Mixed Integer Nonlinear Programming Algorithm Research Articles

A distributed economic model predictive control-based FPPT scheme for large-scale solar farm

Increasing solar-photovoltaic-power penetration necessitates the implementation of flexible power point tracking (FPPT) in solar farms. However, coordinating multiple solar photovoltaic (PV) power generation systems (SPVPGSs) poses a significant challenge due to their inherent intermittency and varying dynamic characteristics, complicating FPPT implementation. To overcome this challenge, this paper proposes a distributed economic model predictive control (DEMPC) scheme to achieve FPPT, while simultaneously enhancing overall economic performance in solar farms. By integrating solar farm control and local control into one optimal control framework, this scheme eliminates the need for power allocation, PV voltage reference calculation, and pulse width modulation. Leveraging the SPVPGS model and soft power constraint, DEMPC controllers are designed to achieve the economic targets of solar farms, which cooperate through a communication network to realize FPPT and economic optimization. Additionally, the strong nonlinearity of SPVPGS causes a non-convex mixed integer nonlinear programming (MINLP) problem, solved by a MINLP algorithm using finite converter switching states. Simulations under step-changed irradiance and power reference, as well as urgent maintenance conditions, demonstrate that the DEMPC-based FPPT scheme significantly outperforms the traditional hierarchical model predictive control-based FPPT scheme, presenting both superior dynamic response and enhanced economic performance in FPPT implementation.

PV/Hydrogen DC microgrid control using distributed economic model predictive control

The integration of hydrogen energy into a photovoltaic-dominated microgrid is now becoming a promising approach to improve the photoconversion efficiency and enhance the operating reliability. However, the energy management and power regulation of the Photovoltaic/Hydrogen DC microgrid face challenges due to the intermittency of photovoltaic (PV) power generation and the randomness of load. In this paper, a distributed economic model predictive control (DEMPC) scheme is developed for a PV/Hydrogen DC microgrid, which integrates the energy management, economic optimization, and power regulation into one optimal control framework. Based on the developed distributed converter-based mathematical model, three local controllers are designed to achieve the economic targets of PV subsystem, alkaline electrolyzer subsystem, and proton exchange membrane fuel cell subsystem respectively, which cooperate with each other through the communication network to realize the power supply–demand balance, DC bus voltage stability, and economic optimization. A mixed integer nonlinear programming algorithm utilizing the finite converter switching states is embedded into the DEMPC to solve the non-convex local optimization problems efficiently. The effectiveness and superiority of the proposed DEMPC scheme are verified by simulations under varying irradiance and load conditions, indicating that the DEMPC can achieve a comparable overall dynamic economic performance with a significantly reduced computation burden and power oscillation, compared to the centralized economic model predictive control (CEMPC).

Mixed integer programming with kriging surrogate model technique for dispersion control of photonic crystal fibers

In this paper, mixed integer nonlinear programming (MINLP) optimization algorithm integrated with kriging surrogate-model is newly formulated to optimize the dispersion characteristics of photonic crystal fibers (PCFs). The MINLP is linked with full vectorial finite difference method (FVFDM) to optimize the modal properties of the PCFs. Through the optimization process, the design parameters can take real and/or integer values. The integer values can be used to selectively fill the PCF air holes to control its dispersion characteristics. However, the other optimization techniques deal with real design parameters where the PCF can be optimized using none or predefined infiltrated air holes. The MINLP algorithm is used to obtain an ultra-flat zero dispersion over a broadband of wavelength range from 1.25 to 1.6 μm using silica PCF selectively infiltrated with Ethanol material. To show the superiority of the proposed algorithm, nematic liquid crystal selectively infiltrated PCFs are also designed with high negative flat dispersion over wide range of wavelengths from 1.25 to 1.6 μm for the quasi transverse magnetic (TM) and the quasi transverse electric (TE) modes. Such designs have negative flat dispersions of − 163 ± 0.9 and − 170 ± 1.2 ps/Km nm, respectively over the studied wavelength range. Therefore, the MINLP algorithms could be used efficiently for the design and optimization of selectively filled photonic devices.

An economic model predictive control-based flexible power point tracking strategy for photovoltaic power generation

In a high solar-power-penetration power grid, photovoltaic (PV) power generation requires to run in a flexible power point tracking (FPPT) mode. However, traditional hierarchical control-based FPPT strategy ignores the dynamic economic performance of PV power generation. To address this issue, an advanced FPPT strategy based on economic model predictive control (EMPC) is proposed to achieve higher dynamic economic performance. This strategy integrates the PV voltage reference calculation, PV voltage control, and pulse width modulation into one optimal control framework, utilizing the economic indices of the PV power generation system (PVPGS) as the cost function to achieve its economic optimization and power tracking. Due to the strong nonlinearity in the PVPGS, the EMPC optimization problem is non-convex, leading to a local optimum. A mixed integer nonlinear programming algorithm is developed, which utilizes a finite number of converter switching states for obtaining the global optimum. Simulations demonstrate that the EMPC-based FPPT strategy enhances the dynamic economic performance compared to the hierarchical control-based FPPT strategy.

Automated patient-robot assignment for a robotic rehabilitation gym: a simplified simulation model

BackgroundA robotic rehabilitation gym can be defined as multiple patients training with multiple robots or passive sensorized devices in a group setting. Recent work with such gyms has shown positive rehabilitation outcomes; furthermore, such gyms allow a single therapist to supervise more than one patient, increasing cost-effectiveness. To allow more effective multipatient supervision in future robotic rehabilitation gyms, we propose an automated system that could dynamically assign patients to different robots within a session in order to optimize rehabilitation outcome.MethodsAs a first step toward implementing a practical patient-robot assignment system, we present a simplified mathematical model of a robotic rehabilitation gym. Mixed-integer nonlinear programming algorithms are used to find effective assignment and training solutions for multiple evaluation scenarios involving different numbers of patients and robots (5 patients and 5 robots, 6 patients and 5 robots, 5 patients and 7 robots), different training durations (7 or 12 time steps) and different complexity levels (whether different patients have different skill acquisition curves, whether robots have exit times associated with them). In all cases, the goal is to maximize total skill gain across all patients and skills within a session.ResultsAnalyses of variance across different scenarios show that disjunctive and time-indexed optimization models significantly outperform two baseline schedules: staying on one robot throughout a session and switching robots halfway through a session. The disjunctive model results in higher skill gain than the time-indexed model in the given scenarios, and the optimization duration increases as the number of patients, robots and time steps increases. Additionally, we discuss how different model simplifications (e.g., perfectly known and predictable patient skill level) could be addressed in the future and how such software may eventually be used in practice.ConclusionsThough it involves unrealistically simple scenarios, our study shows that intelligently moving patients between different rehabilitation robots can improve overall skill acquisition in a multi-patient multi-robot environment. While robotic rehabilitation gyms are not yet commonplace in clinical practice, prototypes of them already exist, and our study presents a way to use intelligent decision support to potentially enable more efficient delivery of technologically aided rehabilitation.

Homotopy continuation enhanced branch and bound algorithms for strongly nonconvex mixed‐integer nonlinear optimization

AbstractLarge‐scale strongly nonlinear and nonconvex mixed‐integer nonlinear programming (MINLP) models frequently appear in optimization‐based process synthesis, integration, intensification, and process control. However, they are usually difficult to solve by existing algorithms within acceptable time. In this study, we propose two robust homotopy continuation enhanced branch and bound (HCBB) algorithms (denoted as HCBB‐FP and HCBB‐RB) where the homotopy continuation method is employed to gradually approach the optimum of the NLP subproblem at a node from the solution at its parent node. A variable step length is adapted to effectively balance feasibility and computational efficiency. The computational results from solving four existing process synthesis problems demonstrate that the proposed HCBB algorithms can find the same optimal solution from different initial points, while the existing MINLP algorithms fail or find much worse solutions. In addition, HCBB‐RB is superior to HCBB‐FP due to much lower computational effort required for the same locally optimal solution.

Automated design of synthetic biocircuits in the stochastic regime

In this work, we present an optimization-based design strategy for gene regulatory networks (GRNs) in the stochastic regime (i.e., in the presence of molecular noise). The approach exploits a recently developed framework for the efficient simulation of stochastic GRNs based on a Partial Integro Differential Equations (PIDE) model formulation, which is here further accelerated with a parallel implementation in GPUs to maximize the performance. The simulator is combined with a global Mixed Integer Nonlinear Programming algorithm to efficiently address the optimization of the design through topology and parameter spaces simultaneously. We illustrate the performance of the proposed methodology through two different case studies: a biocircuit with a pre-defined target dynamics, and a biocircuit with a stationary bi-modal distribution fulfilling a number of requirements (in terms of distance and ratios of probabilities between modes).

Polyhedral approximation strategies for nonconvex mixed-integer nonlinear programming in SHOT

Different versions of polyhedral outer approximation are used by many algorithms for mixed-integer nonlinear programming (MINLP). While it has been demonstrated that such methods work well for convex MINLP, extending them to solve nonconvex problems has traditionally been challenging. The Supporting Hyperplane Optimization Toolkit (SHOT) is a solver based on polyhedral approximations of the nonlinear feasible set of MINLP problems. SHOT is an open source COIN-OR project, and is currently one of the most efficient global solvers for convex MINLP. In this paper, we discuss some extensions to SHOT that significantly extend its applicability to nonconvex problems. The functionality include utilizing convexity detection for selecting the nonlinearities to linearize, lifting reformulations for special classes of functions, feasibility relaxations for infeasible subproblems and adding objective cuts to force the search for better feasible solutions. This functionality is not unique to SHOT, but can be implemented in other similar methods as well. In addition to discussing the new nonconvex functionality of SHOT, an extensive benchmark of deterministic solvers for nonconvex MINLP is performed that provides a snapshot of the current state of nonconvex MINLP.

Two linear approximation algorithms for convex mixed integer nonlinear programming

We present two new algorithms for convex Mixed Integer Nonlinear Programming (MINLP), both based on the well known Extended Cutting Plane (ECP) algorithm proposed by Weterlund and Petersson. Our first algorithm, Refined Extended Cutting Plane (RECP), incorporates additional cuts to the MILP relaxation of the original problem, obtained by solving linear relaxations of NLP problems considered in the Outer Approximation algorithm. Our second algorithm, Linear Programming based Branch-and-Bound (LP-BB), applies the strategy of generating cuts that is used in RECP, to the linear approximation scheme used by the LP/NLP based Branch-and-Bound algorithm. Our computational results show that RECP and LP-BB are highly competitive with the most popular MINLP algorithms from the literature, while keeping the nice and desirable characteristic of ECP, of being a first-order method.

LORM: Learning to Optimize for Resource Management in Wireless Networks With Few Training Samples

Effective resource management plays a pivotal role in wireless networks, which, unfortunately, typically results in challenging mixed-integer nonlinear programming (MINLP) problems. Machine learning-based methods have recently emerged as a disruptive way to obtain near-optimal performance for MINLPs with affordable computational complexity. There have been some attempts in applying such methods to resource management in wireless networks, but these attempts require huge amounts of training samples and lack the capability to handle constrained problems. Furthermore, they suffer from severe performance deterioration when the network parameters change, which commonly happens and is referred to as the task mismatch problem. In this paper, to reduce the sample complexity and address the feasibility issue, we propose a framework of Learning to Optimize for Resource Management (LORM). In contrast to the end-to-end learning approach adopted in previous studies, LORM learns the optimal pruning policy in the branch-and-bound algorithm for MINLPs via a sample-efficient method, namely, imitation learning . To further address the task mismatch problem, we develop a transfer learning method via self-imitation in LORM, named LORM-TL , which can quickly adapt a pre-trained machine learning model to the new task with only a few additional unlabeled training samples. Numerical simulations demonstrate that LORM outperforms specialized state-of-the-art algorithms and achieves near-optimal performance, while providing significant speedup compared with the branch-and-bound algorithm. Moreover, LORM-TL, by relying on a few unlabeled samples, achieves comparable performance with the model trained from scratch with sufficient labeled samples.

Mixed-integer bilinear and piecewise-linear models for designing switching strategies of multilevel power converters

With the adoption of multilevel converters in the industry, the importance of their optimal operation has increased. Able to operate with more than three voltage levels, multilevel converters are often modulated by vectors defined in time, which can be optimized to reduce the number of commutations or total harmonic distortion. This papers presents two formulations that do not use trigonometric functions for selecting optimal switching angles, focusing on the minimization of the current weighted total harmonic distortion. The first relies on bilinear terms to yield a mixed-integer nonlinear programming (MINLP) formulation that can be tackled with MINLP algorithms. The second uses piecewise-linear approximation to enable the application of mixed-integer quadratic programming. Computational results from the application of these formulations are reported, in conjunction with a simulation of the optimal angles that were computed for a 5-level Neutral-Point Clamped converter.

A new box selection criterion in interval Bernstein global optimization algorithm for MINLPs

Mixed-integer nonlinear programming (MINLP) problems are known to be challenging due to the involvement of nonlinear mathematical relations and combinatorial complexities. In this paper, an interval form-based Bernstein global optimization algorithm is presented to solve polynomial MINLP problems. The major contribution of this paper is a new box selection criterion for this interval form-based Bernstein global optimization algorithm. This new box selection criterion promotes fast convergence of the Bernstein algorithm. Moreover, the new box selection criterion allows the construction of a hybrid branch-and-bound framework for the Bernstein algorithm, wherein a local search method is used to obtain a good upper bound on the global minimum, and the Bernstein form is employed to obtain a valid lower bound on the global minimum. We show with numerical results that it is possible to obtain a significant reduction in the number of subdivisions required with this new box selection criterion. Furthermore, the Bernstein algorithm with this new box selection criterion is evaluated numerically on a variety of small- to medium-size MINLP problem instances and its performance is compared with the previously reported Bernstein global optimization algorithm as well as with a few state-of-the-art MINLP solvers. The results of the numerical studies demonstrate satisfactory performance in terms of the chosen performance metrics. Finally, the trim-loss minimization problem from the paper industry is solved to demonstrate the practical applicability of the Bernstein algorithm.

A center-cut algorithm for quickly obtaining feasible solutions and solving convex MINLP problems

Here we present a center-cut algorithm for convex mixed-integer nonlinear programming (MINLP) that can either be used as a primal heuristic or as a deterministic solution technique. Like several other algorithms for convex MINLP, the center-cut algorithm constructs a linear approximation of the original problem. The main idea of the algorithm is to use the linear approximation differently in order to find feasible solutions within only a few iterations. The algorithm chooses trial solutions as the center of the current linear outer approximation of the nonlinear constraints, making the trial solutions more likely to satisfy the constraints. The ability to find feasible solutions within only a few iterations makes the algorithm well suited as a primal heuristic, and we prove that the algorithm finds the optimal solution within a finite number of iterations. Numerical results show that the algorithm obtains feasible solutions quickly and is able to obtain good solutions.

A new approach to developing a conceptual topside process design for an offshore platform

This study introduces a new approach for the conceptual design of an offshore topside process, satisfying environmental standards, saving utility consumption, and consequently, maximizing economic profit. Twelve individual processes are modeled as a case study, based on sets of combinations between four topside process configurations and three individual production scenarios (i.e., peak oil, peak gas, and peak water) over the life cycle of an oil reservoir. Then, the simulation results of these models are analyzed based on economic profit. In particular, the simulation program is integrated with a mixed-integer non-linear programming algorithm to optimize the design and operating variables (e.g., operating pressures of the multi-stage separators) in order to maximize the economic profit of the platform. Lastly, an economic feasibility study is performed for the design of a profitable and eco-friendly offshore platform.

MIDACO Parallelization Scalability on 200 MINLP Benchmarks

Abstract This contribution presents a numerical evaluation of the impact of parallelization on the performance of an evolutionary algorithm for mixed-integer nonlinear programming (MINLP). On a set of 200 MINLP benchmarks the performance of the MIDACO solver is assessed with gradually increasing parallelization factor from one to three hundred. The results demonstrate that the efficiency of the algorithm can be significantly improved by parallelized function evaluation. Furthermore, the results indicate that the scale-up behaviour on the efficiency resembles a linear nature, which implies that this approach will even be promising for very large parallelization factors. The presented research is especially relevant to CPU-time consuming real-world applications, where only a low number of serial processed function evaluation can be calculated in reasonable time.

Global Optimization Algorithm for Large-Scale Refinery Planning Models with Bilinear Terms

We propose a global optimization algorithm for mixed-integer nonlinear programming (MINLP) problems arising from oil refinery planning. It relies on tight mixed-integer linear programming (MILP) relaxations that discretize the bilinear terms dynamically using either piecewise McCormick (PMCR) or normalized multiparametric disaggregation (NMDT). Tight relaxations help finding a feasible solution of the original problem via a local nonlinear solver, with the novelty being the generation of multiple starting points from CPLEX’s solution pool and the parallel execution. We show that optimality-based bound tightening (OBBT) is essential for large-scale problems, even though it is computationally expensive. To reduce execution times, OBBT is implemented in parallel. The results for a refinery case study, featuring units with alternative operating modes, intermediate storage tanks, and single- and multiple-period supply and demand scenarios, show that the algorithm’s performance is comparable to commercial solve...

Two-stage MINLP algorithm for the optimal synthesis and design of networks of CHP units

This paper proposes a Mixed Integer Non Linear Programming (MINLP) model and two-stage optimization algorithm for determining the most profitable synthesis and design of Combined Heat and Power units within a district heating network with heat storage while taking into account the optimal scheduling of the units over the year. A two-stage algorithm for tackling the challenging MINLP problem is devised: at the upper level the selection and sizing of the units is optimized by means of specifically selected evolutionary algorithms, while at the lower level the operational scheduling problem is linearized and optimized with a commercial Mixed Integer Linear Programing solver. Three different approaches, based on two different evolutionary algorithms and discrete variable relaxation, are devised and compared to tackle the upper level problem. Moreover a bounding technique is proposed to limit the computational time required to solve the lower-level problem. The overall algorithm is tested on an industrial scale problem to find the two system designs leading to the minimum energy consumption and the minimum total annual cost. Computational results indicate that the continuous relaxation of the plant sizes significantly helps to improve the convergence rate of the tested evolutionary algorithm and to find improved solutions. For the considered test case, the design optimized for the minimum energy consumption allows to save 64% of primary energy compared to the minimum total annual cost solution, but with a 28% higher total annual cost.

An adaptive discretization MINLP algorithm for optimal synthesis of decentralized energy supply systems

Abstract Decentralized energy supply systems (DESS) are highly integrated and complex systems designed to meet time-varying energy demands, e.g., heating, cooling, and electricity. The synthesis problem of DESS addresses combining various types of energy conversion units, choosing their sizing and operations to maximize an objective function, e.g., the net present value. In practice, investment costs and part-load performances are nonlinear. Thus, this optimization problem can be modeled as a nonconvex mixed-integer nonlinear programming (MINLP) problem. We present an adaptive discretization algorithm to solve such synthesis problems containing an iterative interaction between mixed-integer linear programs (MIPs) and nonlinear programs (NLPs). The proposed algorithm outperforms state-of-the-art MINLP solvers as well as linearization approaches with regard to solution quality and computation times on a test set obtained from real industrial data, which we made available online.

A hybrid stochastic–deterministic optimization approach for integrated solvent and process design

The best solution to computer-aided solvent and process design problems can be only achieved by the simultaneous optimization of solvent molecules and process operating conditions. In this contribution, a hybrid stochastic-deterministic optimization approach is proposed for integrated solvent and process design. It is a combination of a genetic algorithm (GA) that optimizes the discrete molecular variables and a gradient-based deterministic algorithm that solves the continuous nonlinear optimization problem of the process at fixed molecular variables as proposed by the GA. The method is demonstrated on a coupled absorption-desorption process where solvent molecular structures as well as the operating conditions of the absorption and desorption columns are optimized simultaneously. While deterministic mixed-integer nonlinear programming (MINLP) algorithms rely on well-selected initial estimates, the proposed hybrid approach can reliably and steadily solve the problem under random initializations. The combination of the advantages of stochastic and deterministic algorithms makes the approach a promising alternative to conventional MINLP algorithms for solving integrated solvent and process design problems.

Fair profit allocation in supply chain optimization with transfer price and revenue sharing: MINLP model and algorithm for cellulosic biofuel supply chains

A mixed‐integer nonlinear programming (MINLP) formulation to simultaneously optimize operational decisions as well as profit allocation mechanisms in supply chain optimization, namely material transfer prices and revenue share policies among the supply chain participants is proposed. The case of cellulosic bioethanol supply chains is specifically considered and the game‐theory Nash bargaining solution approach is employed to achieve fair allocation of profit among the collection facilities, biorefineries, and distribution centers. The structural advantages of certain supply chain participants can be taken into account by specifying different values of the negotiation‐power indicators in the generalized Nash‐type objective function. A solution strategy based on a logarithm transformation and a branch‐and‐refine algorithm for efficient global optimization of the resulting nonconvex MINLP problem is proposed. To demonstrate the application of the proposed framework, an illustrative example and a state‐wide county‐level case study on the optimization of a potential cellulosic bioethanol supply chain in Illinois are presented. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3211–3229, 2014