Nicholas J. A. Harvey, Christopher Liaw, Sikander Randhawa
Consider the problem of minimizing functions that are Lipschitz and convex, but not necessarily differentiable. We construct a function from this class for which the T th iterate of subgradient descent has error Ω(log( T ) / √ T ) . This matches a known upper bound of O (log( T ) / √ T ) . We prove analogous results for functions that are additionally strongly convex. There exists such a function for which the error of the T th iterate of subgradient descent has error Ω(log( T ) /T ) , matching a known upper bound of O (log( T ) /T ) . These results resolve a question posed by Shamir (2012)
考虑最小化函数的问题，这些函数具有 Lipschitz 特性和凸性，但不一定可微。我们从这类函数中构造出一个函数，对于这个函数，子梯度下降的第 T 次迭代误差为 Ω(log( T ) / √ T ) 。这与已知的 O (log( T ) / √ T ) 上限相吻合。我们将证明强凸函数的类似结果。存在这样一个函数，它的子梯度下降的第 T 次迭代的误差为 Ω(log( T ) /T ) ，与已知的上界 O (log( T ) /T ) 匹配。这些结果解决了沙米尔（2012）提出的一个问题
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Could continuous optimization address efficiently logical constraints? We propose a continuous-optimization alternative to the usual discrete-optimization (big-M and complementary) formulations of logical constraints, that can lead to effective practical methods. Based on the simple idea of guiding the search of a continuous-optimization descent method towards the parts of the domain where the logical constraint is satisfied, we introduce a smooth penalty-function formulation of logical constraints, and related theoretical results. This formulation allows a direct use of state-of-the-art continuous optimization solvers. The effectiveness of the continuous quadrant penalty formulation is demonstrated on an aircraft conflict avoidance application.
{"title":"The continuous quadrant penalty formulation of logical constraints","authors":"Sonia Cafieri, Andrew Conn, Marcel Mongeau","doi":"10.5802/ojmo.28","DOIUrl":"https://doi.org/10.5802/ojmo.28","url":null,"abstract":"Could continuous optimization address efficiently logical constraints? We propose a continuous-optimization alternative to the usual discrete-optimization (big-M and complementary) formulations of logical constraints, that can lead to effective practical methods. Based on the simple idea of guiding the search of a continuous-optimization descent method towards the parts of the domain where the logical constraint is satisfied, we introduce a smooth penalty-function formulation of logical constraints, and related theoretical results. This formulation allows a direct use of state-of-the-art continuous optimization solvers. The effectiveness of the continuous quadrant penalty formulation is demonstrated on an aircraft conflict avoidance application.","PeriodicalId":477184,"journal":{"name":"Open journal of mathematical optimization","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135011359","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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