Nonlinear Optimization Formulation Research Articles

Minimizing Urban Flooding by Optimal Design of Drainage System Using Sequential Least Squares Quadratic Programming and Spatial Datasets

Urban flooding is caused due to poor drainage design and excessive rain. It severely affects the road infrastructure. Existing hydrologic software tools to examine the extent of urban flooding primarily require walking through a series of manual steps and address each study area individually, preventing a collective review of poor storm-drains in an efficient manner. Previous methods for optimal drainage design were inefficient and lacked the ability of solving the underlying optimization problem due to the inherent nonlinearity of the decision variables. In this paper, we develop a nonlinear optimization formulation to minimize urban flooding using underground pipe size as a decision variable. We propose a solution algorithm using sequential least squares quadratic programming and spatial datasets. The proposed method eliminates the need to examine each study area manually using existing hydrologic tools. An example using the storm-drain system for the Baltimore County is performed. The results show that the model is effective in identifying storm-drain deficiencies and correcting them by choosing appropriate storm-drain inlet types to minimize flooding. Future works may include using large datasets and a more sophisticated modeling approach for estimating rainfall intensity based on extreme weather patterns. The method can be applied to other jurisdictions if relevant hydrological and underdrain piping network data were available.

Optimizing Sensor Placement for Temperature Mapping during Ablation Procedures.

Accurately mapping the temperature during ablation is crucial for improving clinical outcomes. While various sensor configurations have been suggested in the literature, depending on the sensors' type, number, and size, a comprehensive understanding of optimizing these parameters for precise temperature reconstruction is still lacking. This study addresses this gap by introducing a tool based on a theoretical model to optimize the placement of fiber Bragg grating sensors (FBG) within the organ undergoing ablation. The theoretical model serves as a general framework, allowing for adaptation to various situations. In practical application, the model provides a foundational structure, with the flexibility to tailor specific optimal solutions by adjusting problem-specific data. We propose a nonlinear and nonconvex (and, thus, only solvable in an approximated manner) optimization formulation to determine the optimal distribution and three-dimensional placement of FBG arrays. The optimization aims to find a trade-off among two objectives: maximizing the variance of the expected temperatures measured by the sensors, which can be obtained from a predictive simulation that considers both the type of applicator used and the specific organ involved, and maximizing the squared sum of the distances between the sensor pairs. The proposed approach provides a trade-off between collecting diverse temperatures and not having all the sensors concentrated in a single area. We address the optimization problem through the utilization of approximation schemes in programming. We then substantiate the efficacy of this approach through simulations. This study tackles optimizing the FBGs' sensor placement for precise temperature monitoring during tumor ablation. Optimizing the FBG placement enhances temperature mapping, aiding in tumor cell eradication while minimizing damage to surrounding tissues.

Convexification approaches for regional route guidance and demand management with generalized MFDs

Traffic congestion is one of the main concerns in big cities with many adverse socioeconomic effects. A promising solution is to simultaneously regulate the admission of vehicles (i.e., demand management) and redistribute traffic flows within the network (i.e., route guidance). In this work, we integrate demand management with route guidance within a Model Predictive Control framework using regional traffic dynamics with generalized Macroscopic Fundamental Diagram (MFD) shapes. Dealing with generalized MFD shapes is challenging due to the resulting nonlinear and non-convex optimization formulation. To tackle this challenge, we develop two real-time solution approaches: (i) a successive convexification approach that constructs convex bounding sets for all nonlinear terms, and (ii) a linear approximation approach that solves the problem using triangular macroscopic fundamental diagram approximations. The proposed approaches offer a trade-off between execution speed and solution quality, as the linear approximation approach runs faster while the successive convexification approach yields better quality and accuracy solutions. Macroscopic simulation results illustrate the efficiency of the successive convexification and linear approximation approaches yielding an optimality gap of less than 3.5% and 10% in all considered cases, respectively. Furthermore, both approaches outperform a state-of-practice nonlinear solver in terms of solution quality and execution time. Finally, substantial gains are also obtained regarding travel time and traffic flow efficiency in a realistic microsimulation environment.

Mixed-integer nonlinear and continuous optimization formulations for aircraft conflict avoidance via heading and speed deviations

We introduce two new optimization models for the aircraft conflict avoidance problem that aims at issuing decisions on both speed and heading-angle deviations to keep aircraft pairwise separated by a given separation distance. The first model is a new mixed-integer nonlinear formulation. The second model is a continuous optimization formulation, less typical in aircraft conflict avoidance. The advantages of the two models are combined within a three-phase method that we propose to solve the problem to global optimality. Computational experiments on various instances from the literature yield very promising results, and show the effectiveness of the proposed models and of the three-phase solution approach.

Gaining or Losing Perspective for Piecewise-Linear Under-Estimators of Convex Univariate Functions

We study mixed-integer nonlinear optimization (MINLO) formulations of the disjunction \(x\in \{0\}\cup [\ell ,u]\), where z is a binary indicator for \(x\in [\ell ,u]\) (\(0 \le \ell <u\)), and y “captures” f(x), which is assumed to be convex and positive on its domain \([\ell ,u]\), but otherwise \(y=0\) when \(x=0\). This model is very useful in nonlinear combinatorial optimization, where there is a fixed cost c for operating an activity at level x in the operating range \([\ell ,u]\), and then, there is a further (convex) variable cost f(x). So the overall cost is \(cz+f(x)\). In applied situations, there can be N 4-tuples \((f,\ell ,u,c)\), and associated (x, y, z), and so, the combinatorial nature of the problem is that for any of the \(2^N\) choices of the binary z-variables, the non-convexity associated with each of the \((f,\ell ,u)\) goes away. We study relaxations related to the perspective transformation of a natural piecewise-linear under-estimator of f, obtained by choosing linearization points for f. Using 3-d volume (in (x, y, z)) as a measure of the tightness of a convex relaxation, we investigate relaxation quality as a function of f, \(\ell \), u, and the linearization points chosen. We make a detailed investigation for convex power functions \(f(x):=x^p\), \(p>1\).

On multivariate randomized classification trees: [formula omitted]-based sparsity, VC dimension and decomposition methods

Decision trees are widely-used classification and regression models because of their interpretability and good accuracy. Classical methods such as CART are based on greedy approaches but a growing attention has recently been devoted to optimal decision trees. We investigate the nonlinear continuous optimization formulation proposed in Blanquero et al. (2020) for training sparse optimal randomized classification trees. Sparsity is important not only for feature selection but also to improve interpretability. We first consider alternative methods to sparsify such trees based on concave approximations of the l0 “norm”. Promising results are obtained on 24 datasets in comparison with the original l1 and l∞ regularizations. Then, we derive bounds on the VC dimension of multivariate randomized classification trees. Finally, since training is computationally challenging for large datasets, we propose a general node-based decomposition scheme and a practical version of it. Experiments on larger datasets show that the proposed decomposition method is able to significantly reduce the training times without compromising the testing accuracy.

A linear Lagrangian model predictive controller of macro- and micro- variable speed limits to eliminate freeway jam waves

Variable speed limits (VSLs) are a common traffic control measure to resolve freeway jam waves. State-of-the-art model predictive control (MPC) approaches of VSLs are developed based on Eulerian Lighthill-Whitham and Richards (LWR) models, where the decision variables are flows between road segments. It is difficult to implement constraints on speeds that are necessary in typical real-world speed limit systems, because converting flow to speed results in nonlinear and non-convex optimization formulations. In this paper, we develop a new MPC of VSLs based on a discrete Lagrangian LWR model, in which the decision variables are average speeds of vehicle groups. This allows formulating speed constraints as control constraints rather than state constraints in the MPC problem. The optimization of vehicle groups speeds is formulated as a linear programming problem which can be solved efficiently. We further integrate the presented MPC to a hierarchical VSL control framework leveraging connected vehicles. The presented MPC decides the optimal target speed of each vehicle group led by a connected automated vehicle (CAV) at the upper macroscopic level with a prediction horizon of 20 min. At the lower microscopic level, CAVs randomly distributed in mixed traffic are regarded as actuators of the upper layer. Microscopic CAV accelerations are optimized in a short horizon of the order 5–10 s so that the human-driven vehicles following them reach the target speed from the upper layer in an efficient and smooth manner. The presented MPC and the hierarchical control approach are tested in microscopic simulation environments. Simulation results show that (i) the presented MPC resolves freeway jam waves efficiently with reasonable safety constraints implemented, and (ii) the presented hierarchical control approach can effectively resolve jam waves in a single-lane freeway, even though the penetration rate of CAVs is as low as 5%.

RegARD: Symmetry-Based Coarse Registration of Smartphone’s Colorful Point Clouds with CAD Drawings for Low-Cost Digital Twin Buildings

Coarse registration of 3D point clouds plays an indispensable role for parametric, semantically rich, and realistic digital twin buildings (DTBs) in the practice of GIScience, manufacturing, robotics, architecture, engineering, and construction. However, the existing methods have prominently been challenged by (i) the high cost of data collection for numerous existing buildings and (ii) the computational complexity from self-similar layout patterns. This paper studies the registration of two low-cost data sets, i.e., colorful 3D point clouds captured by smartphones and 2D CAD drawings, for resolving the first challenge. We propose a novel method named ‘Registration based on Architectural Reflection Detection’ (RegARD) for transforming the self-symmetries in the second challenge from a barrier of coarse registration to a facilitator. First, RegARD detects the innate architectural reflection symmetries to constrain the rotations and reduce degrees of freedom. Then, a nonlinear optimization formulation together with advanced optimization algorithms can overcome the second challenge. As a result, high-quality coarse registration and subsequent low-cost DTBs can be created with semantic components and realistic appearances. Experiments showed that the proposed method outperformed existing methods considerably in both effectiveness and efficiency, i.e., 49.88% less error and 73.13% less time, on average. The RegARD presented in this paper first contributes to coarse registration theories and exploitation of symmetries and textures in 3D point clouds and 2D CAD drawings. For practitioners in the industries, RegARD offers a new automatic solution to utilize ubiquitous smartphone sensors for massive low-cost DTBs.

Global exact optimization for covering a rectangle with 6 circles

We address the problem of covering a rectangle with six identical circles, whose radius is to be minimized. We focus on open cases from Melissen and Schuur (Discrete Appl Math 99:149–156, 2000). Depending on the rectangle side lengths, different configurations of the circles, corresponding to the different ways they are placed, yield the optimal covering. We prove the optimality of the two configurations corresponding to open cases. For the first one, we propose a mathematical mixed-integer nonlinear optimization formulation, that allows one to compute global optimal solutions. For the second one, we provide an analytical expression of the optimal radius as a function of one of the rectangle side lengths. All open cases are thus closed for the optimal covering of a rectangle with six circles.

Uncertainty-Aware Deployment of Mobile Energy Storage Systems for Distribution Grid Resilience

With the spatial flexibility exchange across the network, mobile energy storage systems (MESSs) offer promising opportunities to elevate power distribution system resilience against emergencies. Despite the remarkable growth in integration of renewable energy sources (RESs) in power distribution systems (PDSs), most recovery and restoration strategies do not unlock the full potential in such resources due to their inherent uncertainty and stochasticity. This paper develops a novel restoration mechanism in PDSs for routing and scheduling of MESSs integrated with stochastic RESs to achieve agile system response and recovery in facing the aftermath of high-impact low-probability (HILP) incidents. The proposed integrated model is presented as a non-convex non-linear stochastic optimization formulation with joint probabilistic constraints (JPCs). The problem is equivalently reformulated to a tractable mixed-integer linear programming (MILP) model that can be solved by commercial off-the-shelf solvers. Case studies on the IEEE 33-node and 123-node test systems demonstrate the effectiveness and scalability of the proposed framework in boosting the system resilience. This is achieved via effective routing and scheduling of MESSs jointly managed with dynamic network reconfiguration in presence of stochastic RESs.

Joint Nonlinear Sparse Error Correction for Robust State Estimation

We study joint nonlinear state estimation with multi-period measurement vectors that are potentially corrupted by sparse gross errors. We consider a nonlinear sparse optimization formulation for joint sparse error correction and robust state estimation in a nonlinear sensing system wherein we exploit the sparsity and short-term invariance properties of gross error locations. We introduce a sequential convex approximation approach that can be used to solve the nonlinear sparse optimization problem with a convergence guarantee. We derive a necessary rank condition that identifiable gross error matrix should satisfy. Then, we present an identifiability-aware version of the proposed algorithm wherein we exploit the aforementioned identifiability condition to improve the accuracy of gross error localization. We demonstrate the efficacy of the proposed approach by applying it to power system nonlinear state estimation of IEEE 14-bus and 118-bus networks.

Automated planning for robotic layup of composite prepreg

Hand layup is a commonly used process for making composite structures from several plies of carbon-fiber prepreg. The process involves multiple human operators manipulating and conforming layers of prepreg to a mold. The manual layup process is ergonomically challenging, tedious, and limits throughput. Moreover, different operators may perform the process differently, and hence introduce inconsistency. We have developed a multi-robot cell to automate the layup process. A human expert provides a sequence to conform to the ply and types of end-effectors to be used as input to the system. The system automatically generates trajectories for the robots that can achieve the specified layup. Using the cell requires the automated generation of different types of plans. This paper addresses two main planning problems: (a) generating plans to grasp and manipulate the ply and (b) generating feasible robot trajectories. We use a hybrid-physics based simulator coupled with a state space search to find grasp plans. The system employs a strategy that applies constraints successively in a non-linear optimization formulation to identify suitable placements of the robots around the mold so that feasible trajectories can be generated. Our system can generate plans in a computationally efficient manner, and it can handle a wide variety of complex parts. We demonstrate the automated layup by conducting physical experiments on an industry-inspired mold using the generated plans.

Optimization of the amines‐CO2 capture process by a nonequilibrium rate‐based modeling approach

AbstractAmine‐based chemical absorption has become the most mature technology among carbon dioxide capture processes, featuring the advantages of high separation efficiencies and its simplicity to attach it to existent industrial facilities. Nevertheless, further improvements on its performance are required in order to implement this technology to a wider extent. Energy expenditure at the solvent regeneration unit has been remarked as the major drawback of the process under discussion; therefore, the main objective of this work is to compute optimal operating policies which ensure minimum heat load at the reboiler along with reasonable removal efficiencies. A deterministic mathematical model of a reported pilot plant was deployed as the basis for the nonlinear programming optimization formulations, using a rate‐based approach and an eNRTL thermodynamic framework. Several optimization scenarios were studied to account for diverse capture targets and degrees of freedom. Furthermore, sensitivity analyses were performed to understand the stand‐alone effect of some variables on the process. Results reflect the importance of optimizing the complete plant to account for significant interactions between variables, as well as choosing an operational arrangement according to the separation goal demanded.

On the Producibility of Composite Position Tolerance Specifications for Patterns of Holes: Machining Experiments

Abstract ASME Y 14.5M-2009 provides three distinct variants of composite position tolerance constraints to exercise control over the position characteristics of patterns of features of sizes such as holes. The practical implementation of this class of position tolerance constraints is hampered by a distinct lack of information relative to the ease or difficulty involved in the actual production and satisfaction of these constraints. In this research, machining experiments are carried out to produce various patterns of holes in industrial grade machines. Coordinate location data of the holes are acquired, and a nonlinear optimization formulation is developed and used to assess the producibility of these variants. Boundary curves for pattern and feature-to-feature tolerances are developed, plotted, and discussed. The influence of various variables on the producibility of these variants is also discussed. These results may help design, and manufacturing engineers exercise a more informed approach to the application and production of these composite position tolerance variants for patterns of features of sizes.

Three dimensional unassisted sit-to-stand prediction for virtual healthy young and elderly individuals

Sit-to-stand (STS) motion is one of the most important tasks in daily life and is one of the key determinants of functional independence, especially for the senior people. The STS motion has been extensively studied in the literature, mostly through experiments. Compared to numerous experimental studies, there are limited simulations with mostly assuming bilateral symmetry for STS tasks. However, it is not true even for healthy individuals to perform STS tasks with a perfect bilateral symmetry. In this study, predictive dynamics is utilized for STS prediction. The problem can be constructed as a nonlinear optimization formulation. The digital human model has 21 degrees of freedom (DOFs) for the unassisted STS tasks. The quartic B-spline interpolation is implemented for representing joint angle profiles. The recursive Lagrangian dynamics approach and the Denavit–Hartenberg method are implemented for the equations of motion. This study is to develop a generic three-dimensional unassisted STS motion prediction method for healthy young and elderly individuals. Results show that trunk joint angle peak values are similar between the two virtual-groups in the sagittal, frontal, and transverse planes. Lower-limbs’ joint angle and velocity profiles and their peak values between the right and left side for both virtual groups are also similar. The normalized peak joint torques are slight differences in each active DOF between the two virtual groups and the peak values are similar. The proposed method has been indirectly validated through the literature experimental results. The developed method has various potential applications in the design of exoskeleton, microelectromechanical system for fall detection, and assistive devices in rehabilitation.

A model predictive approach to protect power systems against cascading blackouts

Large scale blackouts normally go through several sequential phases according to the propagation speed of branch outages and the disruption level of power grids. It is crucial to eliminate the propagation of cascading outages in its infancy. In this paper, a model predictive approach is proposed to protect power grids against cascading blackouts. First of all, the cascading dynamics of power grids is described by the outage model of transmission lines and the DC power flow equation, which allows us to predict the cascading failure path. Then a nonlinear convex optimization formulation is established to terminate the cascading outages by adjusting the injected power on buses. Afterwards two protection schemes are designed according to the optimization formulation: one scheme carries out protective actions to terminate cascading outages once for all, while the other takes protection measures in two consecutive steps. Saddle point dynamics is employed to provide a numerical solution to the proposed optimization problem, and its global convergence is guaranteed in theory. Finally, numerical simulations on IEEE test systems are implemented to validate the proposed approach.

More Virtuous Smoothing

In the context of global optimization of mixed-integer nonlinear optimization formulations, we consider smoothing univariate functions $f$ that satisfy $f(0)=0$, $f$ is increasing and concave on $[...

On a nonconvex MINLP formulation of the Euclidean Steiner tree problem in n-space: missing proofs

We supply proofs for a few key results concerning smoothing square roots and model strengthening for a mixed-integer nonlinear-optimization formulation of the the Euclidean Steiner tree problem.

A parallel method for solving the DC security constrained optimal power flow with demand uncertainties

The security constrained optimal power flow (SCOPF) is a fundamental tool to analyze the security and economy of a power system. To ensure the safe and economic operation of a system considering demand uncertainties and to acquire economic and reliable solutions, in this paper, a parallel method for solving the interval DC SCOPF with demand uncertainties is presented. By using the interval optimization method, the uncertain nodal load can be expressed as interval variables and integrated into the DC SCOPF model, which is then formed as a large scale nonlinear interval optimization formulation. According to the theory of interval matching and selection of the extreme value intervals, the interval DC SCOPF problem can be transformed into two deterministic nonlinear programming problems and solved by alternating direction method of multipliers (ADMM) to obtain the range information of interval formulation variables. Using ADMM, the above two deterministic problems, which are large in scale because of the large number of preconceived contingencies, all can be split into independent sub-problems corresponding to pre-contingency status and each individual post-contingency cases. These small-scale sub-problems can be solved in parallel to improve the computing speed. Compared with the Monte Carlo (MC) method, the simulation results of the IEEE 30-, 57- and 118-bus systems validate the effectiveness of the proposed method.

Optimal design of deep-groove ball bearings based on multitude of objectives using evolutionary algorithms

PurposeIn optimum designs of deep-groove ball bearings (DDGBs), an extended service life is one of the vital criteria. The life of a bearing depends on several factors. The purpose of this paper is to sequentially optimize three prime objectives for DDGB, i.e. the dynamic capacity (Cd), the maximum bearing temperature (Tmax) and the elasto-hydrodynamic minimum film thickness (Hmin).Design/methodology/approachFor solving constrained non-linear optimization formulations with multitude of objectives, an optimal design methodology has been put forth with the help of artificial bee colony algorithms. A study on the constraint violation has been carried out. By the Monte Carlo simulation method, a sensitivity investigation of diverse design variables has been done to examine variations in three objective functions and violation of constraints.FindingsExcellent improvement in the dynamic capacity (Cd), the maximum bearing temperature (Tmax) and the elasto-hydrodynamic minimum film thickness (Hmin) have been found in optimized bearing designs.Originality/valueBall bearing design has been done based on multi-discipline objectives that are based on strength, tribology and thermal consideration. This type of design is essential in practical scenario where these physical phenomena will be present simultaneously.