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Several manifold optimization schemes are presented and analyzed for solving a specialized inverse structured symmetric matrix problem with prescribed spectrum. Some entries in the desired matrix are assigned in advance and cannot be altered. The rest of the entries are free, some of them preferably away from zero. The reconstructed matrix must satisfy these requirements and its eigenvalues must be the given ones. This inverse eigenvalue problem is related to the problem of determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our optimization schemes are based on considering the eigenvector matrix as the only unknown and iteratively moving on the manifold of orthogonal matrices, forcing the additional structural requirements through a change of variables and a convenient differentiable objective function in the space of square matrices. We propose Riemannian gradient-type methods combined with two different well-known retractions, and with two well-known constrained optimization strategies: penalization and augmented Lagrangian. We also present a block alternating technique that takes advantage of a proper separation of variables. Convergence properties of the penalty alternating approach are established. Finally, we present initial numerical results to demonstrate the effectiveness of our proposals. [ABSTRACT FROM AUTHOR]

We propose a GPU-based iterative method for accelerated elastodynamic simulation with the log-barrier-based contact model. While Newton's method is a conventional choice for solving the interior-point system, the presence of ill-conditioned log barriers often necessitates a direct solution at each linearized substep and costs substantial storage and computational overhead. Moreover, constraint sets that vary in each iteration present additional challenges in algorithm convergence. Our method employs a novel barrier-augmented Lagrangian method to improve system conditioning and solver efficiency by adaptively updating an augmentation constraint sets. This enables the utilization of a scalable, inexact Newton-PCG solver with sparse GPU storage, eliminating the need for direct factorization. We further enhance PCG convergence speed with a domain-decomposed warm start strategy based on an eigenvalue spectrum approximated through our in-time assembly. Demonstrating significant scalability improvements, our method makes simulations previously impractical on 128 GB of CPU memory feasible with only 8 GB of GPU memory and orders-of-magnitude faster. Additionally, our method adeptly handles stiff problems, surpassing the capabilities of existing GPU-based interior-point methods. Our results, validated across various complex collision scenarios involving intricate geometries and large deformations, highlight the exceptional performance of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose an augmented Lagrangian method with Backtracking Line Search for solving nonconvex composite optimization problems including both nonlinear equality and inequality constraints. In case the variable spaces are homogeneous, our setting yields a generic nonlinear mathematical programming model. When some variables belong to the real Hilbert space and others to the integer space, one obtains a nonconvex mixed-integer/-binary nonlinear programming model for which the nonconvexity is not limited to the integrality constraints. Together with the formal proof of its iteration complexity, the proposed algorithm is then numerically evaluated to solve a multi-constrained network design problem. Extensive numerical executions on a set of instances extracted from the SNDlib repository are then performed to study its behavior and performance as well as identify potential improvement of this method. Finally, analysis of the results and their comparison against those obtained when solving its convex relaxation using mixed-integer programming solvers are reported. [ABSTRACT FROM AUTHOR]

Material heterogeneity gives composite constructions unique mechanical and physical qualities. Combining multiple materials takes full use of these features in stress‐constrained topology optimization. Traditional research in this field often assumes a consistent yield criterion for all possible materials but adapts their stiffness and strengths accordingly. To cope with this challenge, an innovative single‐variable interpolation approach is proposed to enable the simultaneous inclusion of distinct yield criteria and material strengths. A stress‐constrained topology optimization formulation is presented based on this yield function interpolation method, which can independently support various materials with different elastic characteristics, material strengths, and yield criteria. Then, the large‐scale problem of local stress constraints can be effectively solved by the Augmented Lagrangian (AL) method. Several two‐dimensional (2D) and three‐dimensional (3D) design scenarios are investigated to reduce the overall mass of the structure while considering stress constraints. The optimal composite designs exhibit several crucial benefits resulting from material heterogeneity, including the enlargement of the design possibilities, the dispersion of stress, and the utilization of asymmetry in tension‐compression strength. [ABSTRACT FROM AUTHOR]

In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically. [ABSTRACT FROM AUTHOR]

This paper considers a risk-averse Markov decision process (MDP) with non-risk constraints as a dynamic optimization framework to ensure robustness against unfavorable outcomes in high-stakes sequential decision-making situations such as disaster response. In this regard, strong duality is proved while making no assumptions on the problem's convexity. This is necessary for some real-world issues, e.g., in the case of deprivation costs in the context of disaster relief, where convexity cannot be ensured. Our theoretical results imply that the problem can be exactly solved in a dual domain where it becomes convex. Based on our duality results, an augmented Lagrangian-based constraint handling mechanism is also developed for risk-averse reinforcement learning algorithms. The mechanism is proved to be theoretically convergent. Finally, we have also empirically established the convergence of the mechanism using a multi-stage disaster response relief allocation problem while using a fixed negative reward scheme as a benchmark. [ABSTRACT FROM AUTHOR]

We propose a new alternating direction method of multipliers (ADMM) with an optimal parameter for the unilateral obstacle problem. We first use the five-point difference scheme to discretize the problem. Then, we present an augmented Lagrangian by introducing an auxiliary unknown, and an ADMM is applied to the corresponding saddle-point problem. Through eliminating the primal and auxiliary unknowns, a pure dual algorithm is then used. The convergence of the proposed method is analyzed, and a simple strategy is presented for selecting the optimal parameter, with the largest and smallest eigenvalues of the iterative matrix. Several numerical experiments confirm the theoretical findings of this study. [ABSTRACT FROM AUTHOR]

We address the problem of minimizing a smooth function under smooth equality constraints. Under regularity assumptions on these constraints, we propose a notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization. Using a smooth exact penalty function known as Fletcher's augmented Lagrangian, we propose an algorithm to minimize the penalized cost function which reaches ε -approximate second-order critical points of the original optimization problem in at most O (ε - 3) iterations. This improves on current best theoretical bounds. Along the way, we show new properties of Fletcher's augmented Lagrangian, which may be of independent interest. [ABSTRACT FROM AUTHOR]

Sensitivity analysis represents a powerful tool for the optimization of multibody system dynamics. The performance of a gradient-based optimization algorithm is strongly tied to the dynamic and the sensitivity formulations considered. The accuracy and efficiency are critical to any optimization problem, thus they are key factors in the selection of the dynamic and sensitivity analysis approaches used to compute an objective function gradient. Semi-recursive methods usually outperform global methods in terms of computational time, even though they involve sometimes demanding recursive procedures. Semi-recursive methods are well suited to be combined with different constraints enforcement schemes as the augmented Lagrangian index-3 formulation with velocity and acceleration projections (ALI3-P), taking advantage of the robustness, accurate fulfillment of constraint equations and the low computational burden. The sensitivity analysis of the semi-recursive ALI3-P formulation is studied in this document by means of the direct differentiation method. As a result, a semi-recursive ALI3-P sensitivity formulation is developed for an arbitrary reference point selection, and then two particular versions are unfolded and implemented in the general purpose multibody library MBSLIM, using as reference point the center of mass (RTdyn0) or the global origin of coordinates (RTdyn1). Besides, the detailed derivatives of the recursive terms are provided, which will be useful not only for the direct sensitivity formulation presented herein, but also for other sensitivity formulations relying on the same recursive expressions. The implementation has been tested in two numerical experiments, a five-bar benchmark problem and a buggy vehicle. [ABSTRACT FROM AUTHOR]

In this two-part study, we develop a general theory of the so-called exact augmented Lagrangians for constrained optimization problems in Hilbert spaces. In contrast to traditional nonsmooth exact penalty functions, these augmented Lagrangians are continuously differentiable for smooth problems and do not suffer from the Maratos effect, which makes them especially appealing for applications in numerical optimization. Our aim is to present a detailed study of various theoretical properties of exact augmented Lagrangians and discuss several applications of these functions to constrained variational problems, problems with PDE constraints, and optimal control problems. The first paper is devoted to a theoretical analysis of an exact augmented Lagrangian for optimization problems in Hilbert spaces. We obtain several useful estimates of this augmented Lagrangian and its gradient, and present several types of sufficient conditions for KKT-points of a constrained problem corresponding to locally/globally optimal solutions to be local/global minimizers of the exact augmented Lagrangian. [ABSTRACT FROM AUTHOR]

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method. Funding: This work was supported by the National Science Foundation [Grant CNS-1545046] and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research [Grant DE-AC02-06CH11347]. [ABSTRACT FROM AUTHOR]

This work is focused on an accelerated global-in-time solution strategy for the Oseen equations, which highly exploits the augmented Lagrangian methodology to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single all-at-once saddle point problem. By elimination of all velocity unknowns, the resulting implicitly defined equation can then be solved using a global-in-time pressure Schur complement (PSC) iteration. To accelerate the convergence behavior of this iterative scheme, the augmented Lagrangian approach is exploited by modifying the momentum equation for all time steps in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the quality of customized PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. Therefore, a highly specialized multigrid solver based on modified intergrid transfer operators and an additive block preconditioner is extended to solution of the all-at-once problem. The potential of the proposed overall solution strategy is discussed in several numerical studies as they occur in commonly used linearization techniques for the incompressible Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

The dislocation hyperbolic augmented Lagrangian algorithm (DHALA) is a new approach to the hyperbolic augmented Lagrangian algorithm (HALA). DHALA is designed to solve convex nonlinear programming problems. We guarantee that the sequence generated by DHALA converges towards a Karush-Kuhn-Tucker point. We are going to observe that DHALA has a slight computational advantage in solving the problems over HALA. Finally, we will computationally illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We discuss several algorithms for solving a network optimization problem of simultaneous routing and bandwidth allocation in green networks in a decomposed way, based on the augmented Lagrangian. The problem is difficult due to the nonconvexity caused by binary routing variables. The chosen algorithms, which are several versions of the Multiplier Method, including the Alternating Direction Method of Multipliers (ADMM), have been implemented in Python and tested on several networks' data. We derive theoretical formulations for the inequality constraints of the Bertsekas, Tatjewski and SALA methods, formulated originally for problems with equality constraints. We also introduce some modifications to the Bertsekas and Tatjewski methods, without which they do not work for an MINLP problem. The final comparison of the performance of these algorithms shows a significant advantage of the augmented Lagrangian algorithms, using decomposition for big problems. In our particular case of the simultaneous routing and bandwidth allocation problem, these algorithms seem to be the best choice. [ABSTRACT FROM AUTHOR]

The predictor corrector proximal multiplier method (PCPM) proposed by Chen and Teboulle in early 1990s is a popular scheme for linearly constrained composite minimization problems. In this paper, we show that the PCPM is equivalent to a special case of the linearized augmented Lagrangian method (ALM). Using this interpretation, we identify the necessary and sufficient convergence condition for its convergence. As a byproduct, the stepsize condition of Chen-Teboulle's PCPM can be improved without adding any further assumptions. We prove the global convergence of the PCPM with this improved condition. We also propose a generalized version of PCPM with convergence guarantee under mild conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study an augmented Lagrangian-type algorithm called the Dislocation Hyperbolic Augmented Lagrangian Algorithm (DHALA), which solves an inequality nonconvex optimization problem. We show that the sequence generated by DHALA converges to a Karush–Kuhn–Tucker (KKT) point under the Mangasarian–Fromovitz constraint qualification. The contribution of our work is to consider a constraint qualification into this algorithm. Finally, we present some computational illustrations to demonstrate the performance our algorithm works.

In this work, a formulation of alternating direction method of multipliers (ADMM) for addressing power amplifiers (PA) modeling and linearization problems is presented. The proposal consists on leveraging the implicit redundancy of the equations in order to achieve a distributed architecture. A detailed theoretical formulation of the method is provided in order to get a better comprehension of its advantages and the approach to integrating the technique in the standard direct learning architecture scheme for digital predistortion. In addition, a discussion on how the implicit regularization helps to deal with numerical problems is presented. It is proven that the implicit regression for the modeling and linearization of PAs can be carried out in a distributed fashion with similar accuracy, enabling the use of resource-constrained devices for digital predistortion. Furthermore, a computational assessment is presented for measuring in terms of arithmetic operations how simpler the devices that implement the proposed ADMM can be, compared with the classical least squares (LS) data-centralized approach. A proof of concept with three scenarios is included: the first two scenarios feature a commercial PA driven by 5G-NR signals with a bandwidth of 30 MHz, and the third scenario involves a commercial Doherty PA with a 100-MHz signal. In a first scenario, experimental results show that an ADMM implementation of a digital predistorter can achieve the same performance as the classical LS solution. The benefits of the proposed regularization are examined by assessing ADMM in a second scenario characterized by significant numerical instabilities. Finally, the robustness of the technique is illustrated through the linearization of the Doherty PA with a 100-MHz OFDM signal.

The dislocation hyperbolic augmented Lagrangian algorithm (DHALA) is a new approach to the hyperbolic augmented Lagrangian algorithm (HALA). DHALA is designed to solve convex nonlinear programming problems. We guarantee that the sequence generated by DHALA converges towards a Karush-Kuhn-Tucker point. We are going to observe that DHALA has a slight computational advantage in solving the problems over HALA. Finally, we will computationally illustrate our theoretical results.

This paper aims to propose a topology optimization method on generating porous structures comprising multiple materials. The mathematical optimization formulation is established under the constraints of individual volume fraction of constituent phase or total mass, as well as the local volume fraction of all phases. The original optimization problem with numerous constraints is converted into a box-constrained optimization problem by incorporating all constraints to the augmented Lagrangian function, avoiding the parameter dependence in the conventional aggregation process. Furthermore, the local volume percentage can be precisely satisfied. The effects including the global mass bound, the influence radius and local volume percentage on final designs are exploited through numerical examples. The numerical results also reveal that porous structures keep a balance between the bulk design and periodic design in terms of the resulting compliance. All results, including those for irregular structures and multiple volume fraction constraints, demonstrate that the proposed method can provide an efficient solution for multiple material infill structures. [ABSTRACT FROM AUTHOR]

Many real-world problems not only have complicated nonconvex functional constraints but also use a large number of data points. This motivates the design of efficient stochastic methods on finite-sum or expectation constrained problems. In this paper, we design and analyze stochastic inexact augmented Lagrangian methods (Stoc-iALM) to solve problems involving a nonconvex composite (i.e. smooth + nonsmooth) objective and nonconvex smooth functional constraints. We adopt the standard iALM framework and design a subroutine by using the momentum-based variance-reduced proximal stochastic gradient method (PStorm) and a postprocessing step. Under certain regularity conditions (assumed also in existing works), to reach an ε -KKT point in expectation, we establish an oracle complexity result of O (ε - 5) , which is better than the best-known O (ε - 6) result. Numerical experiments on the fairness constrained problem and the Neyman–Pearson classification problem with real data demonstrate that our proposed method outperforms an existing method with the previously best-known complexity result. [ABSTRACT FROM AUTHOR]

Physics-based optical flow models have been successful in capturing the deformities in fluid motion arising from digital imagery. However, a common theoretical framework analyzing several physics-based models is missing. In this regard, we formulate a general framework for fluid motion estimation using a constraint-based refinement approach. We demonstrate that for a particular choice of constraint, our results closely approximate the classical continuity equation-based method for fluid flow. This closeness is theoretically justified by augmented Lagrangian method in a novel way. The convergence of Uzawa iterates is shown using a modified bounded constraint algorithm. The mathematical well-posedness is studied in a Hilbert space setting. Further, we observe a surprising connection to the Cauchy-Riemann operator that diagonalizes the system leading to a diffusive phenomenon involving the divergence and the curl of the flow. Several numerical experiments are performed and the results are shown on different datasets. Additionally, we demonstrate that a flow-driven refinement process involving the curl of the flow outperforms the classical physics-based optical flow method without any additional assumptions on the image data. [ABSTRACT FROM AUTHOR]

This article proposes an augmented Lagrangian-based topology optimization approach to minimize the volume fraction subject to multiple nodal displacement constraints. The proposed method puts the multiple constraint equations in the objective function and transforms it into a sequence of unconstrained optimization problems. This study explains the theoretical aspects of the employed augmented Lagrangian approach in depth. Sensitivity expression in terms of design variables is identified by exercising the differentiate-then-discretize mechanism, and utilizes a moving asymptote algorithm to sort out a sequence of optimization subproblems. The optimized results are compared and analysed with those from conventional aggregation-based topology optimization. However, the predefined aggregation approach indicates a dependency on the supplied parameters. The numerical two- and three-dimensional examples reveal the viability and reliability of the suggested augmented Lagrangian scheme, which shows advantages in terms of robustness and efficiency. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to provide an efficient solution algorithm based on the augmented Lagrangian framework for optimization problems with a stochastic objective function and deterministic constraints. Our main contribution is combining the augmented Lagrangian framework with adaptive sampling, resulting in an efficient optimization methodology validated with practical examples. To achieve the presented efficiency, we consider inexact solutions for the augmented Lagrangian subproblems, and through an adaptive sampling mechanism, we control the variance in the gradient estimates. Furthermore, we analyze the theoretical performance of the proposed scheme by showing equivalence to a gradient descent algorithm on a Moreau envelope function, and we prove sublinear convergence for convex objectives and linear convergence for strongly convex objectives with affine equality constraints. The worst-case sample complexity of the resulting algorithm, for an arbitrary choice of penalty parameter in the augmented Lagrangian function, is O (ϵ − 3 − δ) , where ϵ > 0 is the expected error of the solution and δ > 0 is a user-defined parameter. If the penalty parameter is chosen to be O (ϵ − 1) , we demonstrate that the result can be improved to O (ϵ − 2) , which is competitive with the other methods employed in the literature. Moreover, if the objective function is strongly convex with affine equality constraints, we obtain O (ϵ − 1 log ⁡ (1 / ϵ)) complexity. Finally, we empirically verify the performance of our adaptive sampling augmented Lagrangian framework in machine learning optimization and engineering design problems, including topology optimization of a heat sink with environmental uncertainty. [ABSTRACT FROM AUTHOR]

In this paper, we propose a unified primal-dual algorithm framework based on the augmented Lagrangian function for composite convex problems with conic inequality constraints. The new framework is highly versatile. First, it not only covers many existing algorithms such as PDHG, Chambolle–Pock, GDA, OGDA and linearized ALM, but also guides us to design a new efficient algorithm called Semi-OGDA (SOGDA). Second, it enables us to study the role of the augmented penalty term in the convergence analysis. Interestingly, a properly selected penalty not only improves the numerical performance of the above methods, but also theoretically enables the convergence of algorithms like PDHG and SOGDA. Under properly designed step sizes and penalty term, our unified framework preserves the O (1 / N) ergodic convergence while not requiring any prior knowledge about the magnitude of the optimal Lagrangian multiplier. Linear convergence rate for affine equality constrained problem is also obtained given appropriate conditions. Finally, numerical experiments on linear programming, ℓ 1 minimization problem, and multi-block basis pursuit problem demonstrate the efficiency of our methods. [ABSTRACT FROM AUTHOR]

We propose a distributed solution for a constrained convex optimizationproblem over a network of clustered agents each consisted of a set ofsubagents. The communication range of the clustered agents is such that theycan form a connected undirected graph topology. The total cost in thisoptimization problem is the sum of the local convex costs of the subagents ofeach cluster. We seek a minimizer of this cost subject to a set of affineequality constraints, and a set of affine inequality constraints specifying thebounds on the decision variables if such bounds exist. We design ourdistributed algorithm in a cluster-based framework which results in asignificant reduction in communication and computation costs. Our proposeddistributed solution is a novel continuous-time algorithm that is linked to theaugmented Lagrangian approach. It converges asymptotically when the local costfunctions are convex and exponentially when they are strongly convex and haveLipschitz gradients. Moreover, we use an $\epsilon$-exact penalty function toaddress the inequality constraints and derive an explicit lower bound on thepenalty function weight to guarantee convergence to $\epsilon$-neighborhood ofthe global minimum value of the cost. A numerical example demonstrates ourresults.

We propose a new alternating direction method of multipliers (ADMM) with an optimal parameter for the unilateral obstacle problem. We first use the five-point difference scheme to discretize the problem. Then, we present an augmented Lagrangian by introducing an auxiliary unknown, and an ADMM is applied to the corresponding saddle-point problem. Through eliminating the primal and auxiliary unknowns, a pure dual algorithm is then used. The convergence of the proposed method is analyzed, and a simple strategy is presented for selecting the optimal parameter, with the largest and smallest eigenvalues of the iterative matrix. Several numerical experiments confirm the theoretical findings of this study.

This paper considers a risk-averse Markov decision process (MDP) with non-risk constraints as a dynamic optimization framework to ensure robustness against unfavorable outcomes in high-stakes sequential decision-making situations such as disaster response. In this regard, strong duality is proved while making no assumptions on the problem’s convexity. This is necessary for some real-world issues, e.g., in the case of deprivation costs in the context of disaster relief, where convexity cannot be ensured. Our theoretical results imply that the problem can be exactly solved in a dual domain where it becomes convex. Based on our duality results, an augmented Lagrangian-based constraint handling mechanism is also developed for risk-averse reinforcement learning algorithms. The mechanism is proved to be theoretically convergent. Finally, we have also empirically established the convergence of the mechanism using a multi-stage disaster response relief allocation problem while using a fixed negative reward scheme as a benchmark.

This paper proposes a logistic regression model with structural sparsity to study the characteristics of innovation and entrepreneurship among college students. The article first analyzes the basic form of the logistic regression model, including the objective function and the selection method for the penalty function. Then, because the ADMM algorithm combines the advantages of augmented Lagrangian and pairwise decomposition, which can reduce the computational difficulty and complexity, based on this advantage, this paper designs the ADMM algorithm solution framework that is favorable for distributed computing. Finally, this paper analyzes the relationship between the development of innovation and entrepreneurship ability of students in R colleges and universities and their gender, grade, academic foundation, experience in clubs and discipline type. The results yielded that college students’ mean value of innovation and entrepreneurship competence in HEI R was 3.734. The mean value of the scores of each sub-competence ranged from 3.531 to 3.918, which puts the overall innovation and entrepreneurship competence of students in this university at an intermediate level. Therefore, this study plays an important role in understanding the innovation and entrepreneurial characteristics of students in higher education.

By revisiting some popular operator splitting ideas, we present several classes of splitting schemes based on the Lagrangian, primal–dual and hybrid formulations, from which can be recovered many existing algorithms, including alternating direction method of multipliers and primal–dual hybrid gradient algorithms. In particular, we show that the generalized proximal point framework is at the root of many past and recent splitting algorithms allowing for an elementary convergence analysis of these methods through a unified scheme. The numerical tests on constrained total variation minimization show that the proposed algorithms could offer more freedom in parameter selections and, thus, achieve faster convergence speed. [ABSTRACT FROM AUTHOR]

Currently, fatigue-resistance topology optimization has received ever increasing attention, in which most of the literature considers this issue as a simple extension of stress-based topology optimization. However, previous approaches may not be applicable when considering general loads, as the conventional uniaxial rainflow counting method, commonly employed in prior studies, can result in significant errors. Furthermore, the inclusion of general loads introduces additional nonlinearity to fatigue-resistant topology optimization, posing challenges in identifying the optimal solution. To this end, a novel methodology for fatigue-resistance topology optimization considering general loads is proposed in this paper. The independent rainflow counting method is utilized during the process of structural damage estimation. The damage penalization model is subsequently adopted to reduce the nonlinearity by scaling the value of fatigue damage. To illustrate the necessity of an independent rainflow counting method, an example of a double L-shaped structure subjected to general loads is presented. The augmented Lagrangian (AL) approach is introduced to transform numerous damage constraint equations into the objective function, generating a sequence of box-constrained optimization sub-problems. After employing the typical SIMP technique, the relative sensitivities of the AL function regarding design variables are derived, which facilitates the efficient solution using the method of moving asymptotes (MMA). Through 2D and 3D numerical tests, the effectiveness of the proposed method is validated in comparison to the traditional method. Further investigation is conducted into the influences of general loads, damage penalization model, and manufacturing error robustness. In addition, the fatigue-resistance performance of a bearing support of a wind turbine is improved by the suggested approach, and its overall weight is decreased by 25.40%. The proposed method addresses the nonlinear and localized nature of fatigue-resistant topology optimization more efficiently. The results indicate that the proposed method can develop a lightweight design for structures under general loads. [ABSTRACT FROM AUTHOR]

The value at risk (VaR) and the conditional value at risk (CVaR) are two popular risk measures to hedge against the uncertainty of data. In this paper, we provide a computational toolbox for solving high-dimensional sparse linear regression problems under either VaR or CVaR measures, the former being nonconvex and the latter convex. Unlike the empirical risk (neutral) minimization models in which the overall losses are decomposable across data, the aforementioned risk-sensitive models have nonseparable objective functions so that typical first order algorithms are not easy to scale. We address this scaling issue by adopting a semismooth Newton-based proximal augmented Lagrangian method of the convex CVaR linear regression problem. The matrix structures of the Newton systems are carefully explored to reduce the computational cost per iteration. The method is further embedded in a majorization–minimization algorithm as a subroutine to tackle the nonconvex VaR-based regression problem. We also discuss an adaptive sieving strategy to iteratively guess and adjust the effective problem dimension, which is particularly useful when a solution path associated with a sequence of tuning parameters is needed. Extensive numerical experiments on both synthetic and real data demonstrate the effectiveness of our proposed methods. In particular, they are about 53 times faster than the commercial package Gurobi for the CVaR-based sparse linear regression with 4,265,669 features and 16,087 observations. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported in part by the NSF, the Division of Computing and Communication Foundations [Grant 2153352], the National Natural Science Foundation of China [Grant 12271187], and the Hong Kong Research Grant Council [Grant 15304019]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1282) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2022.0012) at (http://dx.doi.org/10.5281/zenodo.7483279). [ABSTRACT FROM AUTHOR]

This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication and decomposition strategy, the method presented here explicitly handles these difficult constraints, thus generating iterates which are feasible with respect to them, while the remaining (standard and supposingly simple) constraints are tackled by sequential penalization. Inexact optimization steps are proven sufficient for the resulting algorithm to work, so that it is employable even with difficult objective functions. The current work is therefore a significant generalization of existing papers on penalty decomposition methods. On the other hand, it is related to some recent publications which use an augmented Lagrangian idea to solve optimization problems with geometric constraints. Compared to these methods, the decomposition idea is shown to be numerically superior since it allows much more freedom in the choice of the subproblem solver, and since the number of certain (possibly expensive) projection steps is significantly less. Extensive numerical results on several highly complicated classes of optimization problems in vector and matrix spaces indicate that the current method is indeed very efficient to solve these problems. [ABSTRACT FROM AUTHOR]

The effects of an included temperature field in the contact process between a piezoelectric body and a rigid foundation with thermal and electrical conductivity are discussed. The constitutive relation of the material is assumed to be thermo-electro-elastic and involves the nonlinear elastic constitutive Hencky's law. The frictional contact is modeled with Signorini's conditions, the regularized Coulomb law, and the regularized electrical and thermal conductivity conditions. The resulting thermo-electromechanical model includes the temperature field as an additional state variable to take into account thermal effects alongside with those of the piezoelectric. The existence of the unique weak solution to the problem is established by using Schauder's fixed point theorem combined with arguments from the theory of variational inequalities involving nonlinear strongly monotone Lipschitz continuous operators. A successive iteration technique to solve the problem numerically is proposed, and its convergence is established. A variant of the Augmented Lagrangian, the so-called Alternating Direction Method of multipliers (ADMM), is used to decompose the original problem into two sub-problems, solve them sequentially and update the dual variables at each iteration. The influence of the thermal boundary condition on the behavior of contact forces and electrical potential is shown through graphical illustrations. [ABSTRACT FROM AUTHOR]

We consider solving nonlinear optimization problems with a stochastic objective and deterministic equality constraints. We assume for the objective that its evaluation, gradient, and Hessian are inaccessible, while one can compute their stochastic estimates by, for example, subsampling. We propose a stochastic algorithm based on sequential quadratic programming (SQP) that uses a differentiable exact augmented Lagrangian as the merit function. To motivate our algorithm design, we first revisit and simplify an old SQP method Lucidi (J. Optim. Theory Appl. 67(2): 227–245, 1990) developed for solving deterministic problems, which serves as the skeleton of our stochastic algorithm. Based on the simplified deterministic algorithm, we then propose a non-adaptive SQP for dealing with stochastic objective, where the gradient and Hessian are replaced by stochastic estimates but the stepsizes are deterministic and prespecified. Finally, we incorporate a recent stochastic line search procedure Paquette and Scheinberg (SIAM J. Optim. 30(1): 349–376 2020) into the non-adaptive stochastic SQP to adaptively select the random stepsizes, which leads to an adaptive stochastic SQP. The global "almost sure" convergence for both non-adaptive and adaptive SQP methods is established. Numerical experiments on nonlinear problems in CUTEst test set demonstrate the superiority of the adaptive algorithm. [ABSTRACT FROM AUTHOR]

Based on augmented Lagrangian, we propose in this paper a new dual for inequality constrained nonconvex generalized fractional programs (GFP). We give duality results under quite weak assumptions. We associate with this dual program, parametric dual subproblems and establish duality results with the usual parametric primal ones. By taking advantage of the concavity of the parametric dual functions, we propose proximal bundle-like methods that approximately solve the parametric dual subproblems, to finally solve this dual program. For some problems, these method converge linearly. [ABSTRACT FROM AUTHOR]

We discuss several algorithms for solving a network optimization problem of simultaneous routing and bandwidth allocation in green networks in a decomposed way, based on the augmented Lagrangian. The problem is difficult due to the nonconvexity caused by binary routing variables. The chosen algorithms, which are several versions of the Multiplier Method, including the Alternating Direction Method of Multipliers (ADMM), have been implemented in Python and tested on several networks’ data. We derive theoretical formulations for the inequality constraints of the Bertsekas, Tatjewski and SALA methods, formulated originally for problems with equality constraints. We also introduce some modifications to the Bertsekas and Tatjewski methods, without which they do not work for an MINLP problem. The final comparison of the performance of these algorithms shows a significant advantage of the augmented Lagrangian algorithms, using decomposition for big problems. In our particular case of the simultaneous routing and bandwidth allocation problem, these algorithms seem to be the best choice.

The maintenance of power system equipment, specifically, the maintenance of generating units, is implicitly related to the reliability and economic operation of the power system. In this study, we established a new technique for the maintenance scheduling of a hydrothermal power system. Two coordination schemes are analysed. The first scheme can be described as a primal optimisation scheme because it maintains a feasible solution at every iteration. The second coordination scheme is defined as a dual optimisation scheme. Two variants of the second scheme are evaluated. These two variants correspond to the Lagrangian relaxation method and augmented relaxation method. The Lagrangian relaxation method uses space dilation to update the system incremental costs. The space dilation method considers the difference of two successive subgradients and aids the Lagrangian relaxation method in producing better convergence and minimum total cost, to date. By realising convergence, hydro maintenance is scheduled. The methodology of economic dispatch is presented and solved for hydrothermal maintenance scheduling. The proposed methodology is applied to modified version of the IEEE–RTS.

A mixed signal with several unknown modes is common in the industry and is hard to decompose. Variational Mode Decomposition (VMD) was proposed to decompose a signal into several amplitude-modulated modes in 2014, which overcame the limitations of Empirical Mode Decomposition (EMD), such as sensitivity to noise and sampling. We propose an improved VMD, which is simplified as iVMD. In the new algorithm, we further study and improve the mathematical model of VMD to adapt to the decomposition of the broad-band modes. In the new model, the ideal flattest response is applied, which is derived from the mathematical integral form and obtained from different-order derivatives of the improved modes' definitions. The harmonics can be treated via synthesis in our new model. The iVMD algorithm can decompose the complex harmonic signal and the broad-band modes. The new model is optimized with the alternate direction method of multipliers, and the modes with adaptive broad-band and their respective center frequencies can be decomposed. the experimental results show that iVMD is an effective algorithm based on the artificial and real data collected in our experiments. [ABSTRACT FROM AUTHOR]

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The main purpose of this paper is the numerical analysis of a class of mathematical models that describe the contact between a thermo-piezoelectric body and a conductive foundation. Under the assumption of a static process, the material's behavior is modeled with a linear thermo-electro-elastic constitutive law and the frictional contact with Signorini's and Tresca's laws. A variational problem is derived and the existence of a unique weak solution is proved by combining arguments from the theory of variational inequalities with linear strongly monotone Lipschitz continuous operators. A successive iteration technique to linearize the problem by transforming it into an incremental recursive form is proposed, and its convergence is established. An Augmented Lagrangian variant, known as the Alternating Direction Multiplier Method (ADMM), is employed to split the original problem into two subproblems, resolve them sequentially, as well as update the dual variables at each iteration. To illustrate the performance of the proposed approach, several numerical simulations on two-dimensional test problems are carried out. [ABSTRACT FROM AUTHOR]

We present a new algorithm for infinite-dimensional optimization with general constraints, called ALESQP. In short, ALESQP is an augmented Lagrangian method that penalizes inequality constraints and solves equality-constrained nonlinear optimization subproblems at every iteration. The subproblems are solved using a matrix-free trust-region sequential quadratic programming (SQP) method that takes advantage of iterative, i.e., inexact linear solvers, and is suitable for large-scale applications. A key feature of ALESQP is a constraint decomposition strategy that allows it to exploit problem-specific variable scalings and inner products. We analyze convergence of ALESQP under different assumptions. We show that strong accumulation points are stationary. Consequently, in finite dimensions ALESQP converges to a stationary point. In infinite dimensions we establish that weak accumulation points are feasible in many practical situations. Under additional assumptions we show that weak accumulation points are stationary. We present several infinite-dimensional examples where ALESQP shows remarkable discretization-independent performance in all of its iterative components, requiring a modest number of iterations to meet constraint tolerances at the level of machine precision. Also, we demonstrate a fully matrix-free solution of an infinite-dimensional problem with nonlinear inequality constraints. [ABSTRACT FROM AUTHOR]

Owing to absorption, reflection, diffraction and, deplorable conditions, the capturing of underwater images present more challenges for the consumer. The proposed work focuses on Enhanced Augmented Lagrangian for image deblurring with additional performance-enhancing optical and sparse derivative prior to model the underwater imaging. The proposed method using Augmented Lagragian with Optical and Sparse Derivative prior is novel in the following ways: (i) usage of optical prior modeled after underwater imaging conditions, taking into account deformations and distortions that accompany the capture of underwater images (ii) sparse derivative prior helps to optimize fast convergence through regularization and faster computation with better edge preservation; (iii) the deblurring begins with the sparsest derivative beforehand and the final deblurred result achieving good dB improvements through sparse regulation. The weights with penalty and regularization ensure that each iteration through steep descent reaches a global minimum. The proposed algorithm performs better as compared to state-of-the-art approaches. [ABSTRACT FROM AUTHOR]

We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a uid governed by the Stokes equations. The mentioned ow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the L2-norm of the curl and the det-grad measure of the uid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields. [ABSTRACT FROM AUTHOR]

The Boussinesq problem, which describes quasi-static indentation of a rigid punch into a deformable body, is studied within the context of nonlinear constitutive equations. By this, the material response expresses the linearized strain in terms of the stress and cannot be inverted in general. A contact area between the punch and the body is unknown a priori, whereas the total contact force is prescribed and yields a non-local integral condition. Consequently, the unilateral indentation problem is stated as a quasi-variational inequality for unknown variables of displacement, stress and indentation depth. The Lagrange multiplier approach is applied in order to establish well-posedness to the underlying physically and geometrically nonlinear problem based on augmented penalty regularization and applying the minimax theorem of Ekeland and Témam. A sufficient solvability condition implies response functions that are bounded, hemi-continuous, coercive and obey a convex potential. A typical example is power-law hardening models for titanium alloys, Norton–Hoff and Ramberg–Osgood materials. This article is part of the theme issue 'Non-smooth variational problems and applications'. [ABSTRACT FROM AUTHOR]