Mixed-integer Dynamic Optimization Problem Research Articles

DRL-Based Secure Aggregation and Resource Orchestration in MEC-Enabled Hierarchical Federated Learning

Federated learning (FL) provides a new paradigm for protecting data privacy by enabling model training at devices and model aggregation at servers. However, data information may be leaked to honest-but-curious aggregation servers by updated model parameters. The existing secure methods do not fully exploit the potentiality of data characteristics in enhancing security, which makes it impossible to optimize limited system resources overall to achieve secure, fair and efficient FL systems. In this paper, a DRL-based joint secure aggregation and resource orchestration scheme is proposed to guarantee security and fairness, and improve efficiency for hierarchical FL assisted by untrusted mobile-edge computing (MEC) servers. We formulate a joint optimization problem of data size, payment and resource orchestration, to maximize the long-term social welfare subject to secure aggregation and limited resources. Since the formulated problem is a complex mixed integer dynamic optimization problem with NP-hardness, where multiple mixed integer optimization variables are highly coupled in time-varying constraints and objective function, it is difficult to obtain its optimal solution via traditional optimization methods. Thus, we propose a hierarchical reward function-based DRL algorithm (MATD3) to guide the agents to approach the optimal policy of secure aggregation and resource orchestration. Simulation results show that the proposed algorithm MATD3 can achieve superior performance over comparison algorithms and the MEC-enabled HFL framework outperforms two-layer FL frameworks.

Mixed-integer minimax dynamic optimization for structure identification of glycerol metabolic network

Cell metabolism is a dynamic regulation process, in which its network structure and/or regulatory mechanisms can change constantly over time due to internal and external perturbations. This paper models glycerol metabolism in continuous fermentation as a nonlinear mixed-integer dynamic system by defining the time-varying metabolic network structure as an integer-valued function. To identify the dynamic network structure and kinetic parameters, we establish a mixed-integer minimax dynamic optimization problem with concentration robustness as its objective functional. By direct multiple shooting strategy and a decomposition approach consisting of convexification, relaxation and rounding strategy, the optimization problem is transformed into a large-scale approximate multistage parameter optimization problem. It is then solved using a competitive particle swarm optimization algorithm. We also show that the relaxation problem yields the best lower bound for the optimization problem, and its solution can be arbitrarily approximated by the solution obtained from rounding strategy. Numerical results indicate that the proposed mixed-integer dynamic system can better describe cellular self-regulation and response to intermediate metabolite inhibitions in continuous fermentation of glycerol. These numerical results show that the proposed numerical methods are effective in solving the large-scale mixed-integer dynamic optimization problems.

Mixed-Integer dynamic optimization of conventional and vapor recompressed batch distillation for economic and environmental objectives

In this contribution, a unique multi-objective mixed-integer dynamic optimization problem considering two conflicting objectives, namely, maximization of amount of product per dollar while minimizing CO2 emission is formulated and solved using the elitist non-dominated genetic algorithm for both conventional batch distillation (CBD) and vapor recompressed batch distillation (VRBD) operating at constant reflux mode. Here, selection of an optimal solution from the Pareto-optimal front is performed by 10 Pareto ranking methods along with entropy weighting. A wide boiling separating system (i.e., acetone and water) is adopted for illustrating the proposed multi-objective optimization of batch distillation. Two separate optimization studies for CBD and VRBD are conducted with the target of either improving an existing plant or setting up a new plant. Results obtained show that most of the popular Pareto ranking methods select same optimal solution for each of these problems. Finally, a comparative analysis is performed to find the benefits of vapor recompression over the conventional scheme.

Hybrid parallel multimethod hyperheuristic for mixed-integer dynamic optimization problems in computational systems biology

This paper describes and assesses a parallel multimethod hyperheuristic for the solution of complex global optimization problems. In a multimethod hyperheuristic, different metaheuristics cooperate to outperform the results obtained by any of them isolated. The results obtained show that the cooperation of individual parallel searches modifies the systemic properties of the hyperheuristic, achieving significant performance improvements versus the sequential and the non-cooperative parallel solutions. Here we present and evaluate a hybrid parallel scheme of the multimethod, using both message-passing (MPI) and shared memory (OpenMP) models. The hybrid parallelization allows to achieve a better trade-off between performance and computational resources, through a compromise between diversity (number of islands) and intensity (number of threads per island). For the performance evaluation, we considered the general problem of reverse engineering nonlinear dynamic models in systems biology, which yields very large mixed-integer dynamic optimization problems. In particular, three very challenging problems from the domain of dynamic modeling of cell signaling were used as case studies. In addition, experiments have been carried out in a local cluster, a large supercomputer and a public cloud, to show the suitability of the proposed solution in different execution platforms.

Simultaneous Design and Control of a Ternary Reactive Distillation Column with Inert Material

AbstractProcess integration means significant challenges in operating a chemical process dynamically due to its complexity. One example is the ternary reactive distillation with inert material. Based on the heuristic approach, chemical process design and process control are performed independently one after another. This may lead to infeasible solutions either in process design and control or in both. The application of simultaneous design and control, formulated as mixed‐integer dynamic optimization (MIDO) problems, increased rapidly in the last decade. Simultaneous process design and control, including control structure selection and control parameters, is devised for the challenging benchmark problem of a ternary reactive distillation with inert material. The improved process design and control is obtained by solving the MIDO problem.

An adaptive wavelet method for solving mixed-integer dynamic optimization problems with discontinuous controls and application to alkali–surfactant–polymer flooding

ABSTRACTThis article presents an adaptive rationalized Haar function approximation method to solve dynamic optimization with mixed-integer and discontinuous controls. Three measures are taken to deal with the discontinuity. First, the problem is converted into a multi-stage optimization problem by non-uniform control vector parameterization. Secondly, an adaptive strategy is proposed to regulate the interval division and the order of Haar function vectors. Thirdly, a structure detection method is presented to refine the subintervals, in which adjacent arcs with the same input type are merged into one to modify redundant subintervals. During this approximation solution, the mixed-integer restriction is realized by the integer truncation strategy. Combined with the Hamiltonian function, a validation principle is shown to verify the optimality of the solution. Finally, the proposed method is applied to solve the enhanced oil recovery for alkali–surfactant–polymer flooding. The effectiveness of the method is illustrated through simulation.

A parallel metaheuristic for large mixed-integer dynamic optimization problems, with applications in computational biology.

BackgroundWe consider a general class of global optimization problems dealing with nonlinear dynamic models. Although this class is relevant to many areas of science and engineering, here we are interested in applying this framework to the reverse engineering problem in computational systems biology, which yields very large mixed-integer dynamic optimization (MIDO) problems. In particular, we consider the framework of logic-based ordinary differential equations (ODEs).MethodsWe present saCeSS2, a parallel method for the solution of this class of problems. This method is based on an parallel cooperative scatter search metaheuristic, with new mechanisms of self-adaptation and specific extensions to handle large mixed-integer problems. We have paid special attention to the avoidance of convergence stagnation using adaptive cooperation strategies tailored to this class of problems.ResultsWe illustrate its performance with a set of three very challenging case studies from the domain of dynamic modelling of cell signaling. The simpler case study considers a synthetic signaling pathway and has 84 continuous and 34 binary decision variables. A second case study considers the dynamic modeling of signaling in liver cancer using high-throughput data, and has 135 continuous and 109 binaries decision variables. The third case study is an extremely difficult problem related with breast cancer, involving 690 continuous and 138 binary decision variables. We report computational results obtained in different infrastructures, including a local cluster, a large supercomputer and a public cloud platform. Interestingly, the results show how the cooperation of individual parallel searches modifies the systemic properties of the sequential algorithm, achieving superlinear speedups compared to an individual search (e.g. speedups of 15 with 10 cores), and significantly improving (above a 60%) the performance with respect to a non-cooperative parallel scheme. The scalability of the method is also good (tests were performed using up to 300 cores).ConclusionsThese results demonstrate that saCeSS2 can be used to successfully reverse engineer large dynamic models of complex biological pathways. Further, these results open up new possibilities for other MIDO-based large-scale applications in the life sciences such as metabolic engineering, synthetic biology, drug scheduling.

Optimal response under partial plant shutdown with discontinuous dynamic models

This work describes mathematical formulations for modeling aspects of partial shutdowns in multiunit plants. The specific type of partial shutdown considered is one that permits the decoupling of affected units from the rest of the plant, thus enabling continued plant operation, albeit in a more limited fashion. Parsimonious and computationally efficient mixed-integer formulations are presented for specific discontinuous phenomena that arise in partial shutdown modeling, such as shutdown thresholds, induced shutdowns, discontinuous costs, and minimum shutdown durations. It is demonstrated that induced shutdowns (secondary shutdowns triggered by the original shutdown) can be correctly penalized in the objective by capturing the shutdown's true discontinuous economic cost. The computed optimal solution is implemented in closed-loop by employing a multitiered model predictive shutdown controller, in which a discrete-time mixed-integer dynamic optimization (MIDO) problem is embedded. Both objectives of maximizing economics and minimizing restoration (shutdown recovery) time are considered.

Integrated Operation and Cyclic Scheduling Optimization for an Ethylene Cracking Furnaces System

Multiple cracking furnaces in an ethylene plant run in parallel to produce ethylene, propylene, and other products from different hydrocarbon feedstocks. Because both coke formation in radiant coils and change of operation conditions with time have significant effects on the performance of cracking furnaces, it is better for the cyclic scheduling to be simultaneously optimized with the operation conditions. To match this real requirement, a mixed-integer dynamic optimization (MIDO) problem is presented for the optimization of both operation conditions and cyclic scheduling simultaneously, through which the optimal assignment, the processing time, the subcycle number, and the optimal operation conditions of different feeds in different cracking furnaces are determined at the same time. To solve the MIDO problem, it is discretized and converted into a large scale mixed-integer nonlinear programming (MINLP) problem. The two scheduling cases of a cracking furnaces system are illustrated showing a remarkable i...

Novel Optimization Model and Efficient Solution Method for Integrating Dynamic Optimization with Process Operations of Continuous Manufacturing Processes

In this paper, we propose a novel mathematical model for the integrated optimization for production planning, scheduling, and dynamic optimization of continuous manufacturing processes. With a novel approach to estimate the inventory cost, the integrated problem is first formulated as a mixed-integer dynamic optimization (MIDO) problem, which is then reformulated into a large scale mixed-integer nonlinear program (MINLP). By using metamodeling to characterize the detailed linking information between different decision layers, we develop an efficient solution method to decompose the integrated MINLP into a mixed-integer linear program (MILP) for an integrated planning and scheduling problem with flexible recipes, and a set of dynamic optimization problems for generating metamodels. We further apply a bilevel decomposition algorithm to improve the computational efficiency of solving the integrated planning and scheduling problem with flexible recipes. The proposed models and algorithms are demonstrated through case studies of methyl methacrylate (MMA) polymerization processes. The results show that the proposed methods reduce the computational time by more than 2 orders of magnitude compared with the approach of solving the entire integrated optimization problem directly.

MINLP Formulation for Simultaneous Planning, Scheduling, and Control of Short-Period Single-Unit Processing Systems

A simultaneous strategy for solving the integrated planning, scheduling, and control problem considering short-term periods is proposed in this paper. The problem is formulated as a mixed integer dynamic optimization problem. The main practical motivation to use a simultaneous rather than a sequential solution strategy is to seek improved optimal solutions by considering the interactions among planning, scheduling, and control. Integration of the three levels of the problem represents a challenge given the very different horizon times involved in the operations, resulting in growth of the problem in terms of the number of equations to be solved and hence on computational requirements to carry it out. A full space approach was used rather than trying a decomposition strategy. Moreover, a nonlinear model predictive control strategy was also used to account for online product transition trajectories. The integrated planning, scheduling, and control is formulated as a mixed integer dynamic optimization model, which is tested using the dynamic models of three continuous stirred tank reactors featuring different nonlinear behavior.

Integrated Planning, Scheduling, and Dynamic Optimization for Batch Processes: MINLP Model Formulation and Efficient Solution Methods via Surrogate Modeling

We solve the challenging problem of integrated planning, scheduling, and dynamic optimization for sequential batch processes with fixed batch sizes. The integrated problem is first formulated into a complicated mixed-integer dynamic optimization (MIDO) problem that is then discretized into a large-scale mixed-integer nonlinear programing (MINLP) problem. There are a planning model, multiple scheduling models in planning periods, and a number of dynamic models describing task execution processes. To efficiently solve the complex MINLP problem, we develop two efficient methods that separate the subproblems using surrogate models to represent the linking functions. The first method decomposes the dynamic optimization problems from the integrated planning and scheduling problem where the surrogate models represent task processing costs dependent on the processing times. The second method further decomposes the scheduling problems from the planning problem where the surrogate models represent production costs dependent on production quantities. Compared to the direct solution approach, the proposed methods reduce the computational time by more than 4 orders of magnitude in the case studies.

Combined simulation–optimization methodology to reduce the environmental impact of pharmaceutical processes: application to the production of Penicillin V

Abstract In this work, we present a systematic method for the optimal design of pharmaceutical manufacturing processes that allows decision-makers to assess economic and environmental concerns in the early stages of the process development. Our approach relies on the combined use of process simulation, multi-objective optimization, economic analysis and life cycle assessment. The pharmaceutical production plant design is mathematically formulated as a multi-objective mixed-integer dynamic optimization (moMIDO) problem, which is solved by a decomposition algorithm that generates as output a set of Pareto optimal solutions. The capabilities of the proposed methodology are illustrated through the production of the antibiotic Penicillin V.

Simultaneous Optimal Design and Control of an Extractive Distillation System for the Production of Fuel Grade Ethanol Using a Mathematical Program with Complementarity Constraints

The simultaneous optimal design and control problem of an extractive distillation system is studied using a rigorous model and solved when the feed is subjected to composition disturbances. First, the problem is formulated as a mixed-integer dynamic optimization (MIDO) problem, and then transformed into a mathematical program with complementarity constraints (MPCC) problem using orthogonal collocation on finite elements. The model allows simultaneous determination of the number of trays, feed tray locations, structural parameters of the column, heat exchanger areas, set points of both the controlled and manipulated variables, and the optimal control policy in order to obtain an optimal design which maximizes profit and also guarantees feasible dynamic operation. The study also demonstrates how the distillation column design problem can be addressed as a MPCC problem instead of as a mixed-integer nonlinear programming (MINLP) problem, containing thousands of continuous variables and constraints, and how can it be solved in reasonable times using state-of-the-art algorithms. The problem is modeled in GAMS and solved using IPOPT. Finally, results obtained by the simultaneous strategy are compared to those obtained by addressing the design and control problem sequentially.

Integration of production scheduling and dynamic optimization for multi-product CSTRs: Generalized Benders decomposition coupled with global mixed-integer fractional programming

Integration of production scheduling and dynamic optimization can improve the overall performance of multi-product CSTRs. However, the integration leads to a mixed-integer dynamic optimization problem, which could be challenging to solve. We propose two efficient methods based on the generalized Bender decomposition framework that take advantage of the special structures of the integrated problem. The first method is applied to a time-slot formulation. The decomposed primal problem is a set of separable dynamic optimization problems and the master problem is a mixed-integer nonlinear fractional program. The master problem is then solved to global optimality by a fractional programming algorithm, ensuring valid Benders cuts. The second decomposition method is applied to a production sequence formulation. Similar to the first method, the second method uses a fractional programming algorithm to solve the master problem. Compared with the simultaneous method, the proposed decomposition methods can reduce the computational time by over two orders of magnitudes for a polymer production process in a CSTR.

Integrated Scheduling and Dynamic Optimization of Complex Batch Processes with General Network Structure Using a Generalized Benders Decomposition Approach

We address the integration of scheduling and dynamic optimization for batch chemical processes. The processes can have complex network structures, allowing material splitting and mixing. The integrated problem is formulated as a mixed-integer dynamic optimization problem where a continuous-time scheduling model is linked to the dynamic models via processing times, processing costs, and batch sizes. To reduce the computational complexity, we develop a tailored and efficient decomposition method based on the framework of generalized Benders decomposition by exploiting the special structure of the integrated problem. The decomposed master problem is a scheduling problem with variable processing times and processing costs, as well as the Benders cuts. The primal problem comprises a set of separable dynamic optimization problems for the processing units. By collaboratively optimizing the process scheduling and the process dynamics, the proposed method substantially improve the overall economic performance of the batch production compared with the conventional sequential method which solves the scheduling problem and the dynamic optimization problems separately. In comparison with the simultaneous method which solves the integrated problem by a general-purpose MINLP solver directly, the proposed method can reduce computational times by orders of magnitude.

Identification of regulatory structure and kinetic parameters of biochemical networks via mixed-integer dynamic optimization

BackgroundRecovering the network topology and associated kinetic parameter values from time-series data are central topics in systems biology. Nevertheless, methods that simultaneously do both are few and lack generality.ResultsHere, we present a rigorous approach for simultaneously estimating the parameters and regulatory topology of biochemical networks from time-series data. The parameter estimation task is formulated as a mixed-integer dynamic optimization problem with: (i) binary variables, used to model the existence of regulatory interactions and kinetic effects of metabolites in the network processes; and (ii) continuous variables, denoting metabolites concentrations and kinetic parameters values. The approach simultaneously optimizes the Akaike criterion, which captures the trade-off between complexity (measured by the number of parameters), and accuracy of the fitting. This simultaneous optimization mitigates a possible overfitting that could result from addition of spurious regulatory interactions.ConclusionThe capabilities of our approach were tested in one benchmark problem. Our algorithm is able to identify a set of plausible network topologies with their associated parameters.

Integration of scheduling and control with online closed-loop implementation: Fast computational strategy and large-scale global optimization algorithm

In this paper, we propose a novel integration method to solve the scheduling problem and the control problem simultaneously. The integrated problem is formulated as a mixed-integer dynamic optimization (MIDO) problem which contains discrete variables in the scheduling problem and constraints of differential equations from the control problem. Because online implementation is crucial to deal with uncertainties and disturbances in operation and control of the production system, we develop a fast computational strategy to solve the integration problem efficiently and allow its online applications. In the proposed integration framework, we first generate a set of controller candidates offline for each possible transition, and then reformulate the integration problem as a simultaneous scheduling and controller selection problem. This is a mixed-integer nonlinear fractional programming problem with a non-convex nonlinear objective function and linear constraints. To solve the resulting large-scale problem within sufficiently short computational time for online implementation, we propose a global optimization method based on the model properties and the Dinkelbach's algorithm. The advantage of the proposed method is demonstrated through four case studies on an MMA polymer manufacturing process. The results show that the proposed integration framework achieves a lower cost rate than the conventional sequential method, because the proposed framework provides a better tradeoff between the conflicting factors in scheduling and control problems. Compared with the simultaneous approach based on the full discretization and reformulation of the MIDO problem, the proposed integration framework is computationally much more efficient, especially for large-scale cases. The proposed method addresses the challenges in the online implementation of the integrated scheduling and control decisions by globally optimizing the integrated problem in an efficient way. The results also show that the online solution is crucial to deal with the various uncertainties and disturbances in the production system.

Decentralized Control System Design under Uncertainty Using Mixed‐Integer Optimization

AbstractDecentralized control system design comprises the selection of a suitable control structure and controller parameters. In this contribution, the optimal control structure and the optimal controller parameters are determined simultaneously using mixed‐integer dynamic optimization (MIDO) under uncertainty, to account for nonlinear process dynamics and various disturbance scenarios. Application of the sigma point method is proposed in order to approximate the expectation and the variance of a chosen performance index with a minimum number of points to solve the MIDO problem under uncertainty. The proposed methodology is demonstrated with a benchmark problem of an inferential control for a reactive distillation column. The results are compared with established heuristic design methods and with previous deterministic approaches.

A Multiobjective Optimization Approach for the Simultaneous Single Line Scheduling and Control of CSTRs

A new multiobjective optimization formulation dealing with simultaneous scheduling and control issues in single line processing systems is proposed. Objective functions featuring economic profit and dynamic performance are considered because normally they are in conflict. Because integer, continuous variables and process dynamic behavior are involved, the bicriterion optimization problem is cast in terms of a mixed-integer dynamic optimization (MIDO) problem. The Pareto front of each problem is computed using the e-constraint method for handling multiobjective problems. The results indicate that improved optimal solutions can be obtained by using multiobjective optimization techniques instead of just simple merging of all the target objective functions into a single objective. The proposed multiobjective approach for handling scheduling and control problems is illustrated using three CSTR examples of varying nonlinear behavior.