Pub Date : 2021-06-17DOI: 10.23952/jano.4.2022.1.02
N. Dinh, M. Goberna, M. López, M. Volle
We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems. To Dinh The Luc on the occasion of his 70th anniversary
{"title":"Relaxed Lagrangian duality in convex infinite optimization: Reverse strong duality and optimality","authors":"N. Dinh, M. Goberna, M. López, M. Volle","doi":"10.23952/jano.4.2022.1.02","DOIUrl":"https://doi.org/10.23952/jano.4.2022.1.02","url":null,"abstract":"We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems. To Dinh The Luc on the occasion of his 70th anniversary","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128599517","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 : 2021-05-12DOI: 10.23952/jano.4.2022.2.02
Heinz H. Bauschke, Hui Ouyang, Xianfu Wang
There are two basic angles associated with a pair of linear subspaces: the Diximier angle and the Friedrichs angle. The Dixmier angle of the pair of orthogonal complements is the same as the Dixmier angle of the original pair provided that the original pair gives rise to a direct (not necessarily orthogonal) sum of the underlying Hilbert space. The Friedrichs angles of the original pair and the pair of the orthogonal complements always coincide. These two results are due to Krein, Krasnoselskii, and Milman and to Solmon, respectively. In 1995, Deutsch provided a very nice survey with complete proofs and interesting historical comments. One key result in Deutsch’s survey was an inequality for Dixmier angles provided by Hundal. In this paper, we present extensions of these results to the case when the linear subspaces are only required to be convex cones. It turns out that Hundal’s result has a nice conical extension while the situation is more technical for the results by Krein et al. and by Solmon. Our analysis is based on Deutsch’s survey and our recent work on angles between convex sets. Throughout, we also provide examples illustrating the sharpness of our results.
{"title":"On angles between convex cones","authors":"Heinz H. Bauschke, Hui Ouyang, Xianfu Wang","doi":"10.23952/jano.4.2022.2.02","DOIUrl":"https://doi.org/10.23952/jano.4.2022.2.02","url":null,"abstract":"There are two basic angles associated with a pair of linear subspaces: the Diximier angle and the Friedrichs angle. The Dixmier angle of the pair of orthogonal complements is the same as the Dixmier angle of the original pair provided that the original pair gives rise to a direct (not necessarily orthogonal) sum of the underlying Hilbert space. The Friedrichs angles of the original pair and the pair of the orthogonal complements always coincide. These two results are due to Krein, Krasnoselskii, and Milman and to Solmon, respectively. In 1995, Deutsch provided a very nice survey with complete proofs and interesting historical comments. One key result in Deutsch’s survey was an inequality for Dixmier angles provided by Hundal. In this paper, we present extensions of these results to the case when the linear subspaces are only required to be convex cones. It turns out that Hundal’s result has a nice conical extension while the situation is more technical for the results by Krein et al. and by Solmon. Our analysis is based on Deutsch’s survey and our recent work on angles between convex sets. Throughout, we also provide examples illustrating the sharpness of our results.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126098263","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 : 2019-09-19DOI: 10.23952/jano.2.2020.1.05
G. Herman
The purpose of this short paper is to identify the mathematical essence of the superiorization methodology. This methodology has been developed in recent years while attempting to solve specific application-oriented problems. Consequently, superiorization is often presented using the terminology of such problems. A more general approach is provided here by discussing ideas related to superiorization in terms of an abstract mathematical concept, referred to as a problem structure.
{"title":"Problem structures in the theory and practice of superiorization","authors":"G. Herman","doi":"10.23952/jano.2.2020.1.05","DOIUrl":"https://doi.org/10.23952/jano.2.2020.1.05","url":null,"abstract":"The purpose of this short paper is to identify the mathematical essence of the superiorization methodology. This methodology has been developed in recent years while attempting to solve specific application-oriented problems. Consequently, superiorization is often presented using the terminology of such problems. A more general approach is provided here by discussing ideas related to superiorization in terms of an abstract mathematical concept, referred to as a problem structure.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132748193","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 : 2019-04-19DOI: 10.23952/jano.1.2019.2.02
Akshay Agrawal, Shane T. Barratt, Stephen P. Boyd, Enzo Busseti, W. M. Moursi
We consider the problem of efficiently computing the derivative of the solution map of a convex cone program, when it exists. We do this by implicitly differentiating the residual map for its homogeneous self-dual embedding, and solving the linear systems of equations required using an iterative method. This allows us to efficiently compute the derivative operator, and its adjoint, evaluated at a vector. These correspond to computing an approximate new solution, given a perturbation to the cone program coefficients (i.e., perturbation analysis), and to computing the gradient of a function of the solution with respect to the coefficients. Our method scales to large problems, with numbers of coefficients in the millions. We present an open-source Python implementation of our method that solves a cone program and returns the derivative and its adjoint as abstract linear maps; our implementation can be easily integrated into software systems for automatic differentiation.
{"title":"Differentiating through a cone program","authors":"Akshay Agrawal, Shane T. Barratt, Stephen P. Boyd, Enzo Busseti, W. M. Moursi","doi":"10.23952/jano.1.2019.2.02","DOIUrl":"https://doi.org/10.23952/jano.1.2019.2.02","url":null,"abstract":"We consider the problem of efficiently computing the derivative of the solution map of a convex cone program, when it exists. We do this by implicitly differentiating the residual map for its homogeneous self-dual embedding, and solving the linear systems of equations required using an iterative method. This allows us to efficiently compute the derivative operator, and its adjoint, evaluated at a vector. These correspond to computing an approximate new solution, given a perturbation to the cone program coefficients (i.e., perturbation analysis), and to computing the gradient of a function of the solution with respect to the coefficients. Our method scales to large problems, with numbers of coefficients in the millions. We present an open-source Python implementation of our method that solves a cone program and returns the derivative and its adjoint as abstract linear maps; our implementation can be easily integrated into software systems for automatic differentiation.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129134477","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 : 2013-02-21DOI: 10.23952/jano.1.2019.3.02
B. Mordukhovich, N. M. Nam
In the early 17th century, Pierre de Fermat proposed the following problem: given three points in the plane, find a point such that the sum of its Euclidean distances to the three given points is minimal. This problem was solved by Evangelista Torricelli and was named the {em Fermat-Torricelli problem}. A more general version of the Fermat-Torricelli problem asks for a point that minimizes the sum of the distances to a finite number of given points in $Bbb R^n$. This is one of the main problems in location science. In this paper we revisit the Fermat-Torricelli problem from both theoretical and numerical viewpoints using some ingredients of convex analysis and optimization.
17世纪初，皮埃尔·德·费马(Pierre de Fermat)提出了这样一个问题:给定平面上的三个点，找出一个点与这三个点的欧氏距离之和最小。这个问题被埃万杰里斯塔·托里拆利解决了，并被命名为费马-托里拆利问题。费马-托里拆利问题的一个更一般的版本要求在$Bbb R^n$中找到一个点，使到有限个给定点的距离和最小。这是定位科学的主要问题之一。本文利用凸分析和最优化的一些成分，从理论和数值两方面重新讨论了费马-托里拆利问题。
{"title":"The Fermat-Torricelli problem and Weiszfeld’s algorithm in the light of convex analysis","authors":"B. Mordukhovich, N. M. Nam","doi":"10.23952/jano.1.2019.3.02","DOIUrl":"https://doi.org/10.23952/jano.1.2019.3.02","url":null,"abstract":"In the early 17th century, Pierre de Fermat proposed the following problem: given three points in the plane, find a point such that the sum of its Euclidean distances to the three given points is minimal. This problem was solved by Evangelista Torricelli and was named the {em Fermat-Torricelli problem}. A more general version of the Fermat-Torricelli problem asks for a point that minimizes the sum of the distances to a finite number of given points in $Bbb R^n$. This is one of the main problems in location science. In this paper we revisit the Fermat-Torricelli problem from both theoretical and numerical viewpoints using some ingredients of convex analysis and optimization.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123555673","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 : 1900-01-01DOI: 10.23952/jano.1.2019.1.08
M. Shang
In this paper, a descent-like method is introduced for solving a fixed point problem of a strict pseudocontraction and a split variational inclusion problem. A strong convergence theorem of common solutions is established in the framework of Hilbert spaces without any compact assumptions on any mapping.
{"title":"A descent-like method for fixed points and split conclusion problems","authors":"M. Shang","doi":"10.23952/jano.1.2019.1.08","DOIUrl":"https://doi.org/10.23952/jano.1.2019.1.08","url":null,"abstract":"In this paper, a descent-like method is introduced for solving a fixed point problem of a strict pseudocontraction and a split variational inclusion problem. A strong convergence theorem of common solutions is established in the framework of Hilbert spaces without any compact assumptions on any mapping.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115282079","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 : 1900-01-01DOI: 10.23952/jano.2.2020.3.02
{"title":"An inertial splitting algorithm for solving inclusion problems and its applications to compressed sensing","authors":"","doi":"10.23952/jano.2.2020.3.02","DOIUrl":"https://doi.org/10.23952/jano.2.2020.3.02","url":null,"abstract":"","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115438159","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 : 1900-01-01DOI: 10.23952/jano.4.2022.2.07
{"title":"Convergence of inexact iterates of strict contractions in metric spaces with graphs","authors":"","doi":"10.23952/jano.4.2022.2.07","DOIUrl":"https://doi.org/10.23952/jano.4.2022.2.07","url":null,"abstract":"","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126832317","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 : 1900-01-01DOI: 10.23952/jano.1.2019.3.06
Mengxue Liu, Yuto Ogata, Tamaki Tanaka
In this paper, we deal with the inheritation of the semicontinuity of set-valued maps via general scalarization for sets, which is regarded as the framework of generalizations of results by Kuwano, Tanaka, and Yamada in 2010. Since the unified scalarization functions for sets satisfy certain desired semicontinuity, our main theorems can be reduced to the results in earlier study.
本文通过集合的一般尺度化处理集值映射的半连续性的继承问题，该问题被认为是Kuwano, Tanaka, and Yamada在2010年推广结果的框架。由于集合的统一标化函数满足一定的期望半连续性，我们的主要定理可以简化为前面研究的结果。
{"title":"Semicontinuity of the composition of set-valued map and scalarization function for sets","authors":"Mengxue Liu, Yuto Ogata, Tamaki Tanaka","doi":"10.23952/jano.1.2019.3.06","DOIUrl":"https://doi.org/10.23952/jano.1.2019.3.06","url":null,"abstract":"In this paper, we deal with the inheritation of the semicontinuity of set-valued maps via general scalarization for sets, which is regarded as the framework of generalizations of results by Kuwano, Tanaka, and Yamada in 2010. Since the unified scalarization functions for sets satisfy certain desired semicontinuity, our main theorems can be reduced to the results in earlier study.","PeriodicalId":205734,"journal":{"name":"Journal of Applied and Numerical Optimization","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114601376","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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