Mixed-integer Nonlinear Programming Solvers Research Articles

Drawdown minimization in asset portfolio selection: MINLP models and efficient cross‐entropy algorithm

AbstractPortfolio management is an important research topic in finance and optimization. Drawdown as one of the measures in evaluating portfolios indicates the relative difference between the portfolio value in the current moment and its maximum value during a given time interval in the recent past. In this paper, first, the importance of this measure is discussed and then two mixed‐integer nonlinear programming (MINLP) models with the objectives of minimizing the expected drawdown and the maximum drawdown under real‐world constraints are presented. Due to the NP‐hardness of this problem, by utilizing the problem structure, an efficient cross‐entropy‐based algorithm is presented to solve it. An effective mechanism is suggested to calibrate the algorithm parameters. Computational results confirm the performance of the proposed algorithm from both solution quality and running time in comparison with MINLP solvers.

Optimizing the design and operation of water networks: Two decomposition approaches

We consider the design and operation of water networks simultaneously. Water network problems can be divided into two categories: the design problem and the operation problem. The design problem involves determining the appropriate pipe sizing and placements of pump stations, while the operation problem involves scheduling pump stations over multiple time periods to account for changes in supply and demand. Our focus is on networks that involve water co-produced with oil and gas. While solving the optimization formulation for such networks, we found that obtaining a primal (feasible) solution is more challenging than obtaining dual bounds using off-the-shelf mixed-integer nonlinear programming solvers. Therefore, we propose two methods to obtain good primal solutions. One method involves a decomposition framework that utilizes a convex reformulation, while the other is based on time decomposition. To test our proposed methods, we conduct computational experiments on a network derived from the PARETO case study.

An optimal flow rate allocation model of the oilfield treated oil pipeline network

Oilfield treated oil pipeline network is the link connecting the upstream oilfields and the downstream refineries. Due to the differences in operating costs and transportation fee between different pipelines and the fluctuation in the demand and sales prices of the treated oil, there is an optimal flow allocation plan for the pipeline network to make the oilfield company obtain the highest social and economic benefit. In this study, a mixed integer nonlinear programming (MINLP) model is developed to determine the optimal flow rate allocation plan of the large-scale and complex treated oil pipeline network, and both the social and economic benefits are considered simultaneously. The optimization objective is the multi-objective which includes the largest user satisfaction and the highest economic benefit. The model constraints include the oilfield production capacity, refinery demand, pipeline transmission capacity, flow, pressure, and temperature of the node and station, and the pipeline hydraulic and thermal calculations. Python 3.7 is utilized for the programming of the off-line calculation procedure and the MINLP model, and GUROBI 9.0.2 is served as the MINLP solver. Moreover, the model is applied to a real treated oil pipeline network located in China, and three optimization scenarios are analyzed. For social benefit, the values of the user satisfaction of each refinery and the total network are 1 before and after optimization for scenarios 1, 2, and 3. For economic benefit, the annual revenue can be increased by 0.227, 0.293, and 0.548 billion yuan after the optimization in scenario 1, 2, and 3, respectively.

Discrete optimal designs for distributed energy systems with nonconvex multiphase optimal power flow

The optimal selection, sizing, and location of small-scale technologies within a grid-connected distributed energy system (DES) can contribute to reducing carbon emissions, consumer costs, and network imbalances. There is a significant lack of studies on how DES designs, especially those with electrified heating systems, impact unbalanced low-voltage distribution networks to which most DES are connected. This is the first study to present an optimisation framework for obtaining discrete technology sizing and selection for grid-connected DES design, while simultaneously considering multiphase optimal power flow (MOPF) constraints to accurately represent unbalanced low-voltage distribution networks. An algorithm is developed to solve the resulting Mixed-Integer Nonlinear Programming (MINLP) formulation. It employs a decomposition based on Mixed-Integer Linear Programming (MILP) and Nonlinear Programming (NLP) and uses integer cuts and complementarity reformulations to obtain discrete designs that are also feasible with respect to the network constraints. A heuristic modification to the original algorithm is also proposed to improve computational speed. Improved formulations for selecting feasible combinations of air source heat pumps (ASHPs) and hot water storage tanks are also presented. Two networks of varying size are used to test the optimisation methods. Designs with electrified heating (ASHPs and tanks) are compared to those with conventional gas boilers. The algorithms outperform one of the existing state-of-the-art commercial deterministic MINLP solvers, which fails to find any solutions in two instances within specified time limits. While feasible solutions were obtained for all cases, convergence was not achieved for all, especially for those involving the larger network. Where converged, the algorithm with the heuristic modification has achieved results up to 70% faster than the original algorithm. Results for case studies suggest that including ASHPs can support up to 16% higher renewable generation capacity compared to gas boilers, albeit with higher ASHP investment costs, as local generation and consumption minimises network violations associated with excess power export. The results also show the importance of including nonlinear power flow constraints in DES design problems. The optimisation framework and results can be used to inform stakeholders such as policymakers and network operators, to increase renewable energy capacity and aid the decarbonisation of domestic heating systems.

Three Solution Approaches to Stochastic Multi-Period AC Optimal Power Flow in Active Distribution Systems

This paper proposes three novel solution approaches (A1, A2, A3) to solve stochastic multi-period AC optimal power flow (S-MP-OPF) for day-ahead flexibility procurement from distributed energy resources (DER) in active distribution systems (ADSs). The S-MP-OPF is a mixed-integer nonlinear programming (MINLP) problem due to the binary variables modeling the operation of storage devices and flexible loads. The proposed three approaches have a shared common first step, which resorts to a new mixed-integer linear programming (MILP) model approximation of the S-MP-OPF problem. The MILP model employs second-order Taylor series expansion of trigonometric terms and formulates the linear approximations relying on variables such as square of voltage magnitude and voltage angle difference. This first step serves also the purpose of fixing the binary variables to the values computed by the MILP problem. Then, the approach A1 only checks the AC feasibility of MILP solution while approaches A2 and A3 further optimize continuous variables. Specifically, the sophisticated heuristic approach A2 employs sequential linear programming and AC power flow while the approach A3 models and solves directly the remaining nonlinear programming (NLP) problem. The performances of these approaches, a benchmark MINLP solver, and a state-of-the-art method are thoroughly compared in three radial or weakly meshed ADSs of 34, 31, and 191 nodes, respectively. The numerical results indicate that, albeit the approach A2 performs best overall, these approaches present distinct accuracy vs speed trade-offs, which make them suitable to problems of different sizes and diverse accuracy or speed requirements.

COMPARISON BETWEEN FIVE MINLP SOLVERS AND NEW RESULTS RELATED TO TRIGONOMETRIC FUNCTIONS

Due to the rapid development of algorithms and techniques that are used to deal with mixed integer nonlinear programming (MINLP) problems, many global MINLP solvers were introduced. In this paper, computational experiments were done to compare between the performances of five of these solvers. Some of these solvers do not support trigonometric functions. Therefore, piecewise linear approximation (PLA) is applied to problems having these function so the solvers can deal with these problems. Additional computational tests were performed on to show how PLA can be useful, even to some powerful global solvers.

Mixed-Integer Nonlinear Programming for Energy-Efficient Container Handling: Formulation and Customized Genetic Algorithm

Energy consumption is expected to be reduced while maintaining high productivity for container handling. This paper investigates a new energy-efficient scheduling problem of automated container terminals, in which quay cranes (QCs) and lift automated guided vehicles (AGVs) cooperate to handle inbound and outbound containers. In our scheduling problem, operation times and task sequences are both to be determined. The underlying optimization problem is mixed-integer nonlinear programming (MINLP). To deal with its computational intractability, a customized and efficient genetic algorithm (GA) is developed to solve the studied MINLP problem, and lexicographic and weighted-sum strategies are further considered. An <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -constraint algorithm is also developed to analyze the Pareto frontiers. Comprehensive experiments are tested on a container handling benchmark system, and the results show the effectiveness of the proposed lexicographic GA, compared to results obtained with two commonly-used metaheuristics, a commercial MINLP solver, and two state-of-the-art methods.

BONMIN solver-based coordination of distributed FACTS compensators and distributed generation units in modern distribution networks

The problems of increasing electricity consumption and atmospheric pollutants emissions are targeted as critical challenges for power transfer in modern distribution networks. In order to tackle this issue, many researchers have proposed the incorporation of Distributed Generator (DG) units, as well as different types of reactive power compensation devices, which will contribute to the increase of networks efficiency through loss minimization, voltage profile improvement, or even increasing transmission line’s capacity. This paper addresses the optimal allocation/coordination of DG, Distributed Static VAR Compensator (D-SVC), and Distributed Thyristor-Controlled Series Compensator (D-TCSC) to reduce total power loss in the distribution network. The allocation problem is formulated as a mixed-integer nonlinear programming (MINLP) and is solved using a basic open-source MINLP (BONMIN) solver embedded in GAMS. The proposed method was tested in IEEE 16-bus and 33-bus distribution test systems, where two cases were considered. Firstly, BONMIN solver is used for allocation of D-SVC, and D-TCSC, respectively, where the results showed that incorporation of these devices leads to reduction of the active power losses, while the minimal losses and highest efficiency increase for both test networks occurred in the case where 3 D-TCSCs allocated. Secondly, a coordinated allocation of DGs and D-SVCs, as well as DGs and D-TCSCs, is presented, where it was shown that coordination of these devices leads to even greater loss reduction, i.e. increased efficiency. Also, it was concluded that DG and D-TCSC coordination is better suited for loss reduction in the 16-bus system, while DG and D-SVC coordination leads to greater loss reduction for the 33-bus system. Furthermore, the comparison between BONMIN and metaheuristic methods was performed for DG allocation and DG and D-TCSC coordination in IEEE 33-bus system, where it was concluded that BONMIN performs better in terms of minimal power loss, compared to the results which were presented in the available literature. Finally, the BONMIN solver’s performance is evaluated using three parameters: number of nodes, number of iterations, and total execution time, where the numerical analysis showed that BONMIN is efficient in solving the addressed problems.

Fast Energy-Optimal Multikernel DNN-Like Application Allocation on Multi-FPGA Platforms

Platforms with multiple Field Programmable Gate Arrays (FPGAs), such as Amazon Web Services (AWS) F1 instances, can efficiently accelerate multi-kernel pipelined applications, e.g., Convolutional Neural Networks for machine vision tasks or transformer networks for Natural Language Processing tasks. To reduce energy consumption when the FPGAs are underutilized, we propose a model to (1) find off-line the minimum-power solution for given throughput constraints, and (2) dynamically reprogram the FPGA at runtime (which is complementary to dynamic voltage and frequency scaling) to match best the workloads when they change. The off-line optimization model can be solved using a Mixed-Integer Non-Linear Programming (MINLP) solver, but it can be very slow. Hence, we provide two heuristic optimization methods that improve result quality within a bounded time. We use several very large designs to demonstrate that both heuristics obtain comparable results to MINLP, when it can find the best solution, and they obtain much better results than MINLP, when it cannot find the optimum within a bounded amount of time. The heuristic methods can also be thousands of times faster than the MINLP solver.

Subsea field layout optimization (part II)–the location-allocation problem of manifolds

The location-allocation problem of manifolds, which is a part of subsea field layout optimization, directly affects the flowline cost. This problem has always been studied as a mixed-integer nonlinear programming (MINLP) problem, or an integer linear programming (ILP) problem when there are location options for the facilities. Making a MINLP model is surely convenient to interpret the optimization problem. However, finding the global optimum of the MINLP model is very hard. Hence, practically, engineers use approximation algorithms to search a good local optimum or give several good location options based on their experience and knowledge to reduce the MINLP model into an ILP model. Nevertheless, the global optimum of the original MINLP model is no longer guaranteed. In this study, enlightened by the graphic theories, we propose a new method in which we reduce the MINLP model into an ILP model---more precisely, a binary linear programming (BLP) model---without compromise of achieving global optimum, but also with extremely high efficiency. The breakthrough in both efficiency and accuracy of our method for the location-allocation problem of manifolds and wellheads is well demonstrated in various cases with comparison to the published methods and the commercial MINLP solver from LINDO. Besides, we also provide our results for larger-scale problems which were considered infeasible for the commercial MINLP solver. More generally, our method can be regarded as a specific MINLP/NIP (nonlinear integer programming) solver which can be used for many other applications. This work is the second of a series of papers which systematically introduce an efficient method for subsea field layout optimization to minimize the development cost.

Solving Hydropower Unit Commitment Problem Using a Novel Sequential Mixed Integer Linear Programming Approach

Hydro Unit Commitment (HUC) is an important problem of power systems and when it is dealt with via a mathematical programming approach and optimization, it leads to the complicated class of mixed-integer nonlinear programming (MINLP). Many attempts have been made to solve the problem efficiently, while there is still ongoing research to come up with better solution schemes in terms of runtime and optimality. Highly nonlinear nature of the relationships and constraints in the optimization problem have forced the researchers to deal with the HUC problem in simplified manners which may result in impractical and unreliable solutions, i.e. schedules. Here in this paper we proposed a new method based on sequential mixed-integer linear programming (MILP) for solving a more realistic version of the HUC problem efficiently. We applied the proposed method to a cascade of two hydropower plants, Karun-3 and Karun-4, located in the Southwest of Iran. The sequential MILP approach was compared with several MINLP solvers of the GAMS optimization package. The results indicated that the proposed methodology outperformed the MINLP solvers in terms of efficiency, with solution time of less than 30 s, compared to 10 min that were given to the solvers, and in terms of optimality with more than 20 thousand cubic meters per day in water release. Additionally, we have explored the effect of penalizing the total number of startups on the total release, convergence of the algorithm, and the computation time. In all of the cases the total number of startups was reduced more than three times.

Team Orienteering with Time-Varying Profit

This paper studies the team orienteering problem, where the arrival time and service time affect the collection of profits. Such interactions result in a nonconcave profit function. This problem integrates the aspect of time scheduling into the routing decision, which can be applied in humanitarian search and rescue operations where the survival rate declines rapidly. Rescue teams are needed to help trapped people in multiple affected sites, whereas the number of people who could be saved depends as well on how long a rescue team spends at each site. Efficient allocation and scheduling of rescue teams is critical to ensure a high survival rate. To solve the problem, we formulate a mixed-integer nonconcave programming model and propose a Benders branch-and-cut algorithm, along with valid inequalities for tightening the upper bound. To solve it more effectively, we introduce a hybrid heuristic that integrates a modified coordinate search (MCS) into an iterated local search. Computational results show that valid inequalities significantly reduce the optimality gap, and the proposed exact method is capable of solving instances where the mixed-integer nonlinear programming solver SCIP fails in finding an optimal solution. In addition, the proposed MCS algorithm is highly efficient compared with other benchmark approaches, whereas the hybrid heuristic is proven to be effective in finding high-quality solutions within short computing times. We also demonstrate the performance of the heuristic with the MCS using instances with up to 100 customers. Summary of Contribution: Motivated by search and rescue (SAR) operations, we consider a generalization of the well-known team orienteering problem (TOP) to incorporate a nonlinear time-varying profit function in conjunction with routing and scheduling decisions. This paper expands the envelope of operations research and computing in several ways. To address the scalability issue of this highly complex combinatorial problem in an exact manner, we propose a Benders branch-and-cut (BBC) algorithm, which allows us to efficiently deal with the nonconcave component. This BBC algorithm is computationally enhanced through valid inequalities used to strengthen the bounds of the BBC. In addition, we propose a highly efficient hybrid heuristic that integrates a modified coordinate search into an iterated local search. It can quickly produce high-quality solutions to this complex problem. The performance of our solution algorithms is demonstrated through a series of computational experiments.

A Mixed-Integer Conic Formulation for Optimal Placement and Dimensioning of DGs in DC Distribution Networks

The problem of the optimal placement and dimensioning of constant power sources (i.e., distributed generators) in electrical direct current (DC) distribution networks has been addressed in this research from the point of view of convex optimization. The original mixed-integer nonlinear programming (MINLP) model has been transformed into a mixed-integer conic equivalent via second-order cone programming, which produces a MI-SOCP approximation. The main advantage of the proposed MI-SOCP model is the possibility of ensuring global optimum finding using a combination of the branch and bound method to address the integer part of the problem (i.e., the location of the power sources) and the interior-point method to solve the dimensioning problem. Numerical results in the 21- and 69-node test feeders demonstrated its efficiency and robustness compared to an exact MINLP method available in GAMS: in the case of the 69-node test feeders, the exact MINLP solvers are stuck in local optimal solutions, while the proposed MI-SOCP model enables the finding of the global optimal solution. Additional simulations with daily load curves and photovoltaic sources confirmed the effectiveness of the proposed MI-SOCP methodology in locating and sizing distributed generators in DC grids; it also had low processing times since the location of three photovoltaic sources only requires 233.16s, which is 3.7 times faster than the time required by the SOCP model in the absence of power sources.

Practical approximations and heuristic approaches for managing shiftable loads in the multi-period optimal power flow framework

The presence of smart buildings (SBs) is expected to grow manifold in the next decade, leading to both challenges and opportunities in managing increasingly “active” distribution systems. SBs are host to a number of devices and technologies capable of providing residential flexibility in order to maintain acceptable operating conditions within the distribution system. A prevalent class of devices with significant potential for flexibility provision are shiftable loads (SLs), which represent a large amount of potential SB-provided flexibility. However, the accurate modelling of SLs within the multi-period optimal power flow (MP-OPF) framework results in complex non-convex mixed-integer nonlinear programming (MINLP) problems. For realistic feeders, MINLP problems are intractable and hence require some heuristic form of circumvention. This work evaluates the limitations of the MINLP formulation and proposes new practical alternatives, namely a flexible NLP approximation and a multi-faceted heuristic algorithm. The proposed approaches are generic and applicable for any problem size or type of SL, while they are shown to work efficiently, both speed-wise and solution-quality-wise, outperforming state-of-the-art MINLP solvers. A simple scoring and ranking scheme is also proposed in order to make a direct comparison between the different approaches.

CNN-on-AWS: Efficient Allocation of Multikernel Applications on Multi-FPGA Platforms

Multi-FPGA platforms, like Amazon AWS F1, can run in the cloud multikernel pipelined applications, like convolutional neural networks (CNNs), with excellent performance and lower energy consumption than CPUs or GPUs. We propose a method to efficiently map these applications on multi-FPGA platforms to maximize the application throughput. Our methodology finds, for the given resources, the optimal number of parallel instances of each kernel in the pipeline and their allocation to one or more among the available FPGAs. We obtain this by formulating and solving a mixed-integer, nonlinear optimization problem, in which we model the performance of each component and the duration of the phases in which the accelerated computation can be split into, namely: 1) data transfer from a host CPU to the DDR memory of each FPGA; 2) data transfer from FPGA DDR to FPGA on-chip memory; 3) kernel computation on the FPGA; 4) data transfer from FPGA on-chip memory to FPGA DDR; and 5) data transfer from FPGA DDR to host. Finding the optimal solution using a mixed-integer nonlinear programming (MINLP) solver is often highly inefficient. Hence, we provide a fast heuristic method that according to our experiments can be much more efficient than the MINLP solver and finds comparable results. For larger problems (more CNN layers), our heuristic method can quickly find (several thousand times faster) much better solutions than the MINLP solver, even if we run the latter for a very long time.

Design optimization of shell and tube heat exchangers sizing with heat transfer enhancement

Heat transfer enhancement (HTE) is an efficient technology to improve the performance of shell and tube heat exchangers (STHEs). While there exist many commercial software packages, heat exchanger design still highly depends on users’ experiences, resulting in long computing time and inferior solutions. Therefore, a systematic optimization methodology is required in order to achieve an efficient and accurate design solution. This paper presents a generalized disjunctive programming (GDP) model for the optimization of STHE design. It is formulated as a mixed integer non-linear programming (MINLP) problem, involving selection for 12 technology combinations (four tube-side techniques, three shell-side techniques), and all discrete decisions for each selection are modeled by disjunctions. The model is then applied to minimize the total capital cost. Both global optimization solver BARON, and general MINLP solver DICOPT are tested. The results show that the developed method provides a better design solution compared with conventional STHE design procedures.

Integration of crude-oil scheduling and refinery planning by Lagrangean Decomposition

In this work, a Mixed-Integer Nonlinear Programming (MINLP) modeling framework for integrating short-term Crude-oil Scheduling (CS) and mid-term Refinery Planning (RP) has been developed and effectively solved by a proposed Lagrangean Decomposition (LD) algorithm. The principles of this integration are based on the fact that both Crude-oil Scheduling and Refinery Planning have their economic net values as their objectives, and that they are physically linked by Crude Distillation Units (CDUs). A multi-scale approach is proposed in the framework to aggregate continuous- and discrete-time formulations in CS and RP, respectively. Compared to hierarchically solving the non-integrated CS and RP, computational results show significant improvement regarding the economic objective values. Moreover, the proposed LD approach requires less CPU time converging to a small (1%-5%) optimality gap when compared to the monolithic approach using state-of-the-art MINLP solvers.

The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming

This paper presents a new two-phase method for solving convex mixed-integer nonlinear programming (MINLP) problems, called Decomposition-based Outer Approximation Algorithm (DECOA). In the first phase, a sequence of linear integer relaxed sub-problems (LP phase) is solved in order to rapidly generate a good linear relaxation of the original MINLP problem. In the second phase, the algorithm solves a sequence of mixed integer linear programming sub-problems (MIP phase). In both phases the outer approximation is improved iteratively by adding new supporting hyperplanes by solving many easier sub-problems in parallel. DECOA is implemented as a part of Decogo (Decomposition-based Global Optimizer), a parallel decomposition-based MINLP solver implemented in Python and Pyomo. Preliminary numerical results based on 70 convex MINLP instances up to 2700 variables show that due to the generated cuts in the LP phase, on average only 2–3 MIP problems have to be solved in the MIP phase.

Approximation Properties and Tight Bounds for Constrained Mixed-Integer Optimal Control

We extend recent work on solving mixed-integer nonlinear optimal control problems (MIOCPs) to the case of integer control functions subject to constraints that involve a pointwise coupling of the state with the integer controls. We extend a theorem due to [S. Sager, H. Bock, and M. Diehl, Math. Program. Ser. A, 133 (2012), pp. 1--23] to the case of MIOCPs with constraints on the integer control and show that the integrality gap vanishes in function space when the coarseness of the rounding grid is driven to zero even after adding constraints of this type. For the time-discretized problem, we extend a sum-up rounding (SUR) scheme due to [S. Sager, C. Reinelt, and H. Bock, Math. Program. Ser. A, 118 (2009), pp. 109--149] to the new problem class. Our scheme permits one to constructively obtain an $\varepsilon$-feasible and $\varepsilon$-optimal binary feasible control. We derive new, tight upper bounds on the integer control approximation error made by SUR. For unconstrained binary controls on equidistant grids, we reduce the approximation error bound from $\mathcal{O}(|\Omega|)$ to $\mathcal{O}(\log |\Omega|)$ asymptotically for $|\Omega| \to \infty$ and a fixed coarseness of the rounding grid, where $|\Omega|$ is the number of binary controls. For constrained binary controls, we show that the approximation problem is more difficult, and we give a proof of an approximation error bound of complexity $\mathcal{O}(|\Omega|)$. A numerical example compares our approach to a state-of-the-art mixed-integer nonlinear programming solver and illustrates the applicability of our results when solving MIOCPs using the direct and simultaneous approach.

Using Two-Dimensional Projections for Stronger Separation and Propagation of Bilinear Terms

One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well-known McCormick relaxation for a product of two variables $x$ and $y$ over a box-constrained domai...