Constrained Nonlinear Programming Problems Research Articles

Continuous Low-Thrust Maneuver Planning for Space Gravitational Wave Formation Reconfiguration Based on Improved Particle Swarm Optimization Algorithm

This study proposes a three-spacecraft formation reconfiguration strategy of minimum fuel for space gravitational wave detection missions in the high Earth orbit (105 km). For solving the limitations of measurement and communication in long baseline formations, a control strategy of a virtual formation is applied. The virtual reference spacecraft provides a desired relative state between the satellites, which is then used to control the motion of the physical spacecraft to maintain the desired formation. A linear dynamics model based on relative orbit elements' parameterization is used to describe the relative motion in the virtual formation, which facilitates the inclusion of J2, SRP, and lunisolar third-body gravity effects and provides a direct insight into the relative motion geometry. Considering the actual flight scenarios of gravitational wave formations, a formation reconfiguration strategy based on continuous low thrust is investigated to achieve the desired state at a given time while minimizing interference to the satellite platform. The reconfiguration problem is considered a constrained nonlinear programming problem, and an improved particle swarm algorithm is developed to solve this problem. Finally, the simulation results demonstrate the performance of the proposed method in improving the maneuver sequence distribution and optimizing maneuver consumption.

Diversification of Time-Varying Tangency Portfolio under Nonlinear Constraints through Semi-Integer Beetle Antennae Search Algorithm

In finance, the most efficient portfolio is the tangency portfolio, which is formed by the intersection point of the efficient frontier and the capital market line. This paper defines and explores a time-varying tangency portfolio under nonlinear constraints (TV-TPNC) problem as a nonlinear programming (NLP) problem. Because meta-heuristics are commonly used to solve NLP problems, a semi-integer beetle antennae search (SIBAS) algorithm is proposed for solving cardinality constrained NLP problems and, hence, to solve the TV-TPNC problem. The main results of numerical applications in real-world datasets demonstrate that our method is a splendid substitute for other evolutionary methods.

Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

Optimization of a two-echelon supply chain with random demand and random defect rate under strict carbon cap policy

PurposeThe yield of defective items and emissions of greenhouse gases in supply chains are areas of concern. Organizations try to reduce the yield defective items and emissions. In this paper, a constrained optimization model is developed with consideration of the yield of defective items and strict carbon cap policy simultaneously and then optimized. Further, sensitivity analyses have been carried out to draw different managerial insights. Precisely, we have tried to address the following research questions: (1) how to optimize the cost for a two-echelon supply chain considering yield of defective items and strict carbon cap policy, (2) how the total expected cost and total expected emissions act with changing parameters.Design/methodology/approachThe mathematical modeling approach has been adopted to develop a model and further optimized it with optimization software. Costs and emissions from different areas of a supply chain have been derived and then the total cost and total emissions have been formulated mathematically. One constrained mixed-integer nonlinear programming (MINLP) problem has been formulated and solved considering emissions-related, velocity and production related-constraints. Further, different sensitivity analyses have been derived to draw some managerial insights.FindingsIn this paper, many decision variables have been calculated with a set of basic values of other parameters. It has been found that both cost and emissions can be controlled by controlling different parameters. It has been also found that some parameters have very little or no influence either on cost or emissions. In most cases, originations may exhaust the given limit of carbon cap to optimize their costs.Originality/valueIn spite of my sincere efforts, no paper has been found that has considered the yield of defective items and strict carbon cap policy simultaneously. In this paper, it is assumed that both demand and defect rates are random in nature. The model, presented in this paper may give insights to develop different supply chain models with consideration of both defective items and strict carbon cap policy. Sensitivity analyses, drawn in this paper may give deep insights to managers and carbon regulatory bodies.

An Efficient Hybrid Optimization Strategy for Surface Reconstruction

AbstractAn efficient surface reconstruction strategy is presented in this study, which is able to approximate non‐convex sets of target points (TPs). The approach is split in two phases: (a) the mapping phase, making use of the shape preserving method (SPM) to get a proper parametrization of each sub‐domain composing the TPs set; (b) the fitting phase, where each patch is fitted by means of a suitable non‐uniform rational basis spline (NURBS) surface by considering, as design variables, all parameters involved in its definition. To this purpose, the surface fitting problem is formulated as a constrained non‐linear programming problem (CNLPP) defined over a domain having changing dimension, wherein both the number and the value of the design variables are optimized. To deal with this CNLPP, the optimization process is split in two steps. Firstly, a special genetic algorithm (GA) optimizes both the value and the number of design variables by means of a two‐level evolution strategy (species and individuals). Secondly, the solution provided by the GA constitutes the initial guess for the deterministic optimization, which aims at improving the accuracy of the fitting surfaces. The effectiveness of the proposed methodology is proven through some meaningful benchmarks taken from the literature.

HNN-based generalized predictive control for turbofan engine direct performance optimization

The aero-engine performance optimization control is to find the appropriate control variables to maximize or minimize the comprehensive index of one or more performance parameters while ensuring its safe operation. Traditional performance seeking control is mainly based on indirect control, with complex structure and large error. This paper develops an HNN (Hopfield Neural Network)-based generalized predictive control strategy for turbofan engine direct performance optimization. With a Hopfield network as the optimization algorithm and a CARIMA model based on generalized prediction criterion as the optimization model, an optimal controller is designed. Together with the identification module and the self-adaptive module, an integrated performance optimization control system is formed. Namely, the adaptive model calculates the unmeasurable performance parameters of the engine for the CARIMA model identification. Then HNN performs the iterative operation to obtain the optimal control variables for a given objective. The main contribution of this paper is to propose a direct performance control structure to avoid the nonlinear conversion of control reference; meanwhile, the linear CARIMA model is combined with HNN algorithm to build a neural network for solving constrained nonlinear programming problems, and mutation operation is introduced to improve its global search capability, which is evident in the minimum turbine temperature optimization mode. The method is evaluated on a turbofan model from the aspects of model verification, performance optimization control simulation at multiple operating points and under degeneration conditions. The results confirm the satisfactory performance of the proposed strategy at the selected points and conditions. And compared with the traditional algorithms, the computer calculation time is greatly shortened.

PrandtlPlane wing-box least-weight design: A multi-scale optimisation approach

The PrandtlPlane (PrP) aircraft wing-box least-weight design is presented in this work. This design problem is formulated as a constrained non-linear programming problem (CNLPP), by integrating static, buckling, fatigue and manufacturability requirements, under different loading conditions. The solution search is carried out by means of a suitable multi-scale optimisation (MSO) approach. The physical responses involved into the CNLPP formulation are evaluated at the wing-box architecture level (macroscopic scale) and at the stiffened panel level (component scale), as well. The scale transition is ensured by means of a suitable global-local (GL) modelling approach, while the CNLPP is solved by means of an in-house genetic algorithm. The effectiveness of the proposed approach is tested on the PrP wing-box structure, but the presented strategy can be easily extended to conventional aircraft wings.

Multi-scale optimisation of thin-walled structures by considering a global/local modelling approach

In this work, a design strategy for optimising thin-walled structures based on a global-local finite element (FE) modelling approach is presented. The preliminary design of thin-walled structures can be stated in the form of a constrained non-linear programming problem (CNLPP) involving requirements of different nature intervening at the different scales of the structure. The proposed multi-scale optimisation (MSO) strategy is characterised by two main features. Firstly, the CNLPP is formulated in the most general sense by including all design variables involved at each pertinent scale of the problem. Secondly, two scales (with the related design requirements) are considered: (a) the structure macroscopic scale, where low-fidelity FE models are used and (b) the structure mesoscopic scale (or component level), where more accurate FE models are involved. In particular, the mechanical responses of the structure are evaluated at both global and local scales, avoiding the use of approximated analytical methods. The MSO is here applied to the least-weight design of an aluminium fuselage barrel of a wide-body aircraft. Fully parametric global and local FE models are interfaced with an in-house metaheuristic algorithm. Refined local FE models are created only for critical regions of the structure, automatically detected during the global analysis, and linked to the global one, thanks to the implementation of a sub-modelling approach. The whole process is completely automated, and once set, it does not need any further user intervention.

A Comparison of Normal Cone Conditions for Homotopy Methods for Solving Inequality Constrained Nonlinear Programming Problems

Homotopy methods are powerful tools for solving nonlinear programming. Their global convergence can be generally established under conditions of the nonemptiness and boundness of the interior of the feasible set, the Positive Linear Independent Constraint Qualification (PLICQ), which is equivalent to the Mangasarian-Fromovitz Constraint Qualification (MFCQ), and the normal cone condition. This paper provides a comparison of the existing normal cone conditions used in homotopy methods for solving inequality constrained nonlinear programming.

A Metamodel Based on Non-Uniform Rational Basis Spline Hyper-Surfaces for Optimisation of Composite Structures

This study presents a new strategy to generate a surrogate model used for design purposes. The metamodel is based on Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces and is able to fit non-convex sets of target points (TPs). The proposed method aims at determining all the parameters involved in the definition of the NURBS hyper-surface, i.e. control points (CPs) coordinates, weights, degrees, CPs number and knot-vector components. To this purpose, the problem of finding a suitable metamodel is formulated as a constrained non-linear programming problem (CNLPP) wherein the above variables are optimised in order to fit a set of TPs. Nevertheless, when the number of CPs and the degrees of the basis functions are included among the design variables, the resulting problem is defined over a space having a variable dimension. This problem is solved by means of a special genetic algorithm able to determine simultaneously the optimum value of both the design space size (related to the integer variables of the NURBS hyper-surface) and the NURBS hyper-surface continuous parameters. The NURBS-based metamodel is then used to emulate the first buckling load of a composite stiffened panel and it is used in the framework of a meaningful design problem.

Non-Uniform Rational Basis Spline hyper-surfaces for metamodelling

This study presents an original metamodelling technique based on Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces. The proposed approach is able to fit general non-convex sets of target points (TPs) by extending the NURBS formalism to the N-dimensional (N-D) case. The shape of such a hyper-surface is tuned by several parameters: the number of control points (CPs), their coordinates and the related weights, the degrees of the blending functions and the knot-vector components defined along each direction. The goal of the proposed strategy is to automatically determine (i.e. without the user’s intervention) the full set of parameters defining the NURBS hyper-surface approximating a given set of TPs, without considering simplifying hypotheses. To this purpose, the problem is formulated as a constrained nonlinear programming problem (CNLPP) wherein the optimization variables are all the parameters tuning the shape of the NURBS hyper-surface. Nevertheless, when the number of CPs and the degrees of the basis functions are included among the design variables, the resulting problem is defined over a space having a changing dimension. This problem is solved by means of an original genetic algorithm able to determine, simultaneously, the optimum value of both the design space size (related to the integer variables of the NURBS hyper-surface) and the NURBS hyper-surface continuous parameters. The effectiveness of the proposed approach is shown by means of two meaningful test cases. In addition, the proposed method has been applied to a benchmark taken from the literature and the results have been compared to those provided by the Proper Generalized Decomposition method.

An Optimal Multivariable Constrained Nonlinear (MVCNL) Stochastic Microgrid Planning and Operation Problem With Renewable Penetration

Microgrids (MGs) are promising archetypes for the penetration of renewable energy sources (RES) into the system for reducing carbon footprints. This article presents a cost-based optimization model using a stochastic programming to obtain the optimal size of distributed energy resources including RES. Uncertainties in the solar output and the load demand data are considered using multiple scenarios with random variations, which are based on beta distribution. The objective function is formulated as a cost-based multivariable constrained nonlinear convex programming problem. Other than planning costs, the objective function also includes per day running or operational costs such as unit commitment costs as well as the switching costs associated with the dispatchable units. The overall problem is simulated under MATLAB environment. Quantitative results indicate the impact of operational paradigm along with the planning of an MG in the form of cost analysis with different reliability conditions. An assessment of results in tabular form is presented, which incorporates the optimal sizing of MG under the different system reliability conditions and constraints. Some additional cases pertain to the generation and the load demand scenarios have also been included in the article.

Optimal Coordination of Directional Overcurrent Relays in Microgrids by Using Cuckoo-Linear Optimization Algorithm and Fault Current Limiter

Nowadays, distributed generations (DGs) and microgrids play a significant role in development of power systems, but DGs produce several protection problems especially in optimal coordination of directional overcurrent relays (DORs). Optimal coordination of DORs is a highly constrained nonlinear programming problem, which has attracted attention of many research studies in recent years. Cuckoo optimization algorithm (COA) is introduced as an optimization tool based on behavior patterns of Cuckoos. In this paper, COA and linear programming (LP) are combined as a new hybrid COA-LP optimization algorithm in order to optimize the coordination protection of DORs in microgrids and find the optimum value of fault current limiter at the point of common coupling. The optimum settings of DORs are obtained for both modes of operation: 1) grid-connected and 2) island mode. The proposed algorithm is applied to Canadian distribution benchmark and a modified IEEE 14 bus networks and compared with particle swarm optimization (PSO) and genetic algorithm (GA). The obtained results show that the computational efficiency of the proposed COA-LP is better than PSO and GA in terms of speed and total operating time of DORs.

Parallel Secure Outsourcing of Large-scale Nonlinearly Constrained Nonlinear Programming Problems

Nonlinearly constrained nonlinear programming (NLC-NLP) problems arise in various optimal decision-making fields, such as financial engineering, urban planning, supply chain management, and power system control. These real-world NLC-NLP problems are usually large-scale because they need to take into account massive variables and constraints. Solving NLC-NLP problems with common algorithms, e.g., the gradient projection method (GPM), generally incurs very high time complexity. This results in the inevitable inability of solving large-scale NLC-NLP problems by using general-purpose computers within a feasible time. To address this challenge, a cost-effective solution is to adopt cloud computing, but this raises security concerns since real-world NLC-NLP problems always carry sensitive information. Previous secure outsourcing algorithms try to protect sensitive information, but still let cloud service tenants bear heavy computation workload. To fill the gap, we develop a practical secure outsourcing algorithm for solving large-scale NLC-NLP problems with the GPM. To be more prominent, we also parallelize the secure outsourcing algorithm for accelerating computations and avoiding possible memory overflowing. We implement the proposed algorithm on the Amazon Elastic Compute Cloud (EC2) and a laptop, and notice that it can significantly reduce the tenant's computing time even for large-scale NLC-NLP problems.

Min–Max Energy-Efficiency Analysis of Green Multiuser Wireless Systems

In this paper, we propose an optimal power allocation scheme that minimizes the energy per bit of worst user (i.e., the user with highest energy per bit) in a multiuser wireless communication network. The problem of determining energy efficient power allocation to improve the worst user is a constrained non-convex nonlinear fractional programming problem. We propose an iterative energy efficient power allocation algorithm that guarantees optimal solution. We use parametric equivalent formulation to get the optimal solution. Numerical solutions obtained using simulations are presented and compared with equal power allocation scheme.

Alternating Direction Method of Solving Nonlinear Programming with Inequality Constrained

This paper, a new class of augmented Lagrange functions with the new NCP function is proposed for the minimization of a smooth function subject to inequality constraints. Under some conditions, we prove of the equivalences of the KKT point and local point and globe point between primal constrained nonlinear programming problem and the new unconstrained problem. By the character of augmented Lagrange function, the algorithm which uses alternating direction method is constructed and proved convergence.

Constrained Ant Colony Optimisation Algorithm for the layout and size optimisation of sanitary sewer networks

The incremental solution building capability of Ant Colony Optimisation Algorithm (ACOA) is used in this paper for the efficient layout and pipe size optimisation of sanitary sewer network. Layout and pipe size optimisation of sanitary sewer networks requires optimal determination of pipe locations, pipe diameters and pipe slopes leading to a highly constrained mixed-integer nonlinear programming (MINLP) problem presenting a challenge even to the modern heuristic search methods. A constrained version of ACOA equipped with a Tree Growing Algorithm (TGA) is proposed in this paper for the simultaneous layout and pipe size determination of sewer networks. The method is based on the assumption that a base layout including all possible links of the network is available. The TGA algorithm is used in an incremental manner to construct feasible tree-like layouts out of the base layout, while the constrained ACOA is used to optimally determine the cover depths of the constructed layout. Proposed formulation is used to solve three hypothetical test examples of different scales and the results are presented and compared with those produced by a conventional application of ACOA in which an ad-hoc engineering concept is used for layout determination. The results indicate the effectiveness and efficiency of the proposed method to optimally solve the problem of layout and size determination of sewer networks.

Hybrid social cognitive optimization algorithm for constrained nonlinear programming

To improve the global convergence speed of social cognitive optimization (SCO) algorithm, a hybrid social cognitive optimization (HSCO) algorithm based on elitist strategy and chaotic optimization is proposed to solve constrained nonlinear programming problems (NLPs). The proposed algorithm partitions learning agents into three groups in proportion: elite learning agents, chaotic learning agents and common learning agents. The common learning agents work in the search way of traditional SCO, chaotic learning agents search via chaotic search (CS) algorithm based on tent map which helps to avoid the premature convergence, elite learning agents search via elitist selection which helps to improve the global searching performance. Additionally, a chaotic search process is incorporated into local searching operation so as to enhance the local searching efficiency in the neighboring areas of the feasible solutions. Simulation results on a set of benchmark problems show that the proposed algorithm has high optimization efficiency, good global performance, and stable optimization outcomes for constrained NLPs.

Newton-conjugate gradient (CG) augmented Lagrangian method for path constrained dynamic process optimization

In this paper, a Newton-conjugate gradient (CG) augmented Lagrangian method is proposed for solving the path constrained dynamic process optimization problems. The path constraints are simplified as a single final time constraint by using a novel constraint aggregation function. Then, a control vector parameterization (CVP) approach is applied to convert the constraints simplified dynamic optimization problem into a nonlinear programming (NLP) problem with inequality constraints. By constructing an augmented Lagrangian function, the inequality constraints are introduced into the augmented objective function, and a box constrained NLP problem is generated. Then, a linear search Newton-CG approach, also known as truncated Newton (TN) approach, is applied to solve the problem. By constructing the Hamiltonian functions of objective and constraint functions, two adjoint systems are generated to calculate the gradients which are needed in the process of NLP solution. Simulation examples demonstrate the effectiveness of the algorithm.

Smooth tool path generation for five-axis flank milling using multi-objective programming

Flank milling provides an efficient way to machine turbo-machinery components. However, the trajectory smoothness is seldom considered in the existing tool positioning strategies. In this paper, the tool trajectory is smoothed with a constraint on the resulting geometric error for five-axis flank milling. Unlike existing methods, this new method simultaneously considers the geometric smoothness and geometric deviation. The geometric smoothness is characterized by the strain energy of the cutter axis trajectory surface ( SA). The geometric deviation is measured by the signed maximal orthogonal distance between the design surface ( SD) and the tool envelope surface ( SE). For finish and semi-finish flank millings, smooth tool path optimizations are then modelled as multi-objective programming (MOP) problems. Given a prescribed geometric tolerance, the MOP problems are reformulated as constrained nonlinear programming (NLP) problems. Based on the Taylor expansions of the strain energy and the signed distance on the differential deformation of SA, the constrained NLP problems are solved efficiently by the sequential quadratic programming (SQP) method. The existence of the optimal solutions is also discussed. The validity of the approach is confirmed by two numerical examples that generate five-axis flank milling tool paths with cylindrical and conical cutters, respectively.