Showing papers on "Nonlinear programming published in 2022"

Trajectory Optimization for High-Speed Trains via a Mixed Integer Linear Programming Approach

Abstract: This paper proposes a trajectory optimization approach for high-speed trains to reduce traction energy consumption and increase riding comfort. Besides, the proposed approach can also achieve energy-saving effects by optimizing the operation time between stations. First, an optimization model is developed by defining the objective function as a trade-off function of the traction energy consumption and riding comfort. In addition to constraints in the classic optimal train control model, three new factors–the discrete throttle settings, neutral zones, and sectionalized tunnel resistance–are considered. Then, the model is discretized and turned into a multi-step decision optimization problem. All the nonlinear constraints are approximated using piecewise affine (PWA) functions, and the trajectory optimization problem is turned into a mixed integer linear programming (MILP) problem which can be solved by existing solvers CPLEX and YALMIP. Finally, some case studies with real-world data sets are conducted to present the effectiveness of the proposed approach. The simulation results are compared with the practical running data of trains, which shows that the proposed model and the optimization approach save energy and improve the riding comfort.

An integrated lot-sizing policy for the inventory management of constrained multi-level supply chains: null-space method

Abstract: The null-space method (NSM) for solving the nonlinear programming (NLP) is used to transform an indefinite system into a symmetric positive definite one of a smaller dimension. In this paper, an NSM optimises the constrained NLP model of the inventories in multi-level supply chains (SCs). The presented NSM can give a direct method with a predictable level of fill without pivoting, decreasing the number of taken iterations to find the optimum solution. We transform the nonlinear equations into an NLP model, which NSM can solved. We also investigate the suitability of using null-space-based factorisations to derive sparse direct methods. Accordingly, an integrated lot-sizing model of the multi-level SC is designed and then optimised using the presented NSM. The paper's objectives are to find the optimum number of stockpiles and the economic period length for inventories. Some numerical examples demonstrate the applicability of the presented NSM to optimise the integrated lot-sizing policy of the multi-level SCs. The presented NSM shows satisfactory performance in optimum solutions, the number of iterations, infeasibility, optimality error, and complementarity.

Mixed-Integer Nonlinear Programming for State-Based Non-Intrusive Load Monitoring

Abstract: Energy disaggregation, known in the literature as Non-Intrusive Load Monitoring (NILM), is the task of inferring the energy consumption of each appliance given the aggregate signal recorded by a single smart meter. In this paper, we propose a novel two-stage optimization-based approach for energy disaggregation. In the first phase, a small training set consisting of disaggregated power profiles is used to estimate the parameters and the power states by solving a mixed integer programming problem. Once the model parameters are estimated, the energy disaggregation problem is formulated as a constrained binary quadratic optimization problem. We incorporate penalty terms that exploit prior knowledge on how the disaggregated traces are generated, and appliance-specific constraints characterizing the signature of different types of appliances operating simultaneously. Our approach is compared with existing optimization-based algorithms both on a synthetic dataset and on three real-world datasets. The proposed formulation is computationally efficient, able to disambiguate loads with similar consumption patterns, and successfully reconstruct the signatures of known appliances despite the presence of unmetered devices, thus overcoming the main drawbacks of the optimization-based methods available in the literature.

Optimal economic-environmental dispatch in MT-HVDC systems via sine-cosine algorithm

Abstract: This paper addresses the problem of optimal economic-environmental dispatch in Multi-Terminal High-Voltage Direct Current (MT-HVDC) networks using the Sine-Cosine Algorithm (SCA). This optimization methodology allows working with nonlinear non-convex large-scale optimization problems via sequential programming. The SCA works with an initial population and rules of advance based on the best current solution and sine and cosine functions that define the direction of the next solution. Three variants of the SCA are evaluated in a standard six-node MT-HVDC system considering a linear combination of the objective functions (i.e., greenhouse emissions and energy production costs). The main advantage of the proposed evolutionary approach lies in its pure algorithmic structure. Thus, it can be easily adapted to any continuous optimization problem. All numerical calculations are performed using MATLAB software.

On the Conic Convex Approximation to Locate and Size Fixed-Step Capacitor Banks in Distribution Networks

Abstract: The problem of the optimal siting and sizing of fixed-step capacitor banks is studied in this research from the standpoint of convex optimization. This problem is formulated through a mixed-integer nonlinear programming (MINLP) model, in which its binary/integer variables are related to the nodes where the capacitors will be installed. Simultaneously, the continuous variables are mainly associated with the power flow solution. The main contribution of this research is the reformulation of the exact MINLP model through a mixed-integer second-order cone programming model (MI-SOCP). This mixed-integer conic model maintains the nonlinearities of the original MINLP model; however, it can be solved efficiently with the branch & bound method combined with the interior point method adapted for conic programming models. The main advantage of the proposed MI-SOCP model is the possibility of finding the global optimum based on the convex nature of the power flow problem for each binary/integer variable combination in the branch & bound search tree. The numerical results in the IEEE 33- and IEEE 69-bus systems demonstrate the effectiveness and robustness of the proposed MI-SOCP model compared to different metaheuristic approaches. The MI-SOCP model finds the final power losses of the IEEE 33- and IEEE 69-bus systems of 138.416kW and 145.397kW, which improves the best literature results reached with the flower pollination algorithm, i.e., 139.075 kW, and 145.860kW, respectively. The simulations are carried out in MATLAB software using its convex optimizer tool known as CVX with the Gurobi solver.

A Shape Optimisation with the Isogeometric Boundary Element Method and Adjoint Variable Method for the Three-Dimensional Helmholtz Equation

Abstract: This paper presents a shape optimisation system to design the shape of an acoustically-hard object in the three-dimensional open space. The boundary element method (BEM) is suitable to analyse such an exterior field. However, the conventional BEM, which is based on piecewise polynomial shape and approximate (interpolation) functions, can require many design variables because they are usually chosen as a part of the nodes of the underlying boundary element mesh. In addition, it is not easy for the conventional method to compute the gradient of the sound pressure on the surface, which is necessary to compute the shape derivative of our interest, of a given object. To overcome these issues, we employ the isogeometric boundary element method (IGBEM), which was developed in our previous work. With using the IGBEM, we can design the shape of surfaces through control points of the NURBS surfaces of the target object. We integrate the IGBEM with the nonlinear programming software through the adjoint variable method (AVM), where the resulting adjoint boundary value problem can be also solved by the IGBEM with a slight modification. The numerical verification and demonstration validate our shape optimisation framework.

On Optimality Conditions for Nonlinear Conic Programming

Abstract: Sequential optimality conditions play a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions are described in conic contexts, in which many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and improves several known results for specific cases, such as semidefinite programming, second-order cone programming, and nonlinear programming. In particular, we show that feasible limit points of sequences generated by the augmented Lagrangian method satisfy the so-called approximate gradient projection optimality condition and, under an additional smoothness assumption, the so-called complementary approximate Karush–Kuhn–Tucker condition. The first result was unknown even for nonlinear programming, and the second one was unknown, for instance, for semidefinite programming.

Robust Low-Thrust Trajectory Optimization Using Convex Programming and a Homotopic Approach

Abstract: A robust algorithm to solve the low-thrust fuel-optimal trajectory optimization problem for interplanetary spacecraft is developed in this article. The original nonlinear optimal control problem is convexified and transformed into a parameter optimization problem using an arbitrary-order Gauss–Lobatto discretization scheme with nonlinear control interpolation. A homotopic approach that considers the energy-to-fuel smoothing path is combined with an adaptive second-order trust-region mechanism to increase performance. The overall robustness is assessed in several fuel-optimal transfers with poor initial guesses. The results show a superior performance in terms of convergence and computational time compared to standard convex programming approaches in the literature.

Robust Interior Point Method for Quantum Key Distribution Rate Computation

Abstract: Security proof methods for quantum key distribution, QKD, that are based on the numerical key rate calculation problem, are powerful in principle. However, the practicality of the methods are limited by computational resources and the efficiency and accuracy of the underlying algorithms for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model for the key rate calculation problems. We use this to develop an efficient, accurate algorithm. The stable reformulation is based on novel forms of facial reduction, FR, for both the linear constraints and nonlinear quantum relative entropy objective function. This allows for a Gauss-Newton type interior-point approach that avoids the need for perturbations to obtain strict feasibility, a technique currently used in the literature. The result is high accuracy solutions with theoretically proven lower bounds for the original QKD from the FR stable reformulation. This provides novel contributions for FR for general SDP. We report on empirical results that dramatically improve on speed and accuracy, as well as solving previously intractable problems.

Outer Approximation for Mixed-Integer Nonlinear Robust Optimization

Abstract: Abstract Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protection. In this work, we develop an algorithm for convex mixed-integer nonlinear robust optimization problems where a key feature is that the method does not rely on a specific structure of the inner worst-case (adversarial) problem and allows the latter to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the non-convex adversarial problem. Our method is able to achieve this up to a tolerance, by requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial via piecewise relaxations and solving the resulting problem up to any requested error as a mixed-integer linear problem. In our approach, we model a robust optimization problem as a nonsmooth mixed-integer nonlinear problem and tackle it by an outer approximation method that requires only inexact function values and subgradients. To deal with the arising nonlinear subproblems, we render an adaptive bundle method applicable to this setting and extend it to generate cutting planes, which are valid up to a known precision. Relying on its convergence to approximate critical points, we prove, as a consequence, finite convergence of the outer approximation algorithm. As an application, we study the gas transport problem under uncertainties in demand and physical parameters on realistic instances and provide computational results demonstrating the efficiency of our method.

Coordinated Receding-Horizon Control of Battery Electric Vehicle Speed and Gearshift Using Relaxed Mixed-Integer Nonlinear Programming

Abstract: In this article, we investigate coordinated receding-horizon control of vehicle speed and transmission gearshift for energy-efficient operation of automated battery electric vehicles (BEVs). The introduction of multispeed transmissions in BEVs enables manipulation of electric motor operating point under given vehicle speed and acceleration command, thus creating the opportunity to further improve BEV energy efficiency. However, co-optimizing vehicle speed and transmission gearshift leads to a mixed-integer nonlinear programming (MINLP) problem, and it is well known that solving MINLP problems is computationally very challenging. To address this challenge, we propose a novel continuous relaxation technique that enables the computation of solutions to the speed and gearshift co-optimization problem using off-the-shelf nonlinear programming solvers. After analyzing theoretical properties of the proposed relaxation technique, we demonstrate its effectiveness through simulation-based case studies, where we show that co-optimizing vehicle speed and transmission gearshift can lead to considerably greater energy efficiency than optimizing them separately or sequentially and the proposed relaxation technique can reduce the computational cost of the co-optimization problem to a level that is comparable to the time budget available for onboard implementation.

Application of the Crow Search Algorithm to the Problem of the Parametric Estimation in Transformers Considering Voltage and Current Measures

Abstract: The problem of the electrical characterization of single-phase transformers is addressed in this research through the application of the crow search algorithm (CSA). A nonlinear programming model to determine the series and parallel impedances of the transformer is formulated using the mean square error (MSE) between the voltages and currents measured and calculated as the objective function. The CSA is selected as a solution technique since it is efficient in dealing with complex nonlinear programming models using penalty factors to explore and exploit the solution space with minimum computational effort. Numerical results in three single-phase transformers with nominal sizes of 20 kVA, 45 kVA, 112.5 kVA, and 167 kVA demonstrate the efficiency of the proposed approach to define the transformer parameters when compared with the large-scale nonlinear solver fmincon in the MATLAB programming environment. Regarding the final objective function value, the CSA reaches objective functions lower than 2.75×10−11 for all the simulation cases, which confirms their effectiveness in minimizing the MSE between real (measured) and expected (calculated) voltage and current variables in the transformer.

Mathematical programming formulations for the alternating current optimal power flow problem

Abstract: Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

Nonlinear analysis, optimization, and testing of the bridge-type compliant displacement amplification mechanism with a single input force for microgrippers

Abstract: For piezoelectric-driven compliant microgrippers, realizing flexible, stable, and accurate manipulation simultaneously is a key challenge. The bridge-type compliant displacement amplification mechanism (CDAM) with a single input force can enable stable parallel grasping and enlarge the grasping stroke by matching the output performance of typical piezoelectric actuators used in microgrippers. In this study, a nonlinear analysis of the bridge-type CDAM with a single input force was conducted to improve the prediction accuracy of the output displacement, and nonlinear optimization and testing were performed. For the expected and parasitic output displacements, a two-step nonlinear modelling method was proposed to conduct the nonlinear analysis. Using Castigliano's second theorem, small deflection-based finite element analysis (FEA), and numerical fitting, small deflection-based modelling considering the shearing effect was first performed. Then, the correction coefficients for the geometrical nonlinearity were modelled by combining geometrical nonlinear FEA and numerical fitting. By restricting the parasitic output displacement, a nonlinear constraint was derived. Thereafter, geometric parameter optimization and structural optimization of the bridge-type CDAM with a single input force was performed. Model analysis indicates that for the optimal CDAM, the expected output displacement increases with the input force, whereas a negative correlation exists between the growth rate and the input force. Finally, simulations and experimental tests were conducted to verify the effectiveness of the nonlinear models, optimization, and model analysis.

Volt/VAR Optimization: A Survey of Classical and Heuristic Optimization Methods

Abstract: Reactive power optimization and voltage control is one of the most critical components of power system operation, impacting both the economy and security of system operation. It is also one of the most complex optimization problems, being highly nonlinear, and comprising both continuous and discrete decision variables. This paper presents the problem formulation, and a thorough literature review and detailed discussion of the various solution methods that have been applied to the Volt/VAR optimization problem. Each optimization method is described in detail, and its strengths and shortcomings are outlined. The review provides detailed information on classical and heuristic methods that have been applied to the Volt/VAR optimization problem. The classical methods reviewed include (i) first- and second-order gradient-based methods, (ii) Quadratic Programming, (iii) Linear Programming, (iv) Interior-Point Methods, (iv) and mixed-integer programming and decomposition methods. The heuristic methods covered include (i) Genetic Algorithm, (ii) Evolutionary Programming, (iii) Particle Swarm Optimization, (iv) Fuzzy Set Theory, and (v) Expert Systems. A comparative analysis of the key characteristics of the classical and heuristic optimization methods is also presented along with the review.

Mathematical Models and Nonlinear Optimization in Continuous Maximum Coverage Location Problem

Abstract: This paper considers the maximum coverage location problem (MCLP) in a continuous formulation. It is assumed that the coverage domain and the family of geometric objects of arbitrary shape are specified. It is necessary to find such a location of geometric objects to cover the greatest possible amount of the domain. A mathematical model of MCLP is proposed in the form of an unconstrained nonlinear optimization problem. Python computational geometry packages were used to calculate the area of partial coverage domain. Many experiments were carried out which made it possible to describe the statistical dependence of the area calculation time of coverage domain on the number of covering objects. To obtain a local solution, the BFGS method with first-order differences was used. An approach to the numerical estimation of the objective function gradient is proposed, which significantly reduces computational costs, which is confirmed experimentally. The proposed approach is shown to solve the maximum coverage problem of a rectangular area by a family of ellipses.

Nonlinear programming problem for strongly E-invex sets and strongly E-preinvex functions

Abstract: The concepts of strongly $E$-invex sets, strongly $E$-invex, strongly $E$-preinvex, and pseudo strongly $E$-preinvex functions are introduced in this paper. We have included several non-trivial examples to support our definitions. The family of strongly $E$-invex sets has been shown to form a vector space over $R$, and other interesting properties have been addressed. The epigraph of strongly $E$-preinvex function has been derived, as well as the relationship between the strongly $E$-preinvex function and the pseudo strongly $E$-preinvex function has been established. To show an important relationship between strongly $E$-invex and strongly $E$-preinvex functions, a new Condition A has been introduced. A nonlinear programming problem for strongly $E$-preinvex functions is explored as an application. Under a few conditions, it has been proved that the local minimum point is the global minimum for nonlinear programming problems.