Pub Date : 2023-05-19DOI: 10.1080/10556788.2023.2189717
Qingsong Wang, Deren Han
ABSTRACT In this paper, we consider solving a broad class of large-scale nonconvex and nonsmooth minimization problems by a Bregman proximal stochastic gradient (BPSG) algorithm. The objective function of the minimization problem is the composition of a differentiable and a nondifferentiable function, and the differentiable part does not admit a global Lipschitz continuous gradient. Under some suitable conditions, the subsequential convergence of the proposed algorithm is established. And under expectation conditions with the Kurdyka-Łojasiewicz (KL) property, we also prove that the proposed method converges globally. We also apply the BPSG algorithm to solve sparse nonnegative matrix factorization (NMF), symmetric NMF via non-symmetric relaxation, and matrix completion problems under different kernel generating distances, and numerically compare it with other algorithms. The results demonstrate the robustness and effectiveness of the proposed algorithm.
{"title":"A Bregman stochastic method for nonconvex nonsmooth problem beyond global Lipschitz gradient continuity","authors":"Qingsong Wang, Deren Han","doi":"10.1080/10556788.2023.2189717","DOIUrl":"https://doi.org/10.1080/10556788.2023.2189717","url":null,"abstract":"ABSTRACT In this paper, we consider solving a broad class of large-scale nonconvex and nonsmooth minimization problems by a Bregman proximal stochastic gradient (BPSG) algorithm. The objective function of the minimization problem is the composition of a differentiable and a nondifferentiable function, and the differentiable part does not admit a global Lipschitz continuous gradient. Under some suitable conditions, the subsequential convergence of the proposed algorithm is established. And under expectation conditions with the Kurdyka-Łojasiewicz (KL) property, we also prove that the proposed method converges globally. We also apply the BPSG algorithm to solve sparse nonnegative matrix factorization (NMF), symmetric NMF via non-symmetric relaxation, and matrix completion problems under different kernel generating distances, and numerically compare it with other algorithms. The results demonstrate the robustness and effectiveness of the proposed algorithm.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"42 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120919786","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 : 2023-05-19DOI: 10.1080/10556788.2023.2189719
T. D. Chuong, Xinghuo Yu, Andrew Craig Eberhard, C. Li, Chen Liu
In this paper, we consider a bilevel polynomial optimization problem, where the constraint functions of both the upper-level and lower-level problems involve uncertain parameters. We employ the deterministic robust optimization approach to examine the bilevel polynomial optimization problem under data uncertainties by providing lower bound approximations and convergences of sum-of-squares (SOS) relaxations for the robust bilevel polynomial optimization problem. More precisely, we show that under the convexity of the lower-level problem and either the boundedness of the feasible set or the coercivity of the objective function, the global optimal values of SOS relaxation problems are lower bounds of the global optimal value of the robust bilevel polynomial problem and they converge to this global optimal value when the degrees of SOS polynomials in the relaxation problems tend to infinity. Moreover, an application to an electric vehicle charging scheduling problem with renewable energy sources demonstrates that using the proposed SOS relaxation schemes, we obtain more stable optimal values than applying a direct solution approach as the SOS relaxations are capable of solving these models involving data uncertainties in dynamic charging price and weather conditions.
{"title":"Convergences for robust bilevel polynomial programmes with applications","authors":"T. D. Chuong, Xinghuo Yu, Andrew Craig Eberhard, C. Li, Chen Liu","doi":"10.1080/10556788.2023.2189719","DOIUrl":"https://doi.org/10.1080/10556788.2023.2189719","url":null,"abstract":"In this paper, we consider a bilevel polynomial optimization problem, where the constraint functions of both the upper-level and lower-level problems involve uncertain parameters. We employ the deterministic robust optimization approach to examine the bilevel polynomial optimization problem under data uncertainties by providing lower bound approximations and convergences of sum-of-squares (SOS) relaxations for the robust bilevel polynomial optimization problem. More precisely, we show that under the convexity of the lower-level problem and either the boundedness of the feasible set or the coercivity of the objective function, the global optimal values of SOS relaxation problems are lower bounds of the global optimal value of the robust bilevel polynomial problem and they converge to this global optimal value when the degrees of SOS polynomials in the relaxation problems tend to infinity. Moreover, an application to an electric vehicle charging scheduling problem with renewable energy sources demonstrates that using the proposed SOS relaxation schemes, we obtain more stable optimal values than applying a direct solution approach as the SOS relaxations are capable of solving these models involving data uncertainties in dynamic charging price and weather conditions.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"207 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124645155","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 : 2023-05-11DOI: 10.1080/10556788.2023.2196726
Mohammad Maleki, Nahidsadat Majlesinasab, A. Sinha
The hub location problem is one of the challenging subjects in location theory. This problem can benefit transportation, telecommunication or delivery systems, in which hub nodes are responsible for receiving, collecting and delivering commodities. This paper discusses the multiple allocation hub maximal covering problem (MAHMCP). In this problem, the flow of an origin–destination pair is transferred via multiple hubs such that the total flow covered by located hubs is maximized. We present a new model for the MAHMCP, which has a significantly smaller number of binary variables compared to previous models. The proposed model is theoretically stronger than past models, and an empirical study using the Australia Post (AP) dataset demonstrates its effectiveness. Our experiments show that the new formulation provides high-quality solutions and fast run times for instances up to 100 nodes.
{"title":"An efficient model for the multiple allocation hub maximal covering problem","authors":"Mohammad Maleki, Nahidsadat Majlesinasab, A. Sinha","doi":"10.1080/10556788.2023.2196726","DOIUrl":"https://doi.org/10.1080/10556788.2023.2196726","url":null,"abstract":"The hub location problem is one of the challenging subjects in location theory. This problem can benefit transportation, telecommunication or delivery systems, in which hub nodes are responsible for receiving, collecting and delivering commodities. This paper discusses the multiple allocation hub maximal covering problem (MAHMCP). In this problem, the flow of an origin–destination pair is transferred via multiple hubs such that the total flow covered by located hubs is maximized. We present a new model for the MAHMCP, which has a significantly smaller number of binary variables compared to previous models. The proposed model is theoretically stronger than past models, and an empirical study using the Australia Post (AP) dataset demonstrates its effectiveness. Our experiments show that the new formulation provides high-quality solutions and fast run times for instances up to 100 nodes.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126794542","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 : 2023-05-09DOI: 10.1080/10556788.2023.2205645
S. Sass, A. Tsoukalas, I. Bell, D. Bongartz, J. Najman, A. Mitsos
We focus on deterministic global optimization (DGO) for nonconvex parameter estimation problems. Realistic and accurate solutions often require a fitting against very large measurement data sets, resulting in intractable size for DGO. Thus, we aim at accelerating the branch-and-bound algorithm by using reduced data sets for constructing valid bounds. We focus on fitting the equation of state for propane, which is of high interest for the chemical industry. The resulting estimation problem is a challenging nonconvex mixed-integer nonlinear optimization problem. We investigate the validity of using reduced data sets by comparing how the lower and upper bounds change when replacing the full data set with different reduced data sets. We account for regions with high and low quality fits by considering the results for the whole feasible region and 100 different subregions. Our results indicate that both regions containing solution candidates and regions containing only low quality fits can be identified based on reduced data sets. Moreover, we observe that the average CPU time per branch-and-bound iteration typically decreases if reduced data sets are used.
{"title":"Towards global parameter estimation exploiting reduced data sets","authors":"S. Sass, A. Tsoukalas, I. Bell, D. Bongartz, J. Najman, A. Mitsos","doi":"10.1080/10556788.2023.2205645","DOIUrl":"https://doi.org/10.1080/10556788.2023.2205645","url":null,"abstract":"We focus on deterministic global optimization (DGO) for nonconvex parameter estimation problems. Realistic and accurate solutions often require a fitting against very large measurement data sets, resulting in intractable size for DGO. Thus, we aim at accelerating the branch-and-bound algorithm by using reduced data sets for constructing valid bounds. We focus on fitting the equation of state for propane, which is of high interest for the chemical industry. The resulting estimation problem is a challenging nonconvex mixed-integer nonlinear optimization problem. We investigate the validity of using reduced data sets by comparing how the lower and upper bounds change when replacing the full data set with different reduced data sets. We account for regions with high and low quality fits by considering the results for the whole feasible region and 100 different subregions. Our results indicate that both regions containing solution candidates and regions containing only low quality fits can be identified based on reduced data sets. Moreover, we observe that the average CPU time per branch-and-bound iteration typically decreases if reduced data sets are used.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126835402","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 : 2023-05-02DOI: 10.1080/10556788.2023.2205646
F. Madera-Ramírez, J. Trejo-Sánchez, J. López-Martínez, J. Ríos-Martínez
Crossing minimization is a natural problem in graph drawing, which consists of drawing it in the plane with a minimum number of crossings on the edges. We present an algorithm to minimize the crossing edges in radial layered graphs. The algorithm determines the optimal location of nodes in their corresponding circle layer and consists of two phases. In the first phase, a counterclockwise rotation of concentric circles is performed; in the next phase, edges are converted into segment paths. Unlike other algorithms with circular drawings, our algorithm does not require the insertion or removal of nodes and edges; only rotation and path construction operations are utilized. Two versions of the algorithm were created; the exhaustive version tests all the possible paths, while the random version tests a few number of paths. The goal is to minimize the crossing edges to obtain a clear visualization using three criteria: rotation, crossing edge detection and segment path construction. The algorithm has been successfully implemented in 30 instances of graphs with 20, 35 and 50 layers, where crossings are minimized in 86–93%, 93–96% and 95–97%, respectively.
{"title":"Crossing edge minimization in radial outerplanar layered graphs using segment paths","authors":"F. Madera-Ramírez, J. Trejo-Sánchez, J. López-Martínez, J. Ríos-Martínez","doi":"10.1080/10556788.2023.2205646","DOIUrl":"https://doi.org/10.1080/10556788.2023.2205646","url":null,"abstract":"Crossing minimization is a natural problem in graph drawing, which consists of drawing it in the plane with a minimum number of crossings on the edges. We present an algorithm to minimize the crossing edges in radial layered graphs. The algorithm determines the optimal location of nodes in their corresponding circle layer and consists of two phases. In the first phase, a counterclockwise rotation of concentric circles is performed; in the next phase, edges are converted into segment paths. Unlike other algorithms with circular drawings, our algorithm does not require the insertion or removal of nodes and edges; only rotation and path construction operations are utilized. Two versions of the algorithm were created; the exhaustive version tests all the possible paths, while the random version tests a few number of paths. The goal is to minimize the crossing edges to obtain a clear visualization using three criteria: rotation, crossing edge detection and segment path construction. The algorithm has been successfully implemented in 30 instances of graphs with 20, 35 and 50 layers, where crossings are minimized in 86–93%, 93–96% and 95–97%, respectively.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123874435","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 : 2023-04-27DOI: 10.1080/10556788.2023.2189716
Na Huang, Yuhong Dai, D. Orban, M. Saunders
The conjugate gradient (CG) method is a classic Krylov subspace method for solving symmetric positive definite linear systems. We analyze an analogous semi-conjugate gradient (SCG) method, a special case of the existing semi-conjugate direction (SCD) methods, for unsymmetric positive definite linear systems. Unlike CG, SCG requires the solution of a lower triangular linear system to produce each semi-conjugate direction. We prove that SCG is theoretically equivalent to the full orthogonalization method (FOM), which is based on the Arnoldi process and converges in a finite number of steps. Because SCG's triangular system increases in size each iteration, Dai and Yuan [Study on semi-conjugate direction methods for non-symmetric systems, Int. J. Numer. Meth. Eng. 60(8) (2004), pp. 1383–1399] proposed a sliding window implementation (SWI) to improve efficiency. We show that the directions produced are still locally semi-conjugate. A counter-example illustrates that SWI is different from the direct incomplete orthogonalization method (DIOM), which is FOM with a sliding window. Numerical experiments from the convection-diffusion equation and other applications show that SCG is robust and that the sliding window implementation SWI allows SCG to solve large systems efficiently.
共轭梯度法(CG)是求解对称正定线性系统的经典Krylov子空间方法。本文分析了一类非对称正定线性系统的类似半共轭梯度法(SCG)，它是现有半共轭方向法(SCD)的一种特例。与CG不同，SCG需要解一个下三角形线性系统来产生每个半共轭方向。我们证明了SCG在理论上等价于基于Arnoldi过程并在有限步内收敛的完全正交化方法(FOM)。由于SCG的三角系统每次迭代都会增大，Dai和Yuan[非对称系统的半共轭方向方法研究，[j]。j .号码。冰毒。Eng. 60(8) (2004)， pp. 1383-1399]提出了一种滑动窗口实现(SWI)来提高效率。我们证明了产生的方向仍然是局部半共轭的。一个反例说明SWI不同于直接不完全正交方法(DIOM)， DIOM是带滑动窗口的FOM。对流扩散方程和其他应用的数值实验表明，SCG具有鲁棒性，滑动窗口实现SWI使SCG能够有效地求解大型系统。
{"title":"Properties of semi-conjugate gradient methods for solving unsymmetric positive definite linear systems","authors":"Na Huang, Yuhong Dai, D. Orban, M. Saunders","doi":"10.1080/10556788.2023.2189716","DOIUrl":"https://doi.org/10.1080/10556788.2023.2189716","url":null,"abstract":"The conjugate gradient (CG) method is a classic Krylov subspace method for solving symmetric positive definite linear systems. We analyze an analogous semi-conjugate gradient (SCG) method, a special case of the existing semi-conjugate direction (SCD) methods, for unsymmetric positive definite linear systems. Unlike CG, SCG requires the solution of a lower triangular linear system to produce each semi-conjugate direction. We prove that SCG is theoretically equivalent to the full orthogonalization method (FOM), which is based on the Arnoldi process and converges in a finite number of steps. Because SCG's triangular system increases in size each iteration, Dai and Yuan [Study on semi-conjugate direction methods for non-symmetric systems, Int. J. Numer. Meth. Eng. 60(8) (2004), pp. 1383–1399] proposed a sliding window implementation (SWI) to improve efficiency. We show that the directions produced are still locally semi-conjugate. A counter-example illustrates that SWI is different from the direct incomplete orthogonalization method (DIOM), which is FOM with a sliding window. Numerical experiments from the convection-diffusion equation and other applications show that SCG is robust and that the sliding window implementation SWI allows SCG to solve large systems efficiently.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128811932","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 : 2023-04-24DOI: 10.1080/10556788.2023.2192494
A. Barbagallo, Serena Guarino Lo Bianco
In this paper, we present a general oligopolistic market equilibrium model in which each firm produces several commodities in a time interval. To this aim, we introduce tensor variational inequalities in Hilbert spaces which are a powerful tool to analyse the model. Indeed we characterize the equilibrium condition by means of a suitable time-dependent tensor variational inequality. In addition, we prove some existence and regularity results and a numerical scheme to compute the solution. Finally we provide a numerical example.
{"title":"Infinite dimensional tensor variational inequalities with applications to an economic equilibrium problem","authors":"A. Barbagallo, Serena Guarino Lo Bianco","doi":"10.1080/10556788.2023.2192494","DOIUrl":"https://doi.org/10.1080/10556788.2023.2192494","url":null,"abstract":"In this paper, we present a general oligopolistic market equilibrium model in which each firm produces several commodities in a time interval. To this aim, we introduce tensor variational inequalities in Hilbert spaces which are a powerful tool to analyse the model. Indeed we characterize the equilibrium condition by means of a suitable time-dependent tensor variational inequality. In addition, we prove some existence and regularity results and a numerical scheme to compute the solution. Finally we provide a numerical example.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124272548","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 : 2023-03-30DOI: 10.1080/10556788.2023.2189718
Xianzhen Jiang, Huihui Yang, Ji Jian, Xiaodi Wu
In this paper, two families of hybrid conjugate gradient methods with restart procedures are proposed. Their hybrid conjugate parameters are yielded by projection or convex combination of the classical parameters. Moreover, their restart procedures are given uniformly, which are determined by the proposed hybrid conjugate parameters. The search directions of the presented families satisfy the sufficient descent condition. Under usual assumption and the weak Wolfe line search, the proposed families are proved to be globally convergent. Finally, choosing a specific parameter for each family to solve large-scale unconstrained optimization problems, convex constrained nonlinear monotone equations and image restoration problems. All the numerical results are reported and analysed, which show that the proposed families of hybrid conjugate gradient methods are promising.
{"title":"Two families of hybrid conjugate gradient methods with restart procedures and their applications","authors":"Xianzhen Jiang, Huihui Yang, Ji Jian, Xiaodi Wu","doi":"10.1080/10556788.2023.2189718","DOIUrl":"https://doi.org/10.1080/10556788.2023.2189718","url":null,"abstract":"In this paper, two families of hybrid conjugate gradient methods with restart procedures are proposed. Their hybrid conjugate parameters are yielded by projection or convex combination of the classical parameters. Moreover, their restart procedures are given uniformly, which are determined by the proposed hybrid conjugate parameters. The search directions of the presented families satisfy the sufficient descent condition. Under usual assumption and the weak Wolfe line search, the proposed families are proved to be globally convergent. Finally, choosing a specific parameter for each family to solve large-scale unconstrained optimization problems, convex constrained nonlinear monotone equations and image restoration problems. All the numerical results are reported and analysed, which show that the proposed families of hybrid conjugate gradient methods are promising.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131386035","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 : 2023-03-29DOI: 10.1080/10556788.2023.2189712
Savin Treanțǎ
In this paper, we establish a minimal criterion of efficiency for a class of multiobjective variational control problems. Thus, some minimal conditions are formulated so that all local efficient solutions are also global. Also, we present an example to illustrate the mathematical developments derived in the paper.
{"title":"On a minimal criterion of efficiency in vector variational control problems","authors":"Savin Treanțǎ","doi":"10.1080/10556788.2023.2189712","DOIUrl":"https://doi.org/10.1080/10556788.2023.2189712","url":null,"abstract":"In this paper, we establish a minimal criterion of efficiency for a class of multiobjective variational control problems. Thus, some minimal conditions are formulated so that all local efficient solutions are also global. Also, we present an example to illustrate the mathematical developments derived in the paper.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"205 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132687714","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: 10.1080/10556788.2023.2189711
G. N. Grapiglia
This paper addresses the worst-case evaluation complexity of a version of the standard quadratic penalty method for smooth nonconvex optimization problems with constraints. The method analysed allows inexact solution of the subproblems and do not require prior knowledge of the Lipschitz constants related with the problem. When an approximate feasible point is used as starting point, it is shown that the referred method takes at most outer iterations to generate an ϵ-approximate KKT point, where is the first penalty parameter. For equality constrained problems, this bound yields to an evaluation complexity bound of , when and suitable first-order methods are used as inner solvers. For problems having only linear equality constraints, an evaluation complexity bound of is established when appropriate p-order methods ( ) are used as inner solvers. Illustrative numerical results are also presented and corroborate the theoretical predictions.
{"title":"Worst-case evaluation complexity of a quadratic penalty method for nonconvex optimization","authors":"G. N. Grapiglia","doi":"10.1080/10556788.2023.2189711","DOIUrl":"https://doi.org/10.1080/10556788.2023.2189711","url":null,"abstract":"This paper addresses the worst-case evaluation complexity of a version of the standard quadratic penalty method for smooth nonconvex optimization problems with constraints. The method analysed allows inexact solution of the subproblems and do not require prior knowledge of the Lipschitz constants related with the problem. When an approximate feasible point is used as starting point, it is shown that the referred method takes at most outer iterations to generate an ϵ-approximate KKT point, where is the first penalty parameter. For equality constrained problems, this bound yields to an evaluation complexity bound of , when and suitable first-order methods are used as inner solvers. For problems having only linear equality constraints, an evaluation complexity bound of is established when appropriate p-order methods ( ) are used as inner solvers. Illustrative numerical results are also presented and corroborate the theoretical predictions.","PeriodicalId":124811,"journal":{"name":"Optimization Methods and Software","volume":"90 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":"126800182","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：