Nonlinear Programming Techniques Research Articles

Assessment of ship manoeuvrability by using a coupling between a nonlinear transient manoeuvring model and mathematical programming techniques

In this paper, a numerical method based on a coupling between a mathematical model of nonlinear transient ship manoeuvring motion in the horizontal plane and Mathematical Programming (MP) techniques is proposed. The aim of the proposed procedure is an efficient estimation of optimal ship hydrodynamic parameters in a dynamic model at the early design stage. The proposed procedure has been validated through turning circle and zigzag manoeuvres based on experimental data of sea trials of the 190 000dwt oil tanker. Comparisons between experimental and computed data show a good agreement of overall tendency in manoeuvring trajectories.

Entropy based solid transportation problems with discounted unit costs under fuzzy random environment

This paper investigates some multi-objective fixed charge solid transportation problems (FCSTPs) under fuzzy-random environment with fuzzy-random fixed charges and transportation times. Here objectives are total transportation cost and time which are minimized. Unit transportation costs are crisp and three types of discount i.e. All Unit Discount (AUD), Incremental Quantity Discount (IQD), IQD within AUD are applied upon these costs. In the first model (Model-I) supplies, demands and capacities of conveyances are assumed to be fuzzy-random in nature. These quantities are fuzzy in the second model (Model-II). Imprecise objectives of the models are transformed into equivalent crisp ones using random expectation and fuzzy possibility/necessity measures on fuzzy event. Imprecise constraints are reduced to equivalent crisp constraints using two different fuzzy-random chance constraint methods. Similarly constraints of Model-II are reduced to equivalent crisp constraints with the use of possibility and necessity measures on fuzzy events. Models are also formulated with and without entropy function. Finally transformed constrained multi-objective deterministic problems are solved using a multi-objective genetic algorithm (MOGA) based on arithmetic crossover and boundary mutation. Moreover, Model-IA1 is converted to a single objective non-linear programming (SONLP) problem following fuzzy non-linear programming (FNLP) technique and the reduced Model-IA1S is solved using generalized reduced gradient (GRG) method (using LINGO software). Numerical examples are used for illustration and comparison of the models.

Measurement-based optimization of batch and repetitive processes using an integrated two-layer architecture

This paper is concerned with optimal control of batch and repetitive processes in the presence of uncertainty. An integrated two-layer optimization strategy is proposed, whereby within-run corrections are performed using a neighboring-extremal update strategy and run-to-run corrections are based on a constraint-adaptation scheme. The latter is appealing since a feasible operating strategy is guaranteed upon convergence, and its combination with neighboring-extremal updates improves the reactivity and convergence speed. Moreover, these two layers are consistent in that they share the same objective function. The proposed optimization scheme is declined into two versions, namely an indirect version based on the Pontryagin maximum principle and a direct version that applies a control parameterization and nonlinear programming techniques. Although less rigorous, the latter approach can deal with singular extremals and path constraints as well as handle active-set changes more conveniently. Two case studies are considered. The indirect approach is demonstrated for a level-control problem in an experimental two-tank system, whereas the direct approach is illustrated in numerical simulation on a fed-batch reactor for acetoacetylation of pyrrole. The results confirm that faster adaptation is possible with the proposed integrated two-layer scheme compared to either constraint adaptation or neighboring-extremal update alone.

A multi-item transportation problem with fuzzy tolerance

This paper presents the recently introduced modified subgradient method for optimization and its effectiveness in a fuzzy transportation model. Here a multi-item balanced transportation problem (MIBTP) is formulated where unit transportation costs are imprecise. Also available spaces and budgets at destinations are limited but imprecise. The objective is to find a shipment schedule for the items that minimizes the total cost subjected to imprecise warehouse and budget constraints at destinations. The proposed model is reduced to a multi-objective optimization problem using tolerances, then to a crisp single-objective one using fuzzy non-linear programming (FNLP) technique and Zimmermann's method. The above fuzzy MIBTP is also reduced to another form of deterministic one using modified sub-gradient method (MSM). These two crisp optimization problems are solved by Genetic Algorithm (GA). As an extension, fuzzy multi-item balanced solid transportation problems (STPs) with and without restrictions on some routes and items are formulated and reduced to deterministic ones following FNLP and Zimmermann's methods. These models are also solved by GA. Models are illustrated numerically, optimum results of fuzzy MIBTP from two deductions are compared. Results are also presented for different GA parameters.

Parameter Identification and Assessment of Nutrient Transporters in AM Symbiosis through Stochastic Simulations

This paper presents a methodology for identifying the kinetic parameters of a biological system and their relationships with the concentration variations of the species by means of in silico experiments. A nonlinear goal programming technique is employed to identify the kinetic parameters that best satisfy the biological hypothesis of the system behaviour. The simulations are performed on a stochastic simulator for the Calculus of Wrapped Compartments. The extraction of parameters with optimization techniques is usually applied to systems modeled by means of differential equations. Its application in a stochastic framework is rather new. The methodology may open a new path for the investigation of a broader class of biochemical systems (including those that cannot be easily modeled by differential equations) when in vitro experimental measurements are unavailable. The methodology is applied to model the nutrient exchanges in arbuscular mycorrhiza, the most wide-spread plant/fungus symbiosis focusing in particular to a newly discovered ammonium transporter: LjAMT2;2. This case study of ammonia and phosphate plant uptake from the fungus has been dissected in known and hypothetical mechanisms. The results of the analysis provide setting of recent experimental results of carrier LjAMT2;2 and support new insights into biological hypotheses to be put in place.

Lower bound limit analysis by bem: Convex optimization problem and incremental approach

Abstract The lower bound limit approach of the classical plasticity theory is rephrased using the Multidomain Symmetric Galerkin Boundary Element Method, under conditions of plane and initial strains, ideal plasticity and associated flow rule. The new formulation couples a multidomain procedure with nonlinear programming techniques and defines the self-equilibrium stress field by an equation involving all the substructures (bem-elements) of the discretized system. The analysis is performed in a canonical form as a convex optimization problem with quadratic constraints, in terms of discrete variables, and implemented using the Karnak.sGbem code coupled with the optimization toolbox by MatLab. The numerical tests, compared with the iterative elastoplastic analysis via the Multidomain Symmetric Galerkin Boundary Element Method, developed by some of the present authors, and with the available literature, prove the computational advantages of the proposed algorithm.

A microscopic nonlinear programming approach to shakedown analysis of cohesive–frictional composites

A microscopic approach together with nonlinear programming technique and finite element method is developed for shakedown analysis of a composite which has cohesive–frictional constituents. The macroscopic shakedown limit of a composite subject to cyclic loading is calculated in a direct way and the macro–micro relation is quantitatively evaluated. First, by means of the homogenization theory, the classical kinematic theorem of shakedown analysis is generalized to incorporate the microstructure – Representative Volume Element (RVE) chosen from a periodic heterogeneous material. Pressure-dependence and non-associated plastic flow of cohesive–frictional constituent materials are formulated into shakedown analysis. Based on the mathematical programming technique and the finite element method, the numerical micro-shakedown model is finally formulated as a nonlinear programming problem subject to only a few equality constraints, which is solved by a generalized Lagrangian-penalty iterative algorithm. The proposed approach provides a direct approach for determining the reduced macroscopic strength domain of heterogeneous or composite materials due to cyclic loading. Meanwhile, it can capture different plastic behaviors of materials and therefore the developed method has a wide applicability.

Offshore wind farm layout optimization using mathematical programming techniques

Offshore wind power is a renewable energy of growing relevance in current electric energy systems, presenting favorable wind conditions in comparison with the sites on land. However, the higher energy yield has to compensate the increment in installation and maintenance costs, thus the importance of optimizing resources. One relevant aspect to increase profitability is the wind farm layout. The aim of this paper is to propose a new method to maximize the expected power production of offshore wind farms by setting the appropriate layout, i.e. minimizing the wake effects. The method uses a sequential procedure for global optimization consisting of two steps: i) an heuristic method to set an initial random layout configuration, and ii) the use of nonlinear mathematical programming techniques for local optimization, which use the random layout as an initial solution. The method takes full advantage of the most up-to-date mathematical programming techniques while performing a global optimization approach, which can be easily parallelized. The performance of the proposed procedure is tested using the German offshore wind farm Alpha Ventus, located in the North Sea, yielding an increment of expected annual power production of 3.52% with respect to the actual configuration. According to current electricity prices in Germany, this constitutes an expected profit increment of almost 1 M€ per year.

Uncertain Reliability of a Series System via Non-Linear Programming

In this paper, a new algorithm has been introduced to construct the membership function and non-membership function of uncertain reliability of a series system via non-linear programming techniques, which having components following different types of intuitionistic uncertain failure rates.

A Displacement-Based Optimal Design Method of RC Frames Subjected to Minor Earthquakes

A displacement-based optimization design method of RC structure was proposed by combining direct displacement-based design method with nonlinear programming technique. To avert the influence of target displacement, the stationary constraint displacement was presented, and the target displacement can be updated during the optimal design process. Principle of virtual work and Gaussian integral method was employed to simplify the explicit relationship between horizontal displacement and the section dimension. Comparison analysis of the local optimal results corresponding to different displacement shapes was conducted to achieve global optimal design. The numerical tests presented demonstrate the computational advantages of the discussed methods and suggesting that the proposed method is a reliably and efficiently tool for displacement-based optimal design.

Random fuzzy bilevel linear programming through possibility-based value at risk model

This article considers bilevel linear programming problems where random fuzzy variables are contained in objective functions and constraints. In order to construct a new optimization criterion under fuzziness and randomness, the concept of value at risk and possibility theory are incorporated. The purpose of the proposed decision making model is to optimize possibility-based values at risk. It is shown that the original bilevel programming problems involving random fuzzy variables are transformed into deterministic problems. The characteristic of the proposed model is that the corresponding Stackelberg problem is exactly solved by using nonlinear bilevel programming techniques under some convexity properties. A simple numerical example is provided to show the applicability of the proposed methodology to real-world hierarchical problems.

Heat flow control in thermo-magnetic convective systems using engineered magnetic fields

We present the design of a magnetically controlled convective heat transfer system. The underlying thermo-magnetic instability phenomenon is described, and enhanced convective fluid flow patterns are determined using non-linear programming techniques plus a design sensitivity analysis. Specifically, the magnetic fluid body force is computed by finding the optimal distribution and magnetization direction of a magnetic field source, where the objective is to minimize the maximum temperature of a closed loop heat transfer system. Sizeable fluid recirculation zones are induced by arranging magnetic field generation elements in configurations similar to Halbach arrays. Applications include improved heat flow control for electromechanical systems.

Approximate Explicit Nonlinear Receding Horizon Control for Decompression of Divers

This paper is based on a comprehensive dynamic mathematical model (Copernicus) of vascular bubble formation and growth during and after decompression from a dive. The model is founded on the statistical correlation between measurable venous gas emboli (VGE) and risk of severe decompression sickness (DCS) where VGE has been shown to be a reliable and sensitive predictor of decompression stress. By using the Copernicus model the diving decompression problem is formulated as a nonlinear optimal control problem, where the objective is to minimize the total ascend time subject to constraints on the maximum bubbles volume in the central venous pool. A recent study reveals that the optimal solution can be obtained by solving the optimization problem with equality constraints. Inspired by which, a simpler approach using barrier function is proposed in this paper, through which we achieve a more efficient and robust numerical implementation. To reduce the complexity of the nonlinear optimization problem this paper also studies the decompression profile parameterization and its effect. Furthermore, by applying multi-parametric nonlinear programming technique, an approximate explicit solution to the nonlinear optimization problem is obtained, which makes the practical implementation on a typical low-cost diving computer possible.

Comparison of Linear and Nonlinear Programming Techniques for Animal Diet

Linear programming techniques have been extensively used for animal diet formulation for more than last fifty years. To overcome the drawback of linear approximation of objective function for diet formulation, a mathematical model based on nonlinear programming technique is proposed to measure animal performance in terms of milk yield and weight gain. At the second step, it compares the result of proposed program with that of linear programming model. Result of proposed model gives better results using nonlinear programming. Thus the study is an attempt to develop a nonlinear programming model for optimal planning and best use of nutrient ingredients.

A computationally efficient hybrid approach for engineering optimization problems

This article proposes a computationally efficient hybrid search approach for solution of constrained engineering problems by hybridizing differential evolution, a very robust evolutionary algorithm, with sequential quadratic programming, a nonlinear programming technique. Moreover, constraint handling method of standard differential evolution was also modified by embedding a pareto based ranking scheme in the standard differential evolution algorithm. The performance of proposed hybrid algorithm was evaluated based on a set of well-known mathematical and engineering benchmark problems. The comparison of results with those of other evolutionary algorithms demonstrates that the proposed method is highly competitive in terms of solution quality and that the proposed algorithm is computationally efficient as it requires much lower number of function evaluations to achieve the same performance.

Nonlinear programming optimization of filler shapes for composite materials with inverse problem technique to maximize heat conductivity

Adding fillers to materials of low thermal conductivity has been recognized as an efficient way to increase the thermal conductivity of composite materials. The filler shapes have great impacts on heat conduction in the materials, which can be analyzed by a direct heat conduction problem. However what is the best shape? This study answers this inverse problem by a nonlinear optimization programming technique. The problem is first standardized to the optimization of a nonlinear objective function with nonlinear constraints. Then it is solved by SQP (sequence quadratic nonlinear programming) numerical scheme. The optimum filler shape parameters are obtained. The results are that the best shapes are “I” shapes. Further, the optimization process can be regarded as a filler growth process. Depending on the fillers thermal conductivity and filler volume contents, the finally formed optimum shapes can be classified into three categories: the handicapped dumbbells, the infant less developed “I” shapes, and adult fully developed “I” shapes. Composite materials made of paraffin wax and steel fillers of six shapes are prepared to validate the optimization results.

An Analytical Placer for VLSI Standard Cell Placement

Placement is the process of determining the exact locations of circuit elements within a chip. It is a crucial step in very large scale integration (VLSI) physical design, because it affects routability, performance, and power consumption of a design. In this paper, we develop a new analytical placer to solve the VLSI standard cell placement problem. The placer consists of two phases, multilevel global placement (GP) and detailed cell placement (DP). In the stage of GP, during the clustering stage, we use a nonlinear programming technique and a best-choice clustering algorithm to take a global view of the whole netlist and placement information, and then use an iterative local refinement technique during the declustering stage to further distribute the cells and reduce the wirelength. In the stage of DP, we develop a fast legalization algorithm to make the solution by global placement legal and use a cell order polishing to improve the legal solution. The proposed algorithm is tested on the IBM standard cell benchmark circuits and Peko suites. Experimental results show that our placer obtains high-quality results in a reasonable running time.

Robust flutter analysis based on genetic algorithm

Robust flutter analysis considering model uncertain parameters is very important in theory and engineering applications. Modern robust flutter solution based on structured singular value subject to real parametric uncertainties may become difficult because the discontinuity and increasing complexity in real mu analysis. It is crucial to solve the worst-case flutter speed accurately and efficiently for real parametric uncertainties. In this paper, robust flutter analysis is formulated as a nonlinear programming problem. With proper nonlinear programming technique and classical flutter analysis method, the worst-case parametric perturbations and the robust flutter solution will be captured by optimization approach. In the derived nonlinear programming problem, the parametric uncertainties are taken as design variables bounded with perturbed intervals, while the flutter speed is selected as the objective function. This model is optimized by the genetic algorithm with promising global optimum performance. The present approach avoids calculating purely real mu and makes robust flutter analysis a plain job. It is illustrated by a special test case that the robust flutter results coincide well with the exhaustive method. It is also demonstrated that the present method can solve the match-point robust flutter solution under constant Mach number accurately and efficiently. This method is implemented in problem with more uncertain parameters and asymmetric perturbation interval.

Computer-Aided Design and Optimization of High-Efficiency LLC Series Resonant Converter

High conversion efficiency is desired in switch mode power supply converters. Computer-aided design optimization is emerging as a promising way to design power converters. In this work a systematic optimization procedure is proposed to optimize LLC series resonant converter full load efficiency. A mode solver technique is proposed to handle LLC converter steady-state solutions. The mode solver utilizes numerical nonlinear programming techniques to solve LLC-state equations and determine operation mode. Loss models are provided to calculate total component losses using the current and voltage information derived from the mode solver. The calculated efficiency serves as the objective function to optimize the converter efficiency. A prototype 300-W 400-V to 12-V LLC converter is built using the optimization results. Details of design variables, boundaries, equality/inequality constraints, and loss distributions are given. An experimental full-load efficiency of 97.07% is achieved compared to a calculated 97.4% efficiency. The proposed optimization procedure is an effective way to design high-efficiency LLC converters.

A continuous approximation approach for the integrated facility-inventory allocation problem

In today’s retail business many companies have a complex distribution network with several national and regional distribution centers. This article studies an integrated facility location and inventory allocation problem for designing a distribution network with multiple distribution centers and retailers. The key decisions are where to locate the regional distribution centers (RDCs), how to assign retail stores to RDCs and what should be the inventory policy at the different locations such that the total network cost is minimized. Due to the complexity of the problem, a continuous approximation (CA) model is used to represent the network. Nonlinear programming techniques are developed to solve the optimization problems. The main contribution of this work lies in developing a new CA modeling technique when the discrete data cannot be modeled by a continuous function and applying this technique to solve an integrated facility location-allocation and inventory-management problem. Our methodology is illustrated with the network from a leading US retailer. Numerical analysis suggests that the total cost is significantly lower in the case of the integrated model as compared with the non-integrated model, where the location-allocation and inventory-management problems are considered separately. This paper also studies the effects of changing parameter values on the optimal solutions and to point out some management implications.