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In this paper, we investigate a class of interval-valued multiobjective optimization problems (IVMOP), and formulate a certain equivalent multiobjective optimization problem (MOP). We establish that any Pareto optimal solution of the equivalent MOP is also an effective solution of IVMOP. We introduce two variants of the quasi-Newton method to solve IVMOP when every component of the objective function has second-order partial generalized Hukuhara derivatives. In the first variant, we assume that the objective function of IVMOP is strongly convex, and approximate Hessian matrix of each component of the objective function of MOP by using the well-known BFGS method. The algorithm generated by this variant exhibits superlinear convergence to a locally effective solution of IVMOP. In the second variant, we approximate the Hessian matrix of each component of MOP by applying the Levenberg–Marquardt method. Moreover, this variant can be applied to non-convex functions, and the sequence generated by this variant converges to a locally effective solution of IVMOP as well. Finally, some numerical examples are furnished to illustrate the effectiveness of our developed methodology. [ABSTRACT FROM AUTHOR]

• Formulate the no-toll equilibrium (NTE) and the congestion and environmental toll (CET) equilibrium problems with heterogeneous users as unconstrained optimization problems. • Modify the BFGS method to solve these problems. • Formulate the optimal congestion toll and CET design problems as bi-level programs. • Develop the double BFGS method to solve the two bi-level programs. We design a congestion and environmental toll (CET) scheme for the morning commute with heterogeneous users in a single OD network with parallel routes. The designed toll scheme relies upon the concept of marginal-cost pricing and is anonymous. The Henderson approach is used to model road congestion and the tolling problem to examine commuter's arrival time and route choice at the CET equilibrium (CETE). Linear interpolation is applied to approximate the emission cost function and the resulting CETE problem is formulated as an unconstrained optimization problem, which is solved by the modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. Unlike the existing approach, this novel approach does not require that the arrival of each group of commuters at the destination at the equilibrium follows a predetermined order, and can handle non-monotone (emission) cost function. As two special cases, no-toll equilibrium (NTE) and the congestion toll equilibrium (CTE) are also examined, and the two resultant equilibrium problems are formulated and solved by the same approach. This approach is shown to be more efficient than the existing approach. Bi-level programming models are proposed to formulate the optimal congestion toll and CET design problems, in which the CTE and CETE problems are the corresponding lower level problem. These models are solved by the double BFGS method, which uses a classical BFGS method to solve the upper level model and the proposed BFGS method to solve the lower level model. Finally, numerical examples are provided to illustrate the properties of the models and the efficiency of the proposed solution algorithms. [ABSTRACT FROM AUTHOR]

The standard BFGS method plays an important role among the quasi-Newton algorithms for constrained/un-constrained optimization problems. However, Dai (2003) constructed a counterexample to demonstrate that this method may fail for non-convex functions with inexact Wolfe line searches, and Mascarenhas (2004) showed the nonconvergence of this method and other methods in the Broyden family, even with exact line search techniques. These works motivate us to try to find another way to obtain another quasi-Newton method with better convergence based on the standard BFGS formula. In this paper, four approaches are used in the designed algorithm: (1) a modified weak Wolfe–Powell line search technique is introduced; (2) if one defined condition is satisfied, then the search direction and its associated step length are accepted, and the next point is designed; (3) otherwise, a parabolic will be presented, and it is regarded as the projection surface to avoid using the failed direction, and the next point x k + 1 is generated by a projection technique; and (4) to easily obtain the global convergence of the proposed algorithm, the projection point is not used in the current BFGS update but rather for all the next iterations. The new algorithm possesses global convergence for general functions under the inexact modified weak Wolfe–Powell line search technique, and it is shown that other methods in the Broyden class also have this property. Numerical results are reported to show the performance of the given algorithm and other similar methods. [ABSTRACT FROM AUTHOR]

Phase equilibrium calculations are one of the main steps in the compositional simulation of hydrocarbon reservoirs, well tubing, oil and gas pipelines as well as separation units. The main phases in all hydrocarbon production systems from a reservoir to a stock tank are gas, oil, and water. In most studies, the aqueous phase is considered bulk and phase equilibrium calculations are conducted only on the oil and gas phases. In some conditions such as thermal processes, the solubility of water in other phases and also hydrocarbon components in the water phase cannot be ignored. The robustness of equilibrium calculations for a wide range of compositions, temperatures, and pressures, including near critical area and saddle point curve, is essential for rapid and stable simulation studies. In this study, we propose the three-phase equilibrium calculations in detail for water, oil and gas phases. All the handling steps and strategies to establish a robust and efficient phase stability testing and flash calculation are explained. Moreover, appropriate tolerance values for decision-makings in the algorithm are defined. We successfully present the effectiveness of our algorithm for some three-phase samples covering a wide range of composition and production conditions. [ABSTRACT FROM AUTHOR]

The BFGS method is one of the most effective quasi-Newton algorithms for optimization problems. However, its global convergence for general functions is still open. In this paper, under a new line search technique, this problem is solved, and it is shown that other methods in the Broyden class also possess this property. Moreover, the global convergence of the PRP method is established in the case of this new line search. Numerical results are reported to show that the new line search technique is competitive to that of the normal line search. [ABSTRACT FROM AUTHOR]

An inverse method is developed to simultaneously estimate unknown temperature-dependent thermal conductivity and specific heat of a brass rod with knowledge of temperatures taken on the specimen. An experimental process with the brass rod is built for measuring temperatures at some locations to verify the thermal conductivity and specific heat. With known temperature data recorded from the experiment, inverse solutions were rapidly obtained through the Broyden–Fletcher–Goldfarb–Shanno (BFGS) combined simple step method. Results show that the proposed method can estimate thermal properties with low iterations and in a significantly short time compared to other methods. In addition, the estimated temperatures are in very good agreement with the measurement temperature. From experimental verification, the estimated thermal properties are quite close to values obtained by simple tests, with several cases of different heat generation magnitudes. The effect of measurement errors and locations, as well as measured point numbers, on the accuracy of inverse solutions was discussed. According to analysis results, the proposed method, an accurate and efficient method for the prediction of unknown thermal conductivity and specific heat, can be applied to accurately estimate thermophysical properties of various materials. [ABSTRACT FROM AUTHOR]

Dynamic user optimal simultaneous route and departure time choice(DUO-SRDTC) problems are usually formulated as variational inequality (VI) problems whose solution algorithms generally require continuous and monotone route travel cost functions to guarantee convergence. However, the monotonicity of the route travel cost functions can not be ensured even if the route travel time functions are monotone. In contrast to traditional formulations, this paper formulates a DUO-SRDT C problem (that can have fixed or elastic demand) as a system of nonlinear equations. The system of nonlinear equations is a function of generalized origin-destination (OD) travel costs rather than route flows and includes a dynamic user optimal (DUO) route choice subproblem with perfectly elastic demand and a quadratic programming (QP) subproblem under certain assumptions. This study also proposes a solution method based on the backtracking inexact Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, the extra gradient algorithm, and the Frank-Wolf e algorithm The BFGS method, the extra gradient algorithm, and the Frank-Wolf e algorithm are used to solve the system of nonlinear equations, the DUO route choice subproblem, and the QP subproblem, respectively. The proposed formulation and solution method can avoid the requirement of monotonicity of the route travel cost functions to obtain a convergent solution and provide a new approach with which to solve DUO-SRDTC problems. Finally, numeric examples are used to demonstrate the performance of the proposed solution method. [ABSTRACT FROM AUTHOR]

An inverse heat transfer problem is investigated in the present study by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to predict the unknown time-dependent heat generation at the weld interface and convection heat transfer coefficient during an ultrasonic metal welding process based on the knowledge of temperature measurements taken on the horn. With known temperature data at some locations on the horn, the inverse solution was rapidly obtained by solving nonlinear direct problem, Central Finite Difference and Simple Step Method. The proposed method which did not need solving adjoint and sensitivity problem revealed the characteristics of high efficiency, lower iterations for a computational algorithm and high accuracy for estimating values even when measurement error was considered. Besides, a comparison of the BFGS method with some previous methods (i.e. CGM, SCGM) was established. These results show that an excellent estimation on interfacial heat generation (or temperature), as well as a convection heat transfer coefficient, can be simultaneously obtained in this study. The current methodology will provide a useful tool to optimize welding conditions in ultrasonic welding. [ABSTRACT FROM AUTHOR]

Abstract: Nonlinear equations in mathematical physics and engineering are solved by linearizing the equations and forming various iterative procedures, then executing the numerical simulation. For strongly nonlinear problems, the solution obtained in the iterative process can diverge due to numerical instability. As a result, the application of numerical simulation for strongly nonlinear problems is limited. Helicopter aeroelasticity involves the solution of systems of nonlinear equations in a computationally expensive environment. Reliable solution methods which do not need Jacobian calculation at each iteration are needed for this problem. In this paper, a comparative study is done by incorporating different methods for solving the nonlinear equations in helicopter trim. Three different methods based on calculating the Jacobian at the initial guess are investigated. [Copyright &y& Elsevier]

Abstract: This paper investigates several gradient local search methods as an enhancement to Part I, to further improve the computational efficiency while achieving the same identification accuracy. The present study is significant in several ways. First, this study reveals the characteristics of structural identification from the optimization perspective. Second, the “switch point” from global search to local search is determined with due consideration in convergence speed and avoidance of local optima. Finally, the combined strategy based on Part I and Part II with a particular local search (BFGS) is found to achieve substantial improvement in identification efficiency and accuracy. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a modified BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with nonmonotone line search for unconstrained optimization. Under some mild conditions, we show that the method is globally convergent without a convexity assumption on the objective function. We also report some preliminary numerical results to show the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: One class of the lately developed methods for solving optimization problems are filter methods. In this paper we attached a multidimensional filter to the Gauss–Newton-based BFGS method given by Li and Fukushima [D. Li, M. Fukushima, A globally and superlinearly convergent Gauss–Newton-based BFGS method for symmetric nonlinear equations, SIAM Journal of Numerical Analysis 37(1) (1999) 152–172] in order to reduce the number of backtracking steps. The proposed filter method for unconstrained minimization problems converges globally under the standard assumptions. It can also be successfully used in solving systems of symmetric nonlinear equations. Numerical results show reasonably good performance of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: A BFGS method, in association with a new backtracking line search technique, is presented for solving symmetric nonlinear equations. The global and superlinear convergences of the given method are established under mild conditions. Preliminary numerical results show that the proposed method is better than the normal technique for the given problems. [Copyright &y& Elsevier]

Abstract: In this note, we point out that in [Y. Xiao, H. Sun, Z. Wang, A globally convergent BFGS method with nonmonotone line search for non-convex minimization, J. Comput. Appl. Math. 230 (2009) 95–106] are not enough to guarantee the global convergence result Theorem 3.1 in . We present a new assumption and show that the proposed method in still converges globally under these three assumptions. [Copyright &y& Elsevier]

• Three different implementation strategies of phase-field damage models in Abaqus are addressed. • The sources codes and input files of all these three implementation schemes can be open accessed. • Representative examples validating the implementation strategies and source codes are presented. • The UEL subroutine together with the BFGS monolithic solver is of best numerical performances. • The UMAT subroutine is viable if the BFGS solver is available in coupled thermal–stress analyses. Despite the versatility in modeling complex crack configurations, phase-field damage models for fracture usually count only on in-house codes, greatly restricting their potential applications. It is thus of vital importance to implement them in those commonly used commercial software packages like Abaqus. However, so far only the less robust Newton's monolithic algorithm or the inefficient staggered solver has been considered. In this work, taking the unified phase-field damage theory (Wu, 2017) as the particular example, we present three distinct strategies of implementing phase-field damage models into Abaqus: (i) UMAT-Newton-M: a thermo-mechanically coupled user-defined material (UMAT) implementation of the modified Newton monolithic solver; (ii) UEL-Staggered: a novel user-defined element (UEL) implementation of the iterative staggered (alternate minimization) algorithm with dummy dofs; (iii) UEL-BFGS: a UEL implementation of the recently advocated BFGS quasi-Newton monolithic algorithm. The aforesaid implementation strategies are then validated against several representative benchmark problems of brittle fracture and quasi-brittle failure. It is found that, the UMAT-Newton-M implementation is the simplest but not robust enough, while the UEL-Staggered implementation is robust but extremely inefficient. Comparatively, in all cases the UEL-BFGS scheme is of the best numerical performance with lest iterations and sufficient robustness. For the sake of reproducibility of the presented numerical results and the promotion of phase-field damage models, the source codes (programmed in the free format syntax of FORTRAN90) are also provided and interested users can download them at https://github.com/jianyingwu/pfczm-abaqus. [ABSTRACT FROM AUTHOR]

Despite the popularity in modeling complex and arbitrary crack configurations in solids, phase-field damage models suffer from burdensome computational cost. This issue arises largely due to the robust but inefficient alternating minimization (AM) or staggered algorithm usually employed to solve the coupled damage–displacement governing equations. Aiming to tackle this difficulty, we propose in this work, for the first time , to use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm to solve in a monolithic manner the system of coupled governing equations, rather than the standard Newton one which is notoriously poor for problems involving non-convex energy functional. It is found that, the BFGS algorithm yields identical results to the AM/staggered solver, and is also robust for both brittle fracture and quasi-brittle failure with a single or multiple cracks. However, much less iterations are needed to achieve convergence. Furthermore, as the system matrix is less reformed per increment, the quasi-Newton monolithic algorithm is much more efficient than the AM/staggered solver. Representative numerical examples show that the saving in CPU time is about factor 3 ∼ 7 , and the larger the problem is, the more saving it gains. As the BFGS monolithic algorithm has been incorporated in many commercial software packages, it can be easily implemented and is thus attractive in the phase-field damage modeling of localized failure in solids. • A robust and efficient quasi-Newton monolithic algorithm is proposed for phase-field damage models. • The robustness comes from the BFGS method with a symmetric and positive-definite stiffness matrix. • The proposed quasi-Newton monolithic algorithm is about 3 ∼ 7 times faster than the staggered solver. • The phase-field regularized cohesive zone model is not numerically affected by the solution algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes combining the augmented Lagrangian method (ALM) with evolutionary heuristic methods, as well as quasi-Newton optimization methods applied to the energy efficiency (EE) maximization in the optical code division multiple access (OCDMA) communication network. The particle swarm optimization (PSO) and a hybridization between the PSO and the gravitational search algorithm (GSA) called PSOGSA have been deployed. The ALM structure replaces the objective function and allows a best fit to the problem, and ultimately provide more information about the solution. Numerical results demonstrate the robustness and low-complexity of hybrid ALM-PSO, while the ALM associated with PSOGSA attains robustness at cost of high-complexity. In turn, the usually ALM combined with Broyden-Fletcher-Goldfarb-Shanno (BFGS) method presents convergence for a restrict scenarios, failing to perform suitably for networks with large numbers of users. [ABSTRACT FROM AUTHOR]