SIAM Journal on Optimization, Volume 34, Issue 1, Page 790-816, March 2024. Abstract. This paper proposes a path-based approach for the minimization of a continuously differentiable function over sparse symmetric sets, which is a hard problem that exhibits a restrictiveness-hierarchy of necessary optimality conditions. To achieve the more restrictive conditions in the hierarchy, state-of-the-art algorithms require a support optimization oracle that must exactly solve the problem in smaller dimensions. The path-based approach developed in this study produces a path-based optimality condition, which is placed well in the restrictiveness-hierarchy, and a method to achieve it that does not require a support optimization oracle and, moreover, is projection-free. In the development process, new results are derived for the regularized linear minimization problem over sparse symmetric sets, which give additional means to identify optimal solutions for convex and concave objective functions. We complement our results with numerical examples.
{"title":"A Path-Based Approach to Constrained Sparse Optimization","authors":"Nadav Hallak","doi":"10.1137/22m1535498","DOIUrl":"https://doi.org/10.1137/22m1535498","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 790-816, March 2024. <br/> Abstract. This paper proposes a path-based approach for the minimization of a continuously differentiable function over sparse symmetric sets, which is a hard problem that exhibits a restrictiveness-hierarchy of necessary optimality conditions. To achieve the more restrictive conditions in the hierarchy, state-of-the-art algorithms require a support optimization oracle that must exactly solve the problem in smaller dimensions. The path-based approach developed in this study produces a path-based optimality condition, which is placed well in the restrictiveness-hierarchy, and a method to achieve it that does not require a support optimization oracle and, moreover, is projection-free. In the development process, new results are derived for the regularized linear minimization problem over sparse symmetric sets, which give additional means to identify optimal solutions for convex and concave objective functions. We complement our results with numerical examples.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"1 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139919219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Haoya Li, Hsiang-Fu Yu, Lexing Ying, Inderjit S. Dhillon
SIAM Journal on Optimization, Volume 34, Issue 1, Page 764-789, March 2024. Abstract. Entropy regularized Markov decision processes have been widely used in reinforcement learning. This paper is concerned with the primal-dual formulation of the entropy regularized problems. Standard first-order methods suffer from slow convergence due to the lack of strict convexity and concavity. To address this issue, we first introduce a new quadratically convexified primal-dual formulation. The natural gradient ascent descent of the new formulation enjoys global convergence guarantee and exponential convergence rate. We also propose a new interpolating metric that further accelerates the convergence significantly. Numerical results are provided to demonstrate the performance of the proposed methods under multiple settings.
{"title":"Accelerating Primal-Dual Methods for Regularized Markov Decision Processes","authors":"Haoya Li, Hsiang-Fu Yu, Lexing Ying, Inderjit S. Dhillon","doi":"10.1137/21m1468851","DOIUrl":"https://doi.org/10.1137/21m1468851","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 764-789, March 2024. <br/> Abstract. Entropy regularized Markov decision processes have been widely used in reinforcement learning. This paper is concerned with the primal-dual formulation of the entropy regularized problems. Standard first-order methods suffer from slow convergence due to the lack of strict convexity and concavity. To address this issue, we first introduce a new quadratically convexified primal-dual formulation. The natural gradient ascent descent of the new formulation enjoys global convergence guarantee and exponential convergence rate. We also propose a new interpolating metric that further accelerates the convergence significantly. Numerical results are provided to demonstrate the performance of the proposed methods under multiple settings.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"22 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139919220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Optimization, Volume 34, Issue 1, Page 742-763, March 2024. Abstract. This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably correct dual bounds. In this work, we continue to leverage this paradigm and extend an exact branch-and-bound framework by separation routines for safe cutting planes, based on the approach first introduced by Cook, Dash, Fukasawa, and Goycoolea in 2009 [INFORMS J. Comput., 21 (2009), pp. 641–649]. Constraints are aggregated safely using approximate dual multipliers from an LP solve, followed by mixed-integer rounding to generate provably valid, although slightly weaker inequalities. We generalize this approach to problem data that is not representable in floating-point arithmetic, add routines for controlling the encoding length of the resulting cutting planes, and show how these cutting planes can be verified according to the VIPR certificate standard. Furthermore, we analyze the performance impact of these cutting planes in the context of an exact MIP framework, showing that we can solve 21.5% more instances to exact optimality and reduce solving times by 26.8% on the MIPLIB 2017 benchmark test set.
{"title":"Safe and Verified Gomory Mixed-Integer Cuts in a Rational Mixed-Integer Program Framework","authors":"Leon Eifler, Ambros Gleixner","doi":"10.1137/23m156046x","DOIUrl":"https://doi.org/10.1137/23m156046x","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 742-763, March 2024. <br/> Abstract. This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably correct dual bounds. In this work, we continue to leverage this paradigm and extend an exact branch-and-bound framework by separation routines for safe cutting planes, based on the approach first introduced by Cook, Dash, Fukasawa, and Goycoolea in 2009 [INFORMS J. Comput., 21 (2009), pp. 641–649]. Constraints are aggregated safely using approximate dual multipliers from an LP solve, followed by mixed-integer rounding to generate provably valid, although slightly weaker inequalities. We generalize this approach to problem data that is not representable in floating-point arithmetic, add routines for controlling the encoding length of the resulting cutting planes, and show how these cutting planes can be verified according to the VIPR certificate standard. Furthermore, we analyze the performance impact of these cutting planes in the context of an exact MIP framework, showing that we can solve 21.5% more instances to exact optimality and reduce solving times by 26.8% on the MIPLIB 2017 benchmark test set.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"35 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Optimization, Volume 34, Issue 1, Page 718-741, March 2024. Abstract. Linear programming on the Stiefel manifold (LPS) is studied for the first time. It aims at minimizing a linear objective function over the set of all [math]-tuples of orthonormal vectors in [math] satisfying [math] additional linear constraints. Despite the classical polynomial-time solvable case [math], general (LPS) is NP-hard. According to the Shapiro–Barvinok–Pataki theorem, (LPS) admits an exact semidefinite programming relaxation when [math], which is tight when [math]. Surprisingly, we can greatly strengthen this sufficient exactness condition to [math], which covers the classical case [math] and [math]. Regarding (LPS) as a smooth nonlinear programming problem, we reveal a nice property that under the linear independence constraint qualification, the standard first- and second-order local necessary optimality conditions are sufficient for global optimality when [math].
{"title":"Linear Programming on the Stiefel Manifold","authors":"Mengmeng Song, Yong Xia","doi":"10.1137/23m1552243","DOIUrl":"https://doi.org/10.1137/23m1552243","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 718-741, March 2024. <br/> Abstract. Linear programming on the Stiefel manifold (LPS) is studied for the first time. It aims at minimizing a linear objective function over the set of all [math]-tuples of orthonormal vectors in [math] satisfying [math] additional linear constraints. Despite the classical polynomial-time solvable case [math], general (LPS) is NP-hard. According to the Shapiro–Barvinok–Pataki theorem, (LPS) admits an exact semidefinite programming relaxation when [math], which is tight when [math]. Surprisingly, we can greatly strengthen this sufficient exactness condition to [math], which covers the classical case [math] and [math]. Regarding (LPS) as a smooth nonlinear programming problem, we reveal a nice property that under the linear independence constraint qualification, the standard first- and second-order local necessary optimality conditions are sufficient for global optimality when [math].","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"14 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Güzin Bayraksan, Francesca Maggioni, Daniel Faccini, Ming Yang
SIAM Journal on Optimization, Volume 34, Issue 1, Page 682-717, March 2024. Abstract. Multistage mixed-integer distributionally robust optimization (DRO) forms a class of extremely challenging problems since their size grows exponentially with the number of stages. One way to model the uncertainty in multistage DRO is by creating sets of conditional distributions (the so-called conditional ambiguity sets) on a finite scenario tree and requiring that such distributions remain close to nominal conditional distributions according to some measure of similarity/distance (e.g., [math]-divergences or Wasserstein distance). In this paper, new bounding criteria for this class of difficult decision problems are provided through scenario grouping using the ambiguity sets associated with various commonly used [math]-divergences and the Wasserstein distance. Our approach does not require any special problem structure such as linearity, convexity, stagewise independence, and so forth. Therefore, while we focus on multistage mixed-integer DRO, our bounds can be applied to a wide range of DRO problems including two-stage and multistage, with or without integer variables, convex or nonconvex, and nested or nonnested formulations. Numerical results on a multistage mixed-integer production problem show the efficiency of the proposed approach through different choices of partition strategies, ambiguity sets, and levels of robustness.
{"title":"Bounds for Multistage Mixed-Integer Distributionally Robust Optimization","authors":"Güzin Bayraksan, Francesca Maggioni, Daniel Faccini, Ming Yang","doi":"10.1137/22m147178x","DOIUrl":"https://doi.org/10.1137/22m147178x","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 682-717, March 2024. <br/> Abstract. Multistage mixed-integer distributionally robust optimization (DRO) forms a class of extremely challenging problems since their size grows exponentially with the number of stages. One way to model the uncertainty in multistage DRO is by creating sets of conditional distributions (the so-called conditional ambiguity sets) on a finite scenario tree and requiring that such distributions remain close to nominal conditional distributions according to some measure of similarity/distance (e.g., [math]-divergences or Wasserstein distance). In this paper, new bounding criteria for this class of difficult decision problems are provided through scenario grouping using the ambiguity sets associated with various commonly used [math]-divergences and the Wasserstein distance. Our approach does not require any special problem structure such as linearity, convexity, stagewise independence, and so forth. Therefore, while we focus on multistage mixed-integer DRO, our bounds can be applied to a wide range of DRO problems including two-stage and multistage, with or without integer variables, convex or nonconvex, and nested or nonnested formulations. Numerical results on a multistage mixed-integer production problem show the efficiency of the proposed approach through different choices of partition strategies, ambiguity sets, and levels of robustness.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"18 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wutao Si, P.-A. Absil, Wen Huang, Rujun Jiang, Simon Vary
SIAM Journal on Optimization, Volume 34, Issue 1, Page 654-681, March 2024. Abstract. In recent years, the proximal gradient method and its variants have been generalized to Riemannian manifolds for solving optimization problems with an additively separable structure, i.e., [math], where [math] is continuously differentiable, and [math] may be nonsmooth but convex with computationally reasonable proximal mapping. In this paper, we generalize the proximal Newton method to embedded submanifolds for solving the type of problem with [math]. The generalization relies on the Weingarten and semismooth analysis. It is shown that the Riemannian proximal Newton method has a local superlinear convergence rate under certain reasonable assumptions. Moreover, a hybrid version is given by concatenating a Riemannian proximal gradient method and the Riemannian proximal Newton method. It is shown that if the switch parameter is chosen appropriately, then the hybrid method converges globally and also has a local superlinear convergence rate. Numerical experiments on random and synthetic data are used to demonstrate the performance of the proposed methods.
{"title":"A Riemannian Proximal Newton Method","authors":"Wutao Si, P.-A. Absil, Wen Huang, Rujun Jiang, Simon Vary","doi":"10.1137/23m1565097","DOIUrl":"https://doi.org/10.1137/23m1565097","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 654-681, March 2024. <br/> Abstract. In recent years, the proximal gradient method and its variants have been generalized to Riemannian manifolds for solving optimization problems with an additively separable structure, i.e., [math], where [math] is continuously differentiable, and [math] may be nonsmooth but convex with computationally reasonable proximal mapping. In this paper, we generalize the proximal Newton method to embedded submanifolds for solving the type of problem with [math]. The generalization relies on the Weingarten and semismooth analysis. It is shown that the Riemannian proximal Newton method has a local superlinear convergence rate under certain reasonable assumptions. Moreover, a hybrid version is given by concatenating a Riemannian proximal gradient method and the Riemannian proximal Newton method. It is shown that if the switch parameter is chosen appropriately, then the hybrid method converges globally and also has a local superlinear convergence rate. Numerical experiments on random and synthetic data are used to demonstrate the performance of the proposed methods.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"1 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Optimization, Volume 34, Issue 1, Page 642-653, March 2024. Abstract. Proximal mappings are essential in splitting algorithms for both convex and nonconvex optimization. In this paper, we show that proximal mappings of every prox-bounded function are nonexpansive if and only if they are firmly nonexpansive if and only if they are averaged if and only if the function is convex. Lipschitz proximal mappings of prox-bounded functions are also characterized via hypoconvex or strongly convex functions. Our results generalize a recent result due to Rockafellar.
{"title":"Various Notions of Nonexpansiveness Coincide for Proximal Mappings of Functions","authors":"Honglin Luo, Xianfu Wang, Xinmin Yang","doi":"10.1137/23m1597009","DOIUrl":"https://doi.org/10.1137/23m1597009","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 642-653, March 2024. <br/> Abstract. Proximal mappings are essential in splitting algorithms for both convex and nonconvex optimization. In this paper, we show that proximal mappings of every prox-bounded function are nonexpansive if and only if they are firmly nonexpansive if and only if they are averaged if and only if the function is convex. Lipschitz proximal mappings of prox-bounded functions are also characterized via hypoconvex or strongly convex functions. Our results generalize a recent result due to Rockafellar.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"7 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ulysse Marteau-Ferey, Francis Bach, Alessandro Rudi
SIAM Journal on Optimization, Volume 34, Issue 1, Page 616-641, March 2024. Abstract. We consider the problem of decomposing a regular nonnegative function as a sum of squares of functions which preserve some form of regularity. In the same way as decomposing nonnegative polynomials as sum of squares of polynomials allows one to derive methods in order to solve global optimization problems on polynomials, decomposing a regular function as a sum of squares allows one to derive methods to solve global optimization problems on more general functions. As the regularity of the functions in the sum of squares decomposition is a key indicator in analyzing the convergence and speed of convergence of optimization methods, it is important to have theoretical results guaranteeing such a regularity. In this work, we show second order sufficient conditions in order for a [math] times continuously differentiable nonnegative function to be a sum of squares of [math] differentiable functions. The main hypothesis is that, locally, the function grows quadratically in directions which are orthogonal to its set of zeros. The novelty of this result, compared to previous works is that it allows sets of zeros which are continuous as opposed to discrete, and also applies to manifolds as opposed to open sets of [math]. This has applications in problems where manifolds of minimizers or zeros typically appear, such as in optimal transport, and for minimizing functions defined on manifolds.
{"title":"Second Order Conditions to Decompose Smooth Functions as Sums of Squares","authors":"Ulysse Marteau-Ferey, Francis Bach, Alessandro Rudi","doi":"10.1137/22m1480914","DOIUrl":"https://doi.org/10.1137/22m1480914","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 616-641, March 2024. <br/> Abstract. We consider the problem of decomposing a regular nonnegative function as a sum of squares of functions which preserve some form of regularity. In the same way as decomposing nonnegative polynomials as sum of squares of polynomials allows one to derive methods in order to solve global optimization problems on polynomials, decomposing a regular function as a sum of squares allows one to derive methods to solve global optimization problems on more general functions. As the regularity of the functions in the sum of squares decomposition is a key indicator in analyzing the convergence and speed of convergence of optimization methods, it is important to have theoretical results guaranteeing such a regularity. In this work, we show second order sufficient conditions in order for a [math] times continuously differentiable nonnegative function to be a sum of squares of [math] differentiable functions. The main hypothesis is that, locally, the function grows quadratically in directions which are orthogonal to its set of zeros. The novelty of this result, compared to previous works is that it allows sets of zeros which are continuous as opposed to discrete, and also applies to manifolds as opposed to open sets of [math]. This has applications in problems where manifolds of minimizers or zeros typically appear, such as in optimal transport, and for minimizing functions defined on manifolds.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"18 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Optimization, Volume 34, Issue 1, Page 590-615, March 2024. Abstract. We introduce novel polyhedral approximation hierarchies for the cone of nonnegative forms on the unit sphere in [math] and for its (dual) cone of moments. We prove computable quantitative bounds on the speed of convergence of such hierarchies. We also introduce a novel optimization-free algorithm for building converging sequences of lower bounds for polynomial minimization problems on spheres. Finally, some computational results are discussed, showcasing our implementation of these hierarchies in the programming language Julia.
{"title":"Harmonic Hierarchies for Polynomial Optimization","authors":"Sergio Cristancho, Mauricio Velasco","doi":"10.1137/22m1484511","DOIUrl":"https://doi.org/10.1137/22m1484511","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 590-615, March 2024. <br/> Abstract. We introduce novel polyhedral approximation hierarchies for the cone of nonnegative forms on the unit sphere in [math] and for its (dual) cone of moments. We prove computable quantitative bounds on the speed of convergence of such hierarchies. We also introduce a novel optimization-free algorithm for building converging sequences of lower bounds for polynomial minimization problems on spheres. Finally, some computational results are discussed, showcasing our implementation of these hierarchies in the programming language Julia.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"8 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139773222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Optimization, Volume 34, Issue 1, Page 563-589, March 2024. Abstract. Given closed convex sets [math], [math], and some nonzero linear maps [math], [math], of suitable dimensions, the multiset split feasibility problem aims at finding a point in [math] based on computing projections onto [math] and multiplications by [math] and [math]. In this paper, we consider the associated best approximation problem, i.e., the problem of computing projections onto [math]; we refer to this problem as the best approximation problem in multiset split feasibility settings (BA-MSF). We adapt the Dykstra’s projection algorithm, which is classical for solving the BA-MSF in the special case when all [math], to solve the general BA-MSF. Our Dykstra-type projection algorithm is derived by applying (proximal) coordinate gradient descent to the Lagrange dual problem, and it only requires computing projections onto [math] and multiplications by [math] and [math] in each iteration. Under a standard relative interior condition and a genericity assumption on the point we need to project, we show that the dual objective satisfies the Kurdyka-Łojasiewicz property with an explicitly computable exponent on a neighborhood of the (typically unbounded) dual solution set when each [math] is [math]-cone reducible for some [math]: this class of sets covers the class of [math]-cone reducible sets, which include all polyhedrons, second-order cone, and the cone of positive semidefinite matrices as special cases. Using this, explicit convergence rate (linear or sublinear) of the sequence generated by the Dykstra-type projection algorithm is derived. Concrete examples are constructed to illustrate the necessity of some of our assumptions.
{"title":"Convergence Rate Analysis of a Dykstra-Type Projection Algorithm","authors":"Xiaozhou Wang, Ting Kei Pong","doi":"10.1137/23m1545781","DOIUrl":"https://doi.org/10.1137/23m1545781","url":null,"abstract":"SIAM Journal on Optimization, Volume 34, Issue 1, Page 563-589, March 2024. <br/> Abstract. Given closed convex sets [math], [math], and some nonzero linear maps [math], [math], of suitable dimensions, the multiset split feasibility problem aims at finding a point in [math] based on computing projections onto [math] and multiplications by [math] and [math]. In this paper, we consider the associated best approximation problem, i.e., the problem of computing projections onto [math]; we refer to this problem as the best approximation problem in multiset split feasibility settings (BA-MSF). We adapt the Dykstra’s projection algorithm, which is classical for solving the BA-MSF in the special case when all [math], to solve the general BA-MSF. Our Dykstra-type projection algorithm is derived by applying (proximal) coordinate gradient descent to the Lagrange dual problem, and it only requires computing projections onto [math] and multiplications by [math] and [math] in each iteration. Under a standard relative interior condition and a genericity assumption on the point we need to project, we show that the dual objective satisfies the Kurdyka-Łojasiewicz property with an explicitly computable exponent on a neighborhood of the (typically unbounded) dual solution set when each [math] is [math]-cone reducible for some [math]: this class of sets covers the class of [math]-cone reducible sets, which include all polyhedrons, second-order cone, and the cone of positive semidefinite matrices as special cases. Using this, explicit convergence rate (linear or sublinear) of the sequence generated by the Dykstra-type projection algorithm is derived. Concrete examples are constructed to illustrate the necessity of some of our assumptions.","PeriodicalId":49529,"journal":{"name":"SIAM Journal on Optimization","volume":"1 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139773221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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