Nonlinear Programming Code Research Articles

Elasto-plastic analysis and optimal design of composite integral abutment bridge extended with limited residual plastic deformation

Due to the growing significance of structural theories concerning the composite structure analysed and designed plastically, this paper introduces a new optimisation method for controlling the plastic behaviour of a full-scale composite integral abutment bridge by employing complementary strain energy of residual forces that existed within the reinforcing rebars. Composite bridges are structures made of components such as steel and concrete, which are frequent and cost-effective building methods. Thus, various objective functions were used in this work when applying optimum elasto-plastic analysing and designing the composite integrated bridge structure that was tested experimentally in the laboratory. In contrast, the plastic deformations were constrained by restricting the complementary strain energy of the residual internal forces aiming to find the maximum applied load and the minimum number of steel bars used to reinforce the concrete column part of the structure. The numerical model employed in this paper was validated and calibrated using experimental results, which were considered inside ABAQUS to produce the validated numerical model, using concrete damage plasticity (CDP) constitutive model and concrete data from laboratory testing to solve the nonlinear programming code provided by the authors. This paper presents a novel optimization method using complementary strain energy to control the plastic behaviour of a full-scale composite integral abutment bridge, with the original contribution being the incorporation of residual forces within reinforcing rebars to limit plastic deformations. Following that, a parametric investigation of the various optimisation problems revealed how models performed variously under different complementary strain energy values, which influenced the general behaviour of the structure as it transitioned from elastic to elasto-plastic to plastic; also results showed how the complementary strain energy value is connected with the amount of damaged accrued in both concrete and steel.

Globally Optimal Algorithm of Superconducting Magnet Design for Gyrotrons

This paper presentsa new hybrid global optimization method to design a 25 T All YBCO solenoid superconducting magnet with a warm-bore diameter of 52 mm for high-frequency gyrotrons in Wuhan National High Magnetic Field Center, China. This method includes 3 main routines: linear programming (LP) code to get the non-rectangular coil design, coil-picking code to get rectangular coil design, and nonlinear programming (NLP) code to get final integer coil design by using rectangular coil design as initial input. The designed magnet includes Inner, Middle and Outer coils which consist of 34, 56 and 68 double pancakes (DP) wound by 4 mm x 0.1 mm bare YBCO tape, respectively. The designed peak-peak field in-homogeneity is 50 ppm within 10 mm Diameter of Spherical Volume (DSV). The designed 5 Gauss line is restricted in a cylinder with the sizes of 10 m in axial direction and 8 m in radial direction, respectively. The maximum hoop stress is limited below 450 MPa. Layer-layer separation happens in the inner coil since non-insulation (NI) winding is adopted. The results show that this methodology is very efficient for High-frequency Gyrotron superconducting magnet design. This methodology also can be used for other solenoid superconducting magnet design, such as Magnetic Resonance Imaging (MRI) superconducting magnet and Nuclear Magnetic Resonance (NMR) superconducting magnet.

Mesh Adaptation Method for Optimal Control With Non-Smooth Control Using Second-Generation Wavelet

A mesh adaptation method is proposed for solving optimal control problems with non-smooth control. The original optimal control problem (OCP) is transcribed into a nonlinear programming (NLP) problem by using the Runge-Kutta discretization method, in which the NLP can be solved by using standard nonlinear programming codes. The method employs collocations from the dyadic background points, which used for the second-generation wavelet (SGW) translation simultaneously. The SGW is used to approximate the control variables and get the wavelet coefficients once they are obtained. In regions contain discontinuities, the magnitude of the relevant wavelet coefficients is large than other regions. The corresponding dyadic background points are reserved as the collocation points. Furthermore, the approximation error of the control and/or state variables can be predicted by a given threshold. Thus, the accuracy and efficiency can be balanced in a simple way. Finally, the method is demonstrated by three numerical examples from the open literature.

Interval Limit Analysis Within a Scaled Boundary Element Framework

The paper proposes a novel approach for the interval limit analysis of rigid-perfectly plastic structures with (nonprobabilistic) uncertain but bounded forces and yield capacities that vary within given continuous ranges. The discrete model is constructed within a polygon-scaled boundary finite element framework, which advantageously provides coarse mesh accuracy even in the presence of stress singularities and complex geometry. The interval analysis proposed is based on a so-called convex model for the direct determination of both maximum and minimum collapse load limits of the structures involved. The formulation for this interval limit analysis takes the form of a pair of optimization problems, known as linear programs with interval coefficients (LPICs). This paper proposes a robust and efficient reformulation of the original LPICs into standard nonlinear programming (NLP) problems with bounded constraints that can be solved using any NLP code. The proposed NLP approach can capture, within a single step, the maximum collapse load limit in one case and the minimum collapse load limit in the other, and thus eliminates the need for any combinatorial search schemes.

Response Bounds of Elastic Structures in the Presence of Interval Uncertainties

A mathematical-programming (MP)-based approach is proposed to directly determine the two extreme [(1) minimum, and (2) maximum] bounds of some static response quantity for elastic structures subject to the simultaneous application of interval applied forces and interval elastic material properties. Such a direct determination scheme reformulates the interval finite-element (FE) analysis into a pair of standard nonlinear programming (NLP) problems that can be solved by any available NLP code. Not only does the proposed method advantageously bypass any computationally expensive combinatorial searches to capture the response limits, but it can also directly accommodate any dependency in the interval data. Some examples are presented to illustrate the robustness and efficiency of the method.

Sample average approximation method for the chance-constrained stochastic programming in the transportation model of emergency management

This study proposes a stochastic programming model for the transportation of emergency resource during the emergency response. Since it is difficult to predict the timing and magnitude of any disaster and its impact on the urban system, resource mobilisation is treated in a random manner, and the resource requirements are represented as random variables. Randomness is represented by the chance constraints in this paper. To deal with the difficulty in calculating the chance constraint function, we use conditional value at risk (CVaR) to approximate the chance constraint, and solve the approximation problem of the chance–constrained stochastic programming by using the sample average approximation (SAA) method. For a given sample, the SAA problem is a deterministic nonlinear programming (NLP) and any appropriate NLP code can be applied to solve the problem. The model and method provide a new way for the emergency logistics management engineering.

A Superlinearly Convergent Penalty Method with Nonsmooth Line Search for Constrained Nonlinear Least Squares

Recently, we have presented a projected structured algorithm for solving constrained nonlinear least squares problems, and established its local two-step Q-superlinear convergence. The approach is based on an adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method. The structured adaptation also makes use of the ideas of Nocedal and Overton for handling the quasi-Newton updates of projected Hessians and appropriates the structuring scheme of Dennis, Martinez and Tapia. Here, for robustness, we present a specific nonsmooth line search strategy, taking account of the least squares objective. We also discuss the details of our new nonsmooth line search strategy, implementation details of the algorithm, and provide comparative results obtained by the testing of our program and three nonlinear programming codes from KNITRO on test problems (both small and large residuals) from Hock and Schittkowski, Lukšan and Vlček and some randomly generated ones due to Bartels and Mahdavi-Amiri. The results indeed affirm the practical relevance of our special considerations for the inherent structure of the least squares.

Constrained Nonlinear Least Squares: A Superlinearly Convergent Projected Structured Secant Method

Numerical solution of nonlinear least-squares problems is an important computational task in science and engineering. Effective algorithms have been developed for solving nonlinear least squares problems. The structured secant method is a class of efficient methods developed in recent years for optimization problems in which the Hessian of the objective function has some special structure. A primary and typical application of the structured secant method is to solve the nonlinear least squares problems. We present an exact penalty method for solving constrained nonlinear leastsquares problems, when the structured projected Hessian is approximated by a projected version of the structured BFGS formula and give its local two-step Q-superlinear convergence. For robustness, we employ a special nonsmooth line search strategy, taking account of the least squares objective. We discuss the comparative results of the testing of our programs and three nonlinear programming codes from KNITRO on some randomly generated test problems due to Bartels and MahdaviAmiri. Numerical results also confirm the practical relevance of our special considerations for the inherent structure of the least squares.

Sample Average Approximation Method for Chance Constrained Stochastic Programming in Transportation Model of Emergency Management

In this paper, an optimal model for the transportation of emergency resource is established on chance constrained stochastic programming. We use Conditional Value at Risk (CVaR) to approximate the chance constraint, and we get the approximation problem of the chance constrained stochastic programming by using the sample average approximation (SAA) method. For a given sample, the SAA problem is a deterministic nonlinear programming (NLP) and any appropriate NLP code can be applied to solve the problem. The model and method provide a new way for the emergency logistics management engineering.

Trajectory Optimization Using Multiresolution Techniques

We present a multi-resolution-based approach for solving trajectory optimization problems. The original optimal control problem is solved using a direct method, thereby being transcribed into a nonlinear programming problem that is solved using standard nonlinear programming codes. The novelty of the proposed approach hinges on the automatic calculation of a suitable nonuniform grid over which the nonlinear programming problem is subsequently solved. This tends to increase numerical efficiency and robustness. Control and/or state constraints are handled with ease and without any additional computational complexity. The proposed algorithm is based on a simple and intuitive method to balance conflicting objectives, such as accuracy of the solution, convergence, and speed of computations. The benefits of the proposed algorithm over uniform grid implementations are demonstrated with the help of several nontrivial examples.

Optimal power split in a hybrid electric vehicle using direct transcription of an optimal control problem

To efficiently operate electromechanical systems powered by two energy sources, it is necessary to determine the instantaneous power split between sources in order to minimize the energy consumption of the whole system. In this work, this problem is posed as a nonlinear finite horizon optimal control problem with control and state constraints and is solved using a direct transcription approach. The problem is fully discretized in time and the resulting finite dimensional optimization problem is solved using a nonlinear programming code. This paper describes the application of direct transcription to the case of the hybrid electric vehicle (HEV) being developed in the Applied Electronics Group (GEA) at the University of Río Cuarto. The statement and discretization of the control problem, the setting for using the nonlinear programming code and several examples and comparisons with those obtained by other approaches are described.

Extremal problems for convex polygons

Consider a convex polygon V n with n sides, perimeter P n , diameter D n , area A n , sum of distances between vertices S n and width W n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.

A polyhedral branch-and-cut approach to global optimization

A variety of nonlinear, including semidefinite, relaxations have been developed in recent years for nonconvex optimization problems. Their potential can be realized only if they can be solved with sufficient speed and reliability. Unfortunately, state-of-the-art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software.In this paper, we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch-and-cut approach. Our algorithm exploits convexity, either identified automatically or supplied through a suitable modeling language construct, in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems. We prove that, if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions, our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function. The convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited.Root-node relaxations are computed for 87 problems from globallib and minlplib, and detailed computational results are presented for globally solving 26 of these problems with BARON 7.2, which implements the proposed techniques. The use of cutting planes for these problems reduces root-node relaxation gaps by up to 100% and expedites the solution process, often by several orders of magnitude.

Optimal Design of Electronic Components by Mixed-Integer Nonlinear Programming

Computer-aided design optimization of electronic components is a powerful tool to reduce development costs on one hand and to improve the performance of the components on the other. In this paper, a mathematical model of an electronic filter is outlined. It depends on certain parameters, some of them of being continuous, others of integer type. The purpose of the paper is to introduce an extension of the well-known sequential quadratic programming (SQP) method to solve the mixed-integer programming problem (MINLP). It is assumed that the integer variables cannot be relaxed to real ones, that the integer range is sufficiently large, and that they possess some physical meaning so that they basically behave like continuous ones. The general idea is to combine an SQP step with a direct search cycle in the integer space. Hessian information is updated based on difference formulae at neighbored grid points. Numerical results are included to show the feasibility of the mixed-integer nonlinear programming code for academic test examples and in addition for the optimal design of an electronic filter.

Application of nonlinear programming to explosively formed projectile design

The methodology of explosively formed projectile(EFP) design by way of nonlinear programming has been investigated. A nonlinear programming model to design an EFP with expected shape has been set up. The shape of the generator of liner surface is described with a function in the form of a circular arc added with amending terms. An efficient BFGS variable metric sequential programming algorithm was implemented into the nonlinear programming code SQP, which was then coupled with the finite element code DYNA2D. With the resulting code, namely SQPDYNA, two EFPs were optimized, one with flared tail and the other without. Both of them met quite well the preset requirements. Measured EFP velocity and contours agree well with the optimization results.

On the numerical solution of a class of unilateral contact structural optimization problems

This paper deals with a special class of structural optimization problems in nonsmooth mechanics. More precisely, it is required to minimize the weight of a structure subject to frictionless unilateral contact conditions and constraints on the magnitudes of contact forces, displacements, stresses and cross-sectional areas. This problem, as is well-known, can be formulated as a special and challenging optimization problem known as a Mathematical Program with Equilibrium Constraints (MPEC), a key feature of which is the presence of complementarity conditions, involving the orthogonality of two sign-constrained vectors. In spite of its inherent nonsmoothness, we attempt to solve the problem using standard nonlinear programming techniques. In particular, we investigate numerically the application of two simple algorithms, both based on the use of the general-purpose nonlinear programming code CONOPT accessed via the powerful GAMS modelling language, for solving the suitably reformulated problem. Application is illustrated by means of three numerical examples.

Nonlinear programming approach for a class of inverse problems in elastoplasticity

This paper deals with a special class of inverse problems in discrete structural plasticity involving the identification of elastic limits and hardening moduli on the basis of information on displacements. The governing equations lead naturally to a special and challenging optimization problem known as a Mathematical Program with Equilibrium Constraints (MPEC), a key feature of which is the orthogonality of two sign-constrained vectors or so-called "complementarity" condition. We investigate numerically the application of two simple algorithms, both based on the use of the general purpose nonlinear programming code CONOPT accessed via the GAMS modeling language, for solving the suitably reformulated problem. Application is illustrated by means of two numerical examples.

Performance of several nonlinear programming software packages on microcomputers

Nonlinear programming (NLP) problems arise often in science, engineering, and business applications. Due to the inherent mathematical limitations of the solution techniques, the existing solution methods are not guaranteed to solve all problems optimally. In the past years, a number of new commercial packages that solve NLP problems on microcomputers have been developed. To grasp some idea of the performance of the NLP packages on microcomputers from the viewpoints of effectiveness, efficiency, accuracy, and ease of use, six popular NLP packages are selected for comparison. The results provide some information regarding the suitable package to use under different considerations. This paper investigates the performance of six NLP packages, including MATLAB, IMSL, AMPL, GAMS, GINO, and SQP, by solving seventy test problems on a PC 486 machine. These packages are categorized as library subroutines, modeling languages, and the conventional NLP codes. The results show that AMPL and GINO have the best ability in solving NLP problems optimally, GAMS is the most efficient package in terms of the execution time, GINO has the most accurate solution, and MATLAB needs the least effort in modeling problems. In sum, GAMS and GINO have the best overall performance, followed by AMPL. All these three packages are of modeling language type.

One-layer real time optimization of LPG production in the FCC unit: procedure, advantages and disadvantages

This paper summarizes the results of the current stage of the real time optimization (RTO) project of Petrobras in Brazil for optimizing its refineries. Here the maximization of the production of LPG in the FCC converter is considered. The general procedure for applying the RTO using the one-layer approach is outlined. Some difficulties that may arise during the implementation are addressed and the strategy is compared with the more traditional two-layers approach. We also stress the need of using a robust nonlinear programming (NLP) code and all the results presented in the paper were obtained by the MISQPSOL code.

Variational solutions for the heat-rate-limited aeroassisted orbital transfer problem

Minimum energy loss trajectories based on Pontryagin's minimum principle are generated for the atmospheric part of an aeroassisted orbital transfer problem. At the initial time all states are given. At the final time the altitude, velocity, and inclination change are prescribed. Additionally, the trajectory is subject to a heating rate limit, which, in the context of optimal control, represents a first-order state inequality constraint. Numerical difficulties implied by the singularly perturbed nature of the problem are avoided by starting the integration in the interior of the trajectory at a point where the heating rate limit is active. The state constraint becomes active always in the form of a state constrained arc. Nontrivial touch points have not been observed in numerical solutions. VER the years a considerable amount of research has been conducted in the area of aeroassisted orbital transfers (AOT). Surveys on the current state of the art in optimization of AOT trajectories can be found in Refs. 1 and 2. The optimization of the atmospheric part of orbital plane changes using aerodynamic controls is discussed in Refs. 3-6. In Ref. 5, a guidance law is developed through regular expansion of the Hamilton-Jacobi-Bellmann equation. A guidance law development through matched asymptotic expansions and through a hybrid regular expansion/collocation approach is presented in Refs. 7 and 8, respectively. In Ref. 9, maximum cross range and maximum orbital plane change trajectories are calculated using a nonlinear programming code. In Ref. 10, the use of multiple pass trajectories is investigated to alleviate the severe aeroheating. In Ref. 11, aerocruise and aeroglide maneuvers are compared, both in presence of a heating rate limit. Recently, variational solutions for the heating rate limited aerocruise maneuver were presented in Ref. 12. Numerical difficulties were reported, and converged solutions could not be obtained below a certain value for the prescribed maximum heating rate. The optimal switching structure was identified as (free, touch point, free). Solutions with state constrained arcs were not obtained. For the same vehicle model and atmospheric model as in Ref. 12 the heating rate limit could be reduced much further in Ref. 13, using direct shooting and direct collocation. The obtained trajectories are physically feasible and involve state constrained arcs. The solutions were reported to be very sensitive, and certain convergence difficulties were overcome by reducing the prescribed precision in the nonlinear program solver. The present paper is based on a dynamical model that is identical to the one used in Ref. 12. The objective is to verify the results reported in Ref. 12, to identify the correct optimal switching structure, and to generate variational benchmark solutions that could be used to verify and to calibrate direct optimization approaches.