Gradient-based Nonlinear Programming Research Articles

Optimization of Planar Phased Arrays for Vehicles

Radar sensors are used for the detection of objects surrounding a vehicle to enable advanced driver assistance systems. Wide angle azimuth field-of-view (FOV) sensor having high angular resolution is beneficial, but presents challenges for low-cost implementation where element-to-element spacing may be larger than one-half wavelength. Irregular placement of array elements in a two-dimensional (2-D) array can achieve low-cost targets, but may generate large sidelobes if done without care. We introduce an optimization method to determine placement of array elements in a vehicular volumetric beam-scanning radar to minimize sidelobe power for typical automotive beam steering angles. Optimization with array constraints is realized using a gradient-based nonlinear programming technique. Optimization is accelerated by extending <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uv</i> -projection planes beyond the unit circle. For a grating lobe-free scan area based on vehicular applications, the azimuth scanning FOV is increased by 11° bidirectionally to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> 54° compared to a reference triangular grid array with 24 elements. The validity of an antenna design generated using the optimization method is established by a three-dimensional (3-D) electromagnetic wave simulation of a representative optimized array configuration.

Omnidirectional acoustic cloaking against airborne sound realized by a locally resonant sonic material

We report that a locally resonant sonic material realizes omnidirectional acoustic invisibility in air. To achieve acoustic cloaking in the low-frequency regime, we axisymmetrically placed elastic rods comprised of silicone rubber and lead around a cloaked object. The radii of the rods are designed to minimize their total scattering cross section for a given frequency. The optimization is performed using an algorithm incorporating multiple scattering theory and gradient-based nonlinear programming. We numerically confirmed that the designed cloaking device suppressed the scattering cross section by almost 92% for all incident directions at the target frequency.

Complementary Sequences and Receiving Filters Design for Suppressing Interrupted Sampling Repeater Jamming

The realistic false targets caused by interrupted sampling repeater jamming (ISRJ) can mask the real target, leading to the failure of radar target detection. In this letter, a method based on jointly designing complementary sequences and receiving filters under the signal-to-noise ratio loss constraint is proposed to suppress ISRJ. A gradient-based nonlinear programming solver and the Lagrange multiplier method are used to optimize the sequences and filters alternately. Numerical results show that the proposed method can generate sequences and receiving filters with good correlation properties and anti-ISRJ performance.

Optimum location of outrigger in tall buildings using finite element analysis and gradient-based optimization method

In this study, the optimum locations of outriggers minimizing the top drift of tall buildings are identified using a gradient-based nonlinear programming approach. Since the proposed optimization method uses finite element analysis to evaluate the objective function for arbitrarily configured buildings, the discontinuity due to design variables (such as outrigger locations) is overcome by piecewise linear and quadratic interpolation functions. A series of optimum designs for three analysis models with different vertical profiles were performed to investigate the relation between outrigger stiffness and optimum location by changing the cross-sectional area of outriggers. As outrigger cross-sectional area increases, the optimum outrigger location moves down in the structure. The design result demonstrates that correlation between outrigger stiffness in a practical range and its optimum location is negligible; thus, for design purposes, variables related to outrigger optimum location and its stiffness can be dealt with separately.

Fuzzy production-inventory models with shortages via ranking technique and credibility measure

In this paper, a production-inventory model for a defective/deteriorating item is developed with imprecise inventory cost parameters. Here, we have used two defuzzification techniques: 1) valuation/weighting function of Yager's ranking method; 2) credibility measure, a compromise between optimistic and pessimistic measures for the deterministic transformation of the proposed model. A weighting function is added to Yager's ranking method so that it can represent the decision preference to solve a production inventory model. The proposed fuzzy inventory models are reduced to the crisp ones applying credibility measure. The problems are formulated as cost minimisation problems and solved using a gradient-based nonlinear programming technique. Finally, the models are illustrated through a numerical experiment with triangular fuzzy cost parameters. The results from two methods are compared and sensitivity analysis for different values of cost parameters and confidence levels (in credibility measure) is presented.

Optimization of Water-Supply and Hydropower Reservoir Operation Using the Charged System Search Algorithm

The Charged System Search (CSS) metaheuristic algorithm is introduced to the field of water resources management and applied to derive water-supply and hydro-power operating policies for a large-scale real-world reservoir system. The optimum algorithm parameters for each reservoir operation problems are also obtained via a tuning procedure. The CSS algorithm is a metaheuristic optimization method inspired by the governing laws of electrostatics in physics and motion from the Newtonian mechanics. In this study, the CSS algorithm’s performance has been tested with benchmark problems, consisting of highly non-linear constrained and/or unconstrained real-valued mathematical models, such as the Ackley’s function and Fletcher–Powell function. The CSS algorithm is then used to optimally solve the water-supply and hydropower operation of “Dez” reservoir in southern Iran over three different operation periods of 60, 240, and 480 months, and the results are presented and compared with those obtained by other available optimization approaches including Genetic Algorithm (GA), Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO) and Constrained Big Bang–Big Crunch (CBB–BC) algorithm, as well as those obtained by gradient-based Non-Linear Programming (NLP) approach. The results demonstrate the robustness and superiority of the CSS algorithm in solving long term reservoir operation problems, compared to alternative methods. The CSS algorithm is used for the first time in the field of water resources management, and proves to be a robust, accurate, and fast convergent method in handling complex problems in this filed. The application of this approach in other water management problems such as multi-reservoir operation and conjunctive surface/ground water resources management remains to be studied.

Multi-Objective Optimization-Simulation for Reliability-Based Inter-Basin Water Allocation

A simulation-optimization framework is presented for reliability-based optimal sizing, operation, and water allocation in the Bashar-to-Zohreh inter-basin water transfer project. The problem was formulated as a mixed integer nonlinear program (MINLP), for which two solution approaches were tested, including gradient-based nonlinear programming and simulation-optimization (SO). The SO framework linked the water evaluation and planning system (WEAP) simulation module, benefiting from fast, single-period linear programming, to the multi-objective particle swarm optimization (MOPSO) for multiperiod optimization. The objective functions were minimizing the sizes of the project’s infrastructures and maximizing the reliability of supplying water to agricultural lands. The combination of nonlinear programming and the branch-and-bound algorithm was not able to solve the resulting MINLP. The results of the MOPSO-WEAP framework indicated that the project can supply water for land development in the Dehdasht and Choram agricultural plains, located in the less developed Kohgiluye and Boyer-Ahmad Province of Iran at an acceptable reliability level. For example, for one of the Pareto solutions found corresponding to maximum land development (30,000 ha), an average volume of 237 million cubic meter (MCM) is transferred annually at a 73.2% reliability level with average sizes of water transfer and storage elements. Further, design-operation and hydropower scenarios were also evaluated, and the Pareto solutions were obtained.

Exact sensitivity analysis of stresses and lightweight design of Timoshenko composite beams

The paper describes the novel optimization techniques for lightweight design of composite beams. The optimization model is to find width and depth of composite beams to minimize the mass of beams under the stiffness, strength and delamination failure constraints. The exact formulae for displacements, stresses and their sensitivities with respect to width and depth are derived using Timoshenko continuous beam theory. The analytical stiffness, strength and delamination failure functions, and their gradients are obtained using the exact expressions of displacements and stresses, and their sensitivities. The mass and its gradient are also expressed analytically. The standard gradient-based nonlinear programming algorithms are employed to solve lightweight design problems of composite beams. The lightweight designs of composite beams are performed using the proposed optimization techniques. Three standard gradient-based optimization methods (SQP, interior-point and active-set) using exact derivatives converge to the same lightweight design. However, gradient-based algorithms using finite difference sensitivities may not lead to optimal lightweight designs. It is necessary to develop exact sensitivity analysis method instead of the difference methods for gradient-based algorithms.

Optimization Alternatives for Robust Model-based Design of Synthetic Biological Circuits

Synthetic biology is reaching the situation where tuning devices by hand is no longer possible due to the complexity of the biological circuits being designed. Thus, mathematical models need to be used in order, not only to predict the behavior of the designed synthetic devices; but to help on the selection of the biological parts, i.e., guidelines for the experimental implementation. However, since uncertainties are inherent to biology, the desired dynamics for the circuit usually requires a trade-off among several goals. Hence, a multi-objective optimization design (MOOD) naturally arises to get a suitable parametrization (or range) of the required kinetic parameters to build a biological device with some desired properties. Biologists have classically addressed this problem by evaluating a set of random Monte Carlo simulations with parameters between an operation range. In this paper, we propose solving the MOOD by means of dynamic programming using both a global multi-objective evolutionary algorithm (MOEA) and a local gradient-based nonlinear programming (NLP) solver. The performance of both alternatives is then checked in the design of a well-known biological circuit: a genetic incoherent feed-forward loop showing adaptive behavior.

A Hybrid Optimization Technique Applied to the Intermediate-Target Optimal Control Problem

The DoD has introduced the concept of Manned-Unmanned Teaming, a subset of which is the loyal wingman. Optimal control techniques have been proposed as a method for rapidly solving the intermediate-target (mid-point constraint) optimal control problem. Initial results using direct orthogonal collocation and a gradient-based method for solving the resulting nonlinear program reveals a tendency to converge to or to get `stuck’ in locally optimal solutions. The literature suggested a hybrid technique in which a particle swarm optimization is used to quickly find a neighborhood of a more globally minimal solution, at which point the algorithm switches to a gradient-based nonlinear programming solver to converge on the globally optimal solution. The work herein applies the hybrid optimization technique to rapidly solve the loyal wingman optimal control problem. After establishing the background and describing the loyal wingman particle swarm optimization algorithm, the problem is solved first using the gradient-based direct orthogonal collocation method, then re-solved using a hybrid approach in which the results of the particle swarm optimization algorithm are used as the initial guess for the gradient-based direct orthogonal collocation method. Results comparing the final trajectory and convergence time, demonstrate the hybrid technique as a reliable method for producing rapid, autonomous, and feasible solutions to the loyal wingman optimal control problem.

Multi-Period Dynamic Optimization for Large-Scale Differential-Algebraic Process Models under Uncertainty

A technique for optimizing large-scale differential-algebraic process models under uncertainty using a parallel embedded model approach is developed in this article. A combined multi-period multiple-shooting discretization scheme is proposed, which creates a significant number of independent numerical integration tasks for each shooting interval over all scenario/period realizations. Each independent integration task is able to be solved in parallel as part of the function evaluations within a gradient-based non-linear programming solver. The focus of this paper is on demonstrating potential computation performance improvement when the embedded differential-algebraic equation model solution of the multi-period discretization is implemented in parallel. We assess our parallel dynamic optimization approach on two case studies; the first is a benchmark literature problem, while the second is a large-scale air separation problem that considers a robust set-point transition under parametric uncertainty. Results indicate that focusing on the speed-up of the embedded model evaluation can significantly decrease the overall computation time; however, as the multi-period formulation grows with increased realizations, the computational burden quickly shifts to the internal computation performed within the non-linear programming algorithm. This highlights the need for further decomposition, structure exploitation and parallelization within the non-linear programming algorithm and is the subject for further investigation.

Multi-item fuzzy-stochastic supply chain models for long-term contracts with a profit sharing scheme

In this study, we develop multi-item multi-objective manufacturer–retailer supply chain models with risk and budget constraints for long-term contracts with a profit sharing scheme in a fuzzy-stochastic environment. The manufacturing costs of the items are fuzzy and the demands for the items are random during each period. We examine two separate models with crisp and probabilistic budget constraints for long-term contracts where the business relationship is repeated, while demand is uncertain in a stochastic sense. The models are formulated as profit maximization problems. Next, the fuzzy compromise programming method, global criteria method, and weighted sum method are used to reduce the multi-objective problems into their corresponding single objective problems. Finally, the problems are solved using a gradient-based nonlinear programming technique: the generalized reduced gradient method. The models are illustrated with numerical examples for different types of stochastic demand and a comparative study is presented.

An integrated model for lot sizing with supplier selection and quantity discounts

Good inventory management is essential for a firm to be cost competitive and to acquire decent profit in the market, and how to achieve an outstanding inventory management has been a popular topic in both the academic field and in real practice for decades. As the production environment getting increasingly complex, various kinds of mathematical models have been developed, such as linear programming, nonlinear programming, mixed integer programming, geometric programming, gradient-based nonlinear programming and dynamic programming, to name a few. However, when the problem becomes NP-hard, heuristics tools may be necessary to solve the problem. In this paper, a mixed integer programming (MIP) model is constructed first to solve the lot-sizing problem with multiple suppliers, multiple periods and quantity discounts. An efficient Genetic Algorithm (GA) is proposed next to tackle the problem when it becomes too complicated. The objectives are to minimize total costs, where the costs include ordering cost, holding cost, purchase cost and transportation cost, under the requirement that no inventory shortage is allowed in the system, and to determine an appropriate inventory level for each planning period. The results demonstrate that the proposed GA model is an effective and accurate tool for determining the replenishment for a manufacturer for multi-periods.

Damage tolerance based design optimisation of a fuel flow vent hole in an aircraft structure

This paper investigates the design optimisation of a fuel flow vent hole (FFVH) located in the wing pivot fitting (WPF) of an F-111 aircraft assuming a damage tolerance design philosophy. The design of the vent hole shape is undertaken considering the basic durability based design objectives of stress, residual (fracture) strength, and fatigue life. Initially, a stress based optimised shape is determined. Damage tolerance based design optimisation is then undertaken to determine the shape of the cutout so as to maximise its residual strength and fatigue life. For stress optimisation, the problem is analysed using the gradient-less biological algorithm and the gradient-based nonlinear programming methods. The optimum designs predicted by the two fundamentally different optimisation algorithms agree well. The optimum shapes of the vent hole are subsequently determined considering residual strength and fatigue life as the distinct design objectives in the presence of numerous 3D cracks located along the vent hole boundary. A number of crack cases are considered to investigate how the crack size affects the optimal shapes. A semi-analytical method is employed for computation of the stress intensity factors (SIF), and an analytical crack closure model is subsequently used to evaluate the fatigue life. The 3D biological algorithm is used for designing the cutout profiles that optimise residual strength and fatigue life of the component. An improved residual strength/fatigue life (depending on the optimisation objective) is achieved for the optimal designs. The variability in SIF/fatigue life around the cutout boundary is reduced, thereby making the shape more evenly fracture/fatigue critical. The vent hole shapes optimised for stress, residual strength, and fatigue life are different from each other for a given nature and size of the flaws. This emphasises the need to consider residual strength and/or fatigue life as the explicit design objective. The durability based optimal vent hole shapes depend on the initial and final crack sizes. It is also shown that a damage tolerance optimisation additionally produces a reduced weight WPF component, which is highly desirable for aerospace industries. The design space near the ‘optimal’ region is found to be flat. This allows us to achieve a considerable enhancement in fatigue performance without precisely identifying the local/global optimum solution, and also enables us to select a reduced weight ‘near optimal’ design rather than the precise optimal shape.

Inverse design techniques for improving composite prosthetic devices

The design and performance of composite prosthetic devices can be improved by tailoring the material properties to achieve a prescribed response. An example of such a response would be displacements and stresses exhibited by healthy, undisturbed femoral bone. In this paper, an inverse design methodology, used in the Volumetrically Controlled Manufacturing (VCM) process, is developed and tested for improving the design of orthopedic prosthetic devices. First, a three-dimensional finite element (FE) model is developed based on available Computed Tomography (CT) data. The FE model is used to evaluate the response of the model subjected to a typical load. Second, as a part of the VCM process, the inverse design process is used to formulate a design problem that is in the form of a constrained least-squares problem. The intent is to find the material properties of the FE model to obtain a known displacement field on the stem-cancellous interface. Third, a solution methodology is developed to solve this constrained least squares problem using the finite element analysis for function evaluations and a gradient-based nonlinear programming (NLP) method to solve the design problem. Two test problems are solved to illustrate the developed methodology. The results indicate that material properties can be tailored to meet specific response requirements.

Solving nonlinear water management models using a combined genetic algorithm and linear programming approach

Gradient-based nonlinear programming (NLP) methods can solve problems with smooth nonlinear objectives and constraints. However, in large and highly nonlinear models, these algorithms can fail to find feasible solutions, or converge to local solutions which are not global. Evolutionary search procedures in general, and genetic algorithms (GAs) specifically, are less susceptible to the presence of local solutions. However, they often exhibit slow convergence, especially when there are many variables, and have problems finding feasible solutions in constrained problems with “narrow” feasible regions. In this paper, we describe strategies for solving large nonlinear water resources models management, which combine GAs with linear programming. The key idea is to identify a set of complicating variables in the model which, when fixed, render the problem linear in the remaining variables. The complicating variables are then varied by a GA. This GA&LP approach is applied to two nonlinear models: a reservoir operation model with nonlinear hydropower generation equations and nonlinear reservoir topologic equations, and a long-term dynamic river basin planning model with a large number of nonlinear relationships. For smaller instances of the reservoir model, the CONOPT2 nonlinear solver is more accurate and faster, but for larger instances, the GA&LP approach finds solutions with significantly better objective values. The multiperiod river basin model is much too large to be solved in its entirety. The complicating variables are chosen here so that, when they are fixed, each period's model is linear, and these models can be solved sequentially. This approach allows sufficient model detail to be retained so that long-term sustainability issues can be explored.

Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: A geometric programming approach

A multi-item inventory model with constant demand and infinite replenishment is developed under the restrictions on storage area, total average shortage cost and total average inventory investment cost. These restrictions may be precise or imprecise. Here, it is assumed that inventory costs are directly proportional to the respective quantities, and unit purchase/production cost is inversely related to the demand. Restricted shortages are allowed but fully backlogged. First, the problem is formulated in crisp environment taking the deterministic and precise inventory parameters. It is solved by both geometric programming (GP) and gradient-based non-linear programming (NLP) methods. Later, the problem is formulated with fuzzy goals on constraints and objectives where impreciseness is introduced through linear membership functions. It is solved using the fuzzy geometric programming (FGP) method. The inventory models are illustrated with numerical values and compared with the crisp results. A sensitivity analysis on the optimum order quantity and average cost is also presented due to the variation in the tolerance of total average inventory investment cost and total average shortage cost following Dutta et al., 1993, Fuzzy Sets and Systems, 55, 133-142.

Enhancements to Genetic Algorithm for Optimal Ground-Water Management

Genetic algorithm (GA) is considered to be a robust technique for solving ground-water optimization problems. Often these problems are difficult to solve using traditional gradient-based techniques as these are nonlinear, nonconvex, and discontinuous. In this manuscript, recent research related to application of GA in solving these problems is critically reviewed, and three areas of potential enhancement to GA are identified and explored. These enhancement methods to GA are fitness reduction method (FRM), search bound sampling method (SBSM), and optimal resource allocation guideline (ORAG). In order to assess these methods, a nonlinear ground-water problem with fixed and variable costs is selected (from literature) where the corresponding optimal solution using a gradient-based nonlinear programming (NLP) technique is available. The problem is resolved using GA coupled with the enhancement methods, and the GA solutions are compared with the NLP solution. In addition, the sensitivity of these methods to various GA parameters are studied. The results of the analysis using the enhancement methods show the following: (1) FRM enhances efficiency of GA in handling constraints; (2) SBSM enhances accuracy of GA in solving problems with fixed costs; (3) ORAG enhances reliability of GA by providing some convergence guarantee for a given computational resource; and (4) when applied with FRM and SBSM, accuracy of GA is marginally increased from near-optimal to global-optimal by tuning any one of the several parameters. There appears to be little need to embark on a full-scale analysis for achieving the increase.

Use of Fracture Mechanics and Shape Optimization for Component Designs

About half of mechanical failures are due to repeated loading. Most of the failures are fatigue related, and the fatigue prohlem has been of major concern in the design of structures. The failure is a function of, among other factors, the external loads, material behavior, geometry of the structure, and crack characteristics. The relationship between structural geometry and number of life cycles to failure is investigated to improve the fatigue life of structural components. The linear elastic fracture mechanics (LEFM) approach is integrated with shape optimal design methodology. The primary ohjective of the design problem is to enhance the life of the structure. The results from LEFM analyses are used in the fatigue model to predict the life of the structure before failure is deemed to have occurred. The shape of the structure is changed using the natural shape optimal design methodology. Gradient-based nonlinear programming techniques are used with the computed sensitivity information to predict the required shape changes. Relevant issues such as problem formulation, finite element modeling, mesh generation, and regeneration are discussed. Two design examples are solved, and the results show that, with proper shape changes, the life of structural systems subjected to fatigue loads can be enhanced significantly.

Application of the first-order perturbation method to optimal structural design

An application of the perturbation method to optimum structural design with random parameters is presented. It is formulated on the basis of the first-order stochastic finite element perturbation method. It also takes into full account the stress, displacement and eigenvalue constraints, together with the rates of change of the random variables. A method for calculating the sensitivity coefficients in regard to the governing equation and the first-order perturbed equation has been derived, by using a direct differentiation approach. A gradient-based nonlinear programming technique is used to solve the problem. The numerical results are specifically noted, where the stiffness parameter and external load are treated as random variables.