-based algorithms

Although the first logic-based algorithms appeared in the 1990s, these are becoming popular due to their increasing accessibility. Furthermore, recent research works have developed new logic-based algorithms aimed to facilitate GDP optimization, including integrated decision-making applications. Logic-based solution approaches take advantage of the GDP form of the problem and its underlying logic relationships. The defining characteristic of logic-based algorithms is that they solve the nonlinear subproblems that result from fixing Boolean variables in a reduced space, i.e., nonlinear constraints and variables related to inactive disjunctions are omitted from the subproblem model. This avoids numerical issues (e.g., singularities) and reduces subproblems’ sizes when compared to MINLP solvers. Logic-based solvers that are currently available include logic-based outer approximation (L-OA) (Türkay and Grossmann 1996), global logic-based outer approximation (GL-OA) (Trespalacios and Grossmann 2016), and logic-based branch and bound (L-BB) (Lee and Grossmann 2000). More recently, logic-based D-SDA (LD-SDA) (Bernal et al. 2022a) and logic-based discrete Benders decomposition (LD-BD) (Liñán and Ricardez-Sandoval 2023) have been proposed in the literature, which incorporate discrete convex analysis and logic-based Benders decomposition (LBBD) (Hooker 2019) principles into the direct solution of GDP problems. Similarly to D-SDA, LD-SDA and LD-BD require disjunctions to represent discrete ordered decisions, e.g., choosing an optimal discrete processing time, or selecting an optimal tray in a distillation column.

Despite their benefits, logic-based solvers have not been extensively used yet in the field of integrated MINLP decision-making. To the authors’ knowledge, LD-SDA and LD-BD are the only recent logic-based algorithms that have been presented for integrated decision making. LD-SDA and LD-BD bring together the benefits of logic-based optimization and discrete convex analysis features, e.g., an exploration of disjunctions in reduced-space that relies on neighborhood verification. These algorithms have been tested in the context of simultaneous scheduling and dynamic optimization of network batch plants under variable processing times, showing promising results when compared to general-purpose MINLP solvers (Liñán and Ricardez-Sandoval 2024b), e.g., a multicut LD-BD strategy performed a more efficient exploration of disjuncts, while general-purpose MINLP solvers sometimes stagnated at the initialization. Note that the development of logic-based methods as general-purpose GDP solvers is in an early (developing) stage compared to general-purpose MINLP solvers. Consequently, there are a variety of future research directions, such as using more sophisticated branching techniques in L-BB, or expanding the range of solvable problems with LD-SDA and LD-BD by enabling inclusive and unordered disjunctions in the formulation.

5.6 Software tools

The expertise, time, and effort needed for the implementation and coding of MINLP algorithms for integrated decision-making may hinder the adoption of these techniques by engineering practitioners. Although many of the MINLP techniques discussed in Sections 4 and 5 require some degree of customization, many algorithmic steps can be automated and incorporated within new software tools and solvers suitable for integrated decision-making. Some examples of readily available software include the parametric optimization and control (PAROC) framework for the multiparametric optimal design, operation, and control of process systems (Burnak et al. 2020); the bilevel-parametric optimization (B-POP) toolbox that incorporates a variety of solvers for bilevel optimization (Avraamidou and Pistikopoulos 2019); the data-driven optimization of bi-level mixed-integer nonlinear problems (DOMINO) framework that approximates bilevel optimization problems as single level problems through a data-driven approximation of one of the problems levels (Beykal et al. 2020); or the reactive distillation toolbox (RD-toolbox), specifically designed for the integrated design and control of reactive distillation systems (Iftakher et al. 2022). Furthermore, automated tools for chemical process flowsheet synthesis that can deal with GDP problems using advanced machine learning frameworks are emerging in the literature (Mann et al. 2024; Reynoso-Donzelli and Ricardez-Sandoval 2024).

In the context of recent MINLP algorithmic developments, the mixed-integer nonlinear decomposition toolbox in Pyomo (MindtPy) incorporates traditional MINLP techniques based on BB and OA, along with recently proposed strategies such as ROA and a feasibility pump for OA (Bernal et al. 2018). GDP modeling and logic-based algorithms are also becoming more accessible to the general public, thanks to recently developed packages for algebraic modeling languages (Chen et al. 2022; Perez et al. 2023). Particularly, the GDPopt solver in Pyomo not only enables GDP modeling and MINLP transformations (e.g., Big-M) but also provides access to logic-based solvers such as LOA, GLOA, and LBB (Chen et al. 2022).