Pub Date : 2023-03-28DOI: 10.1080/10556788.2023.2189713
Xuyang Wu, K. C. Sou, Jie Lu
In this paper, we develop a regularized Fenchel dual gradient method (RFDGM), which allows nodes in a time-varying undirected network to find a common decision, in a fully distributed fashion, for minimizing the sum of their local objective functions subject to their local constraints. Different from most existing distributed optimization algorithms that also cope with time-varying networks, RFDGM is able to handle problems with general convex objective functions and distinct local constraints, and still has non-asymptotic convergence results. Specifically, under a standard network connectivity condition, we show that RFDGM is guaranteed to reach ϵ-accuracy in both optimality and feasibility within iterations. Such iteration complexity can be improved to if the local objective functions are strongly convex but not necessarily differentiable. Finally, simulation results demonstrate the competence of RFDGM in practice.
{"title":"A Fenchel dual gradient method enabling regularization for nonsmooth distributed optimization over time-varying networks","authors":"Xuyang Wu, K. C. Sou, Jie Lu","doi":"10.1080/10556788.2023.2189713","DOIUrl":"https://doi.org/10.1080/10556788.2023.2189713","url":null,"abstract":"In this paper, we develop a regularized Fenchel dual gradient method (RFDGM), which allows nodes in a time-varying undirected network to find a common decision, in a fully distributed fashion, for minimizing the sum of their local objective functions subject to their local constraints. Different from most existing distributed optimization algorithms that also cope with time-varying networks, RFDGM is able to handle problems with general convex objective functions and distinct local constraints, and still has non-asymptotic convergence results. Specifically, under a standard network connectivity condition, we show that RFDGM is guaranteed to reach ϵ-accuracy in both optimality and feasibility within iterations. Such iteration complexity can be improved to if the local objective functions are strongly convex but not necessarily differentiable. Finally, simulation results demonstrate the competence of RFDGM in practice.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126956561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-28DOI: 10.1080/10556788.2021.2022147
Henrik A. Friberg
The exponential function and its logarithmic counterpart are essential corner stones of nonlinear mathematical modelling. In this paper, we treat their conic extensions, the exponential cone and the relative entropy cone, in primal, dual and polar form, and show that finding the nearest mapping of a point onto these convex sets all reduce to a single univariate root-finding problem. This leads to a fast projection algorithm shown numerically robust over a wide range of inputs.
{"title":"Projection onto the exponential cone: a univariate root-finding problem","authors":"Henrik A. Friberg","doi":"10.1080/10556788.2021.2022147","DOIUrl":"https://doi.org/10.1080/10556788.2021.2022147","url":null,"abstract":"The exponential function and its logarithmic counterpart are essential corner stones of nonlinear mathematical modelling. In this paper, we treat their conic extensions, the exponential cone and the relative entropy cone, in primal, dual and polar form, and show that finding the nearest mapping of a point onto these convex sets all reduce to a single univariate root-finding problem. This leads to a fast projection algorithm shown numerically robust over a wide range of inputs.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124134740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-28DOI: 10.1080/10556788.2023.2189715
Deliang Wei, Peng Chen, Fang Li, Xiangyun Zhang
A growing number of training methods for generative adversarial networks (GANs) are differential games. Different from convex optimization problems on single functions, gradient descent on multiple objectives may not converge to stable fixed points (SFPs). In order to improve learning dynamics in such games, many recently proposed methods utilize the second-order information of the game, such as the Hessian matrix. Unfortunately, these methods often suffer from the enormous computational cost of Hessian, which hinders their further applications. In this paper, we present efficient second-order optimization (ESO), in which only a part of Hessian is updated in each iteration, and the algorithm is derived. Furthermore, we give the local convergence of the method under reasonable assumptions. In order to further speed up the training process of GANs, we propose efficient second-order optimization with predictions (ESOP) using a novel accelerator. Basic experiments show that the proposed learning methods are faster than some state-of-art methods in GANs, while applicable to many other n-player differential games with local convergence guarantee.
{"title":"Efficient second-order optimization with predictions in differential games","authors":"Deliang Wei, Peng Chen, Fang Li, Xiangyun Zhang","doi":"10.1080/10556788.2023.2189715","DOIUrl":"https://doi.org/10.1080/10556788.2023.2189715","url":null,"abstract":"A growing number of training methods for generative adversarial networks (GANs) are differential games. Different from convex optimization problems on single functions, gradient descent on multiple objectives may not converge to stable fixed points (SFPs). In order to improve learning dynamics in such games, many recently proposed methods utilize the second-order information of the game, such as the Hessian matrix. Unfortunately, these methods often suffer from the enormous computational cost of Hessian, which hinders their further applications. In this paper, we present efficient second-order optimization (ESO), in which only a part of Hessian is updated in each iteration, and the algorithm is derived. Furthermore, we give the local convergence of the method under reasonable assumptions. In order to further speed up the training process of GANs, we propose efficient second-order optimization with predictions (ESOP) using a novel accelerator. Basic experiments show that the proposed learning methods are faster than some state-of-art methods in GANs, while applicable to many other n-player differential games with local convergence guarantee.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129685614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-16DOI: 10.1080/10556788.2022.2152023
S. Adly, M. Haddou, Manh Hung Le
In this paper, we propose to solve Pareto eigenvalue complementarity problems by using interior-point methods. Precisely, we focus the study on an adaptation of the Mehrotra Predictor Corrector Method (MPCM) and a Non-Parametric Interior Point Method (NPIPM). We compare these two methods with two alternative methods, namely the Lattice Projection Method (LPM) and the Soft Max Method (SM). On a set of data generated from the Matrix Market, the performance profiles highlight the efficiency of MPCM and NPIPM for solving eigenvalue complementarity problems. We also consider an application to a concrete and large size situation corresponding to a geomechanical fracture problem. Finally, we discuss the extension of MPCM and NPIPM methods to solve quadratic pencil eigenvalue problems under conic constraints.
{"title":"Interior point methods for solving Pareto eigenvalue complementarity problems","authors":"S. Adly, M. Haddou, Manh Hung Le","doi":"10.1080/10556788.2022.2152023","DOIUrl":"https://doi.org/10.1080/10556788.2022.2152023","url":null,"abstract":"In this paper, we propose to solve Pareto eigenvalue complementarity problems by using interior-point methods. Precisely, we focus the study on an adaptation of the Mehrotra Predictor Corrector Method (MPCM) and a Non-Parametric Interior Point Method (NPIPM). We compare these two methods with two alternative methods, namely the Lattice Projection Method (LPM) and the Soft Max Method (SM). On a set of data generated from the Matrix Market, the performance profiles highlight the efficiency of MPCM and NPIPM for solving eigenvalue complementarity problems. We also consider an application to a concrete and large size situation corresponding to a geomechanical fracture problem. Finally, we discuss the extension of MPCM and NPIPM methods to solve quadratic pencil eigenvalue problems under conic constraints.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123640158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-16DOI: 10.1080/10556788.2022.2157002
Amélie Lambert
We consider the exact solution of Problem (P) which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit description of new general triangle inequalities that are derived from the ranges of the variables of (P). We show that they extend the triangle inequalities, introduced for the binary case, to variables that belong to a generic interval. We also prove that these inequalities cut feasible solutions of McCormick envelopes, and we relate them to the literature. We then introduce (SDP), a strong semidefinite relaxation of (P), that we call ‘Shor's plus RLT plus Triangle’, which includes both the McCormick envelopes and the general triangle inequalities. We further show how to compute a convex relaxation whose optimal value reaches the value of (SDP). In order to handle these inequalities in the solution of (SDP), we solve it by a heuristic that also serves as a separation algorithm. We then solve (P) to global optimality with a branch-and-bound based on . Finally, we show that our method outperforms the compared solvers.
{"title":"Using general triangle inequalities within quadratic convex reformulation method","authors":"Amélie Lambert","doi":"10.1080/10556788.2022.2157002","DOIUrl":"https://doi.org/10.1080/10556788.2022.2157002","url":null,"abstract":"We consider the exact solution of Problem (P) which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit description of new general triangle inequalities that are derived from the ranges of the variables of (P). We show that they extend the triangle inequalities, introduced for the binary case, to variables that belong to a generic interval. We also prove that these inequalities cut feasible solutions of McCormick envelopes, and we relate them to the literature. We then introduce (SDP), a strong semidefinite relaxation of (P), that we call ‘Shor's plus RLT plus Triangle’, which includes both the McCormick envelopes and the general triangle inequalities. We further show how to compute a convex relaxation whose optimal value reaches the value of (SDP). In order to handle these inequalities in the solution of (SDP), we solve it by a heuristic that also serves as a separation algorithm. We then solve (P) to global optimality with a branch-and-bound based on . Finally, we show that our method outperforms the compared solvers.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128543266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-05DOI: 10.1080/10556788.2022.2157001
Fatemeh Keshavarz-Kohjerdi, A. Bagheri
The Hamiltonian cycle problem is one of the most important problems in graph theory that has many applications. This problem is NP-complete for general grid graphs. For solid grid graphs, there are polynomial-time algorithms. Existence of polynomial-time algorithms for grid graphs with few holes has been asked in the literature. In this paper, we give a linear-time algorithm for rectangular grid graphs with two rectangular holes. This extends the previous result for rectangular grid graphs with one rectangular hole. We first present the forbidden conditions in which there is no Hamiltonian cycle for any grid graphs, including rectangular grid graphs with rectangular holes. We then show that if these forbidden conditions do not hold, then there exists a Hamiltonian cycle. The proof is constructive, therefore, it gives an algorithm. An application of the problem is in off-line robot exploration.
{"title":"A linear-time algorithm for finding Hamiltonian cycles in rectangular grid graphs with two rectangular holes","authors":"Fatemeh Keshavarz-Kohjerdi, A. Bagheri","doi":"10.1080/10556788.2022.2157001","DOIUrl":"https://doi.org/10.1080/10556788.2022.2157001","url":null,"abstract":"The Hamiltonian cycle problem is one of the most important problems in graph theory that has many applications. This problem is NP-complete for general grid graphs. For solid grid graphs, there are polynomial-time algorithms. Existence of polynomial-time algorithms for grid graphs with few holes has been asked in the literature. In this paper, we give a linear-time algorithm for rectangular grid graphs with two rectangular holes. This extends the previous result for rectangular grid graphs with one rectangular hole. We first present the forbidden conditions in which there is no Hamiltonian cycle for any grid graphs, including rectangular grid graphs with rectangular holes. We then show that if these forbidden conditions do not hold, then there exists a Hamiltonian cycle. The proof is constructive, therefore, it gives an algorithm. An application of the problem is in off-line robot exploration.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131374964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-05DOI: 10.1080/10556788.2022.2142583
Zhaosong Lu, Xiaorui Li, S. Xiang
In this paper, we consider a class of constrained optimization problems whose constraints involve a cardinality or rank constraint. The penalty formulation based on a partial regularization has recently been promoted in the literature to approximate these problems, which usually outperforms the penalty formulation based on a full regularization in terms of solution quality. Nevertheless, the relation between the penalty formulation with a partial regularizer and the original problem was not much studied yet. Under some suitable assumptions, we show that the penalty formulation based on a partial regularization is an exact reformulation of the original problem in the sense that they both share the same global minimizers. We also show that a local minimizer of the original problem is that of the penalty reformulation. These results provide some theoretical justification for the often-observed superior performance of the penalty model based on a partial regularizer over a corresponding full regularizer.
{"title":"Exact penalization for cardinality and rank-constrained optimization problems via partial regularization","authors":"Zhaosong Lu, Xiaorui Li, S. Xiang","doi":"10.1080/10556788.2022.2142583","DOIUrl":"https://doi.org/10.1080/10556788.2022.2142583","url":null,"abstract":"In this paper, we consider a class of constrained optimization problems whose constraints involve a cardinality or rank constraint. The penalty formulation based on a partial regularization has recently been promoted in the literature to approximate these problems, which usually outperforms the penalty formulation based on a full regularization in terms of solution quality. Nevertheless, the relation between the penalty formulation with a partial regularizer and the original problem was not much studied yet. Under some suitable assumptions, we show that the penalty formulation based on a partial regularization is an exact reformulation of the original problem in the sense that they both share the same global minimizers. We also show that a local minimizer of the original problem is that of the penalty reformulation. These results provide some theoretical justification for the often-observed superior performance of the penalty model based on a partial regularizer over a corresponding full regularizer.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129081174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-05DOI: 10.1080/10556788.2022.2142585
Maryam Khoshsimaye-Bargard, A. Ashrafi
The prominent computational features of the Hestenes–Stiefel parameter as one of the fundamental members of conjugate gradient methods have attracted the attention of many researchers. Yet, as a weak stop, it lacks global convergence for general functions. To overcome this defect, a family of spectral version of Hestenes–Stiefel conjugate gradient methods is introduced. To compute the spectral parameter, in the account of worthy properties of quasi-Newton methods, we minimize the distance between the search direction matrix of the spectral conjugate gradient method and the BFGS (Broyden–Fletcher–Goldfarb–Shanno) update. To achieve sufficient descent property, the search direction is projected in the orthogonal subspace to the gradient of the objective function. The convergence analysis of the proposed method is carried out under standard assumptions for general functions. Finally, the practical merits of the suggested method are investigated by numerical experiments on a set of CUTEr test functions using the Dolan–Moré performance profile. The results show the computational efficiency of the proposed method.
{"title":"A descent family of the spectral Hestenes–Stiefel method by considering the quasi-Newton method","authors":"Maryam Khoshsimaye-Bargard, A. Ashrafi","doi":"10.1080/10556788.2022.2142585","DOIUrl":"https://doi.org/10.1080/10556788.2022.2142585","url":null,"abstract":"The prominent computational features of the Hestenes–Stiefel parameter as one of the fundamental members of conjugate gradient methods have attracted the attention of many researchers. Yet, as a weak stop, it lacks global convergence for general functions. To overcome this defect, a family of spectral version of Hestenes–Stiefel conjugate gradient methods is introduced. To compute the spectral parameter, in the account of worthy properties of quasi-Newton methods, we minimize the distance between the search direction matrix of the spectral conjugate gradient method and the BFGS (Broyden–Fletcher–Goldfarb–Shanno) update. To achieve sufficient descent property, the search direction is projected in the orthogonal subspace to the gradient of the objective function. The convergence analysis of the proposed method is carried out under standard assumptions for general functions. Finally, the practical merits of the suggested method are investigated by numerical experiments on a set of CUTEr test functions using the Dolan–Moré performance profile. The results show the computational efficiency of the proposed method.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131574033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-04DOI: 10.1080/10556788.2022.2142581
M. Pfetsch, Andreas Schmitt
ABSTRACT This paper addresses the optimal design of resilient systems, in which components can fail. The system can react to failures and its behaviour is described by general mixed integer nonlinear programs, which allows for applications to many (technical) systems. This then leads to a three-level optimization problem. The upper level designs the system minimizing a cost function, the middle level represents worst-case failures of components, i.e. interdicts the system, and the lowest level operates the remaining system. We describe new inequalities that characterize the set of resilient solutions and allow to reformulate the problem. The reformulation can then be solved using a nested branch-and-cut approach. We discuss several improvements, for instance, by taking symmetry into account and strengthening cuts. We demonstrate the effectiveness of our implementation on the optimal design of water networks, robust trusses, and gas networks, in comparison to an approach in which the failure scenarios are directly included into the model.
{"title":"A generic optimization framework for resilient systems","authors":"M. Pfetsch, Andreas Schmitt","doi":"10.1080/10556788.2022.2142581","DOIUrl":"https://doi.org/10.1080/10556788.2022.2142581","url":null,"abstract":"ABSTRACT This paper addresses the optimal design of resilient systems, in which components can fail. The system can react to failures and its behaviour is described by general mixed integer nonlinear programs, which allows for applications to many (technical) systems. This then leads to a three-level optimization problem. The upper level designs the system minimizing a cost function, the middle level represents worst-case failures of components, i.e. interdicts the system, and the lowest level operates the remaining system. We describe new inequalities that characterize the set of resilient solutions and allow to reformulate the problem. The reformulation can then be solved using a nested branch-and-cut approach. We discuss several improvements, for instance, by taking symmetry into account and strengthening cuts. We demonstrate the effectiveness of our implementation on the optimal design of water networks, robust trusses, and gas networks, in comparison to an approach in which the failure scenarios are directly included into the model.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126405604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-21DOI: 10.1080/10556788.2022.2117354
C. Petra, M. Troya, N. Petra, Youngsoo Choi, G. Oxberry, D. Tortorelli
ABSTRACT We present a quasi-Newton interior-point method appropriate for optimization problems with pointwise inequality constraints in Hilbert function spaces. Among others, our methodology applies to optimization problems constrained by partial differential equations (PDEs) that are posed in a reduced-space formulation and have bounds or inequality constraints on the optimized parameter function. We first introduce the formalization of an infinite-dimensional quasi-Newton interior-point algorithm using secant BFGS updates and then proceed to derive a discretized interior-point method capable of working with a wide range of finite element discretization schemes. We also discuss and address mathematical and software interface issues that are pervasive when existing off-the-shelf PDE solvers are to be used with off-the-shelf nonlinear programming solvers. Finally, we elaborate on the numerical and parallel computing strengths and limitations of the proposed methodology on several classes of PDE-constrained problems.
{"title":"On the implementation of a quasi-Newton interior-point method for PDE-constrained optimization using finite element discretizations","authors":"C. Petra, M. Troya, N. Petra, Youngsoo Choi, G. Oxberry, D. Tortorelli","doi":"10.1080/10556788.2022.2117354","DOIUrl":"https://doi.org/10.1080/10556788.2022.2117354","url":null,"abstract":"ABSTRACT We present a quasi-Newton interior-point method appropriate for optimization problems with pointwise inequality constraints in Hilbert function spaces. Among others, our methodology applies to optimization problems constrained by partial differential equations (PDEs) that are posed in a reduced-space formulation and have bounds or inequality constraints on the optimized parameter function. We first introduce the formalization of an infinite-dimensional quasi-Newton interior-point algorithm using secant BFGS updates and then proceed to derive a discretized interior-point method capable of working with a wide range of finite element discretization schemes. We also discuss and address mathematical and software interface issues that are pervasive when existing off-the-shelf PDE solvers are to be used with off-the-shelf nonlinear programming solvers. Finally, we elaborate on the numerical and parallel computing strengths and limitations of the proposed methodology on several classes of PDE-constrained problems.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123843479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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