An augmented Lagrangian method for discrete large‐slip contact problems

We present in this paper new results on the existence of saddle points of augmented Lagrangian functions for constrained nonconvex optimization. Four classes of augmented Lagrangian functions are considered: the essentially quadratic augmented Lagrangian, the exponential-type augmented Lagrangian, the modified barrier augmented Lagrangian, and the penalized exponential-type augmented Lagrangian. We first show that under second-order sufficiency conditions, all these augmented Lagrangian functions possess local saddle points. We then prove that global saddle points of these augmented Lagrangian functions exist under certain mild additional conditions. The results obtained in this paper provide a theoretical foundation for the use of augmented Lagrangians in constrained global optimization. Our findings also give new insights to the role played by augmented Lagrangians in local duality theory of constrained nonconvex optimization.

Complementary energy approach to contact problems based on consistent augmented Lagrangian formulation

Estimation of Optimal Well Controls Using the Augmented Lagrangian Function with Approximate Derivatives

Duality in nonlinear programs using augmented Lagrangian functions

Contact problems are widely encountered in geotechnical engineering, such as the contact between soils and concrete used in earth and rockfill dams, tunnels and coastal levees. Due to the unknown contact region and contact forces, the contact problems have strong boundary nonlinearity. In addition, soils have been recognized as heterogeneous materials in geotechnical engineering. The existence of the soil heterogeneity increases the nonlinearity of the contact problems. Currently, the contact problems are mostly analysed without considering the soil heterogeneity, which may not reflect the contact behavior well. In order to investigate the influence of soil heterogeneity on the contact problems, in this paper, a simple plane-strain contact problem is analysed as an example. In this example, Young’s modulus is taken to be a spatially variable. The local average subdivision (LAS) is used to model the heterogeneity of Young’s modulus. The penalty method is utilised to determine the contact behavior. By the first use of linking the penalty method with the LAS, the proposed approach can be used to analyse the contact problems considering soil heterogeneity. The results show that the influence of soil heterogeneity on the elastic contact problems is significant. The contact forces of the heterogeneous case present apparent variation compared to the results of the homogeneous case. The distribution of the contact force at a specific point is also normal when Young’s modulus is normally distributed, moreover, the coefficient of variation (COV) and the horizontal scale of fluctuation of Young’s modulus affect the extent of variation of the normal contact forces. The standard deviation of the normal contact force increases with the increase of the COV and decreases with the increase of the horizontal scale of fluctuation of Young’s modulus. From the analyses, to better predict the deformation/stress in the contact problems, heterogeneity needs to be considered.

Stochastic optimization methods have been widely employed to find solutions to structural design optimization problems in the past two decades, especially for truss structures. The primary aim of this study is to introduce a design optimization method combining an augmented Lagrangian function and teaching–learning-based optimization for truss and nontruss structural design optimization. The augmented Lagrangian function serves as a constraint-handling tool in the proposed method and converts a constrained optimization problem into an unconstrained one. On the other hand, teaching–learning-based optimization is employed to resolve the transformed, unconstrained optimization problems. Since the proper values of the Lagrangian multipliers and penalty factors are unknown in advance, the proposed method is implemented in an iterative way to avoid the issue of selecting them, i.e. the Lagrangian multipliers and penalty factors are automatically updated according to the violation level of all constraints. To examine the performance of the proposed method, it is applied on a group of benchmark truss optimization problems and a group of nontruss optimization problems of aircraft wing structures. The computational results obtained by the proposed method are compared to the results produced by both other version of teaching–learning-based optimization and stochastic optimization methods.

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

The paper describes a new approach to the optimal-power-flow problem based on Newton's method which it operates with an augmented Lagrangian function associated with the original problem. The function aggregates all the equality and inequality constraints. The first-order necessary conditions for optimality are reached by Newton's method, and by updating the dual variables and the penalty terms associated with the inequality constraints. The proposed approach does not have to identify the set of binding constraints and can be utilised for an infeasible starting point. The sparsity of the Hessian matrix of the augmented Lagrangian is completely exploited in the computational implementation. Tests results are presented to show the good performance of this approach.

In this paper, we introduce two primal-dual active-set methods for solving large-scale constrained optimization problems. The first method minimizes a sequence of primal-dual augmented Lagrangian functions subject to bounds on the primal variables and artificial bounds on the dual variables. The basic structure is similar to the well-known optimization package Lancelot (Conn, et al. in SIAM J Numer Anal 28:545---572, 1991), which uses the traditional primal augmented Lagrangian function. Like Lancelot, our algorithm may use gradient projection-based methods enhanced by subspace acceleration techniques to solve each subproblem and therefore may be implemented matrix-free. The artificial bounds on the dual variables are a unique feature of our method and serve as a form of dual regularization. Our second algorithm is a two-phase method. The first phase computes iterates using our primal-dual augmented Lagrangian algorithm, which benefits from using cheap gradient projections and matrix-free linear CG calculations. The final iterate produced during this phase is then used as input for phase two, which is a stabilized sequential quadratic programming method (Gill and Robinson in SIAM J Opt 1---45, 2013). Obtaining superlinear local convergence under weak assumptions is an important benefit of the transition to a stabilized sequential quadratic programming algorithm. Interestingly, the bound-constrained subproblem used in phase one is equivalent to the stabilized subproblem used in phase two under certain assumptions. This fact makes our choice of algorithms a natural one.

Dynamic economic dispatch (DED) over multiple time periods is a large-scale coupled spatial-temporal optimization problem. Therefore, the Lagrangian relaxation method has been widely used to split the large-scale optimization problem with coupled structure into several small sub-problems. In order to bring robustness for updating the dual multipliers and yielding convergence without strong assumptions, the augmented Lagrangian relaxation method is introduced in this paper. However, the added penalty term in an augmented Lagrangian function is non-separable, which leads to the difficulty in achieving full decomposition for parallel computation. To address this problem, a diagonal quadratic approximation method is employed to yield an approximated block separation of the non-separable penalty term. Furthermore, the ramp rate constraints are relaxed in this paper, so that the DED model is decomposed into several single-period economic dispatch models that can be efficiently handled in parallel, called the parallel augmented Lagrangian relaxation method. Particularly, the proposed relaxation strategy has a high separability feature which theoretically leads to sound convergence property. Numerical results on the IEEE 118-bus and a practical Polish 2383-bus test system over a different number of time periods show the effectiveness of the proposed method. In addition, the proposed method can be extended to other coupled spatial-temporal scheduling problems in power systems, such as energy storage dispatch.

We consider two quasistatic frictionless contact problems for piezoelectric bodies. For the first problem the contact is modelled with Signorini′s conditions and for the second one is modelled with normal compliance. In both problems the material′s behavior is electroelastic and the adhesion of the contact surfaces is taken into account and is modelled with a surface variable, the bonding field. We provide variational formulations for the problems and prove the existence of a unique weak solution to each model. The proofs are based on arguments of time‐dependent variational inequalities, differential equations, and fixed point. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solution of the contact problem with normal compliance as the stiffness coefficient of the foundation converges to infinity.

In this paper, we investigate a novel model problem involving a viscoelastic, frictionless contact model characterized by history-dependent operators. The constitutive relation is formulated based on a time-fractional Kelvin–Voigt model. We represent the contact by incorporating normal compliance within the framework of a time-fractional derivative. In addressing the contact problem, we begin by formulating its weakness and subsequently verify the existence of a solution within this framework. Finally, we study the solution’s behavior concerning the integral term and its relation to the solution, while also presenting convergence results.

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.

Studies of contact problem have been widely executed by researchers with variable scopes, methods and definitions. A common problem occurs while handling contact phenomena is sliding through element boundary [1], due to the discontinuity of the local coordinate between elements and a contact point [2] [3]. The common problem that occurs at an element boundary is a stable convergence result is hard to achieve [4], thus inspires authors to make a comparison of two beam methods which are Euler-Bernoulli beam theory and Timoshenko beam theory for frictionless contact problem. Authors have been investigated geometrically non-linear analysis with extremely large displacements by using Tangent Stiffness Method (TSM) [5], a robust non-linear analysis method to execute analysis and produce results with high accuracy. In this study, authors propose the modification of the beam elements with three nodes by considering the adaptation of shear deformation by Timoshenko beam theory. The modification enables the contact point to slide through the element edge smoothly and some numerical examples are provided in this study.

Solving contact forces with the competition between potential energy and entropy in elastic mechanics