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A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces 希尔伯特空间非光滑多目标优化的后裔方法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-09-12 DOI: 10.1007/s10957-024-02520-4
Konstantin Sonntag, Bennet Gebken, Georg Müller, Sebastian Peitz, Stefan Volkwein
{"title":"A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces","authors":"Konstantin Sonntag, Bennet Gebken, Georg Müller, Sebastian Peitz, Stefan Volkwein","doi":"10.1007/s10957-024-02520-4","DOIUrl":"https://doi.org/10.1007/s10957-024-02520-4","url":null,"abstract":"<p>The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from Gebken and Peitz (J Optim Theory Appl 188:696–723, 2021) is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the Clarke subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"46 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Expected Residual Minimization Formulation for Stochastic Absolute Value Equations 随机绝对值方程的期望残差最小化公式
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-09-12 DOI: 10.1007/s10957-024-02527-x
Jingyong Tang, Jinchuan Zhou
{"title":"Expected Residual Minimization Formulation for Stochastic Absolute Value Equations","authors":"Jingyong Tang, Jinchuan Zhou","doi":"10.1007/s10957-024-02527-x","DOIUrl":"https://doi.org/10.1007/s10957-024-02527-x","url":null,"abstract":"<p>In this paper we investigate a class of stochastic absolute value equations (SAVE). After establishing the relationship between the stochastic linear complementarity problem and SAVE, we study the expected residual minimization (ERM) formulation for SAVE and its Monte Carlo sample average approximation. In particular, we show that the ERM problem and its sample average approximation have optimal solutions under the condition of <span>(R_0)</span> pair, and the optimal value of the sample average approximation has uniform exponential convergence. Furthermore, we prove that the solutions to the ERM problem are robust for SAVE. For a class of SAVE problems, we use its special structure to construct a smooth residual and further study the convergence of the stationary points. Finally, a smoothing gradient method is proposed by simultaneously considering sample sampling and smooth techniques for solving SAVE. Numerical experiments exhibit the effectiveness of the method.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"7 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Weak Maximum Principle for Discrete Optimal Control Problems with Mixed Constraints 具有混合约束条件的离散最优控制问题的弱最大原则
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-09-10 DOI: 10.1007/s10957-024-02524-0
Roberto Andreani, John Frank Matos Ascona, Valeriano Antunes de Oliveira
{"title":"A Weak Maximum Principle for Discrete Optimal Control Problems with Mixed Constraints","authors":"Roberto Andreani, John Frank Matos Ascona, Valeriano Antunes de Oliveira","doi":"10.1007/s10957-024-02524-0","DOIUrl":"https://doi.org/10.1007/s10957-024-02524-0","url":null,"abstract":"<p>In this study, first-order necessary optimality conditions, in the form of a weak maximum principle, are derived for discrete optimal control problems with mixed equality and inequality constraints. Such conditions are achieved by using the Dubovitskii–Milyutin formalism approach. Nondegenerate conditions are obtained under the constant rank of the subspace component (CRSC) constraint qualification, which is an important generalization of both the Mangasarian–Fromovitz and constant rank constraint qualifications. Beyond its theoretical significance, CRSC has practical importance because it is closely related to the formulation of optimization algorithms. In addition, an instance of a discrete optimal control problem is presented in which CRSC holds while other stronger regularity conditions do not.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"25 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gradient Descent Provably Escapes Saddle Points in the Training of Shallow ReLU Networks 在浅层 ReLU 网络的训练中梯度下降可避开鞍点
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-09-10 DOI: 10.1007/s10957-024-02513-3
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
{"title":"Gradient Descent Provably Escapes Saddle Points in the Training of Shallow ReLU Networks","authors":"Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek","doi":"10.1007/s10957-024-02513-3","DOIUrl":"https://doi.org/10.1007/s10957-024-02513-3","url":null,"abstract":"<p>Dynamical systems theory has recently been applied in optimization to prove that gradient descent algorithms bypass so-called strict saddle points of the loss function. However, in many modern machine learning applications, the required regularity conditions are not satisfied. In this paper, we prove a variant of the relevant dynamical systems result, a center-stable manifold theorem, in which we relax some of the regularity requirements. We explore its relevance for various machine learning tasks, with a particular focus on shallow rectified linear unit (ReLU) and leaky ReLU networks with scalar input. Building on a detailed examination of critical points of the square integral loss function for shallow ReLU and leaky ReLU networks relative to an affine target function, we show that gradient descent circumvents most saddle points. Furthermore, we prove convergence to global minima under favourable initialization conditions, quantified by an explicit threshold on the limiting loss.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"58 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Optimized Fractional-Order Type-2 Fuzzy PID Attitude Controller for Fixed-Wing Aircraft 用于固定翼飞机的优化分数阶 2 型模糊 PID 姿态控制器
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-09-09 DOI: 10.1007/s10957-024-02512-4
Wenfan Wang, Jun Zhang, Ruili Jiao
{"title":"Optimized Fractional-Order Type-2 Fuzzy PID Attitude Controller for Fixed-Wing Aircraft","authors":"Wenfan Wang, Jun Zhang, Ruili Jiao","doi":"10.1007/s10957-024-02512-4","DOIUrl":"https://doi.org/10.1007/s10957-024-02512-4","url":null,"abstract":"<p>This paper addresses the design of attitude controller for a fixed-wing unmanned aerial vehicle. To address the complexity of the coupled nonlinear model of a fixed-wing aircraft, this paper introduces a Fractional-Order Type-2 Fuzzy PID (FOTFPID) controller. The adoption of interval valued type-2 fuzzy sets, as an extension of conventional fuzzy sets, has endowed decision makers with the ability to assign membership and non-membership values as intervals. This enhanced capability facilitates more resilient decision-making processes. The Bat optimization algorithm is also employed to fine-tune the membership functions, scaling factors, and primary controller parameters, aiming to minimize the integrated absolute error index. Numerical simulations are conducted to demonstrate effectiveness of the proposed controllers in comparison to classical PID controllers, while subjecting the aircraft system to various disturbance conditions.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"32 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Convergence-Accelerated Fixed-Time Dynamical Methods for Absolute Value Equations 绝对值方程的收敛加速固定时间动态方法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-09-09 DOI: 10.1007/s10957-024-02525-z
Xu Zhang, Cailian Li, Longcheng Zhang, Yaling Hu, Zheng Peng
{"title":"Convergence-Accelerated Fixed-Time Dynamical Methods for Absolute Value Equations","authors":"Xu Zhang, Cailian Li, Longcheng Zhang, Yaling Hu, Zheng Peng","doi":"10.1007/s10957-024-02525-z","DOIUrl":"https://doi.org/10.1007/s10957-024-02525-z","url":null,"abstract":"<p>Two new accelerated fixed-time stable dynamic systems are proposed for solving absolute value equations (AVEs): <span>(Ax-|x|-b=0)</span>. Under some mild conditions, the equilibrium point of the proposed dynamic systems is completely equivalent to the solution of the AVEs under consideration. Meanwhile, we have introduced a new relatively tighter global error bound for the AVEs. Leveraging this finding, we have separately established the globally fixed-time stability of the proposed methods, along with providing the conservative settling-time for each method. Compared with some existing state-of-the-art dynamical methods, preliminary numerical experiments show the effectiveness of our methods in solving the AVEs.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"6 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Metric Subregularity and $$omega (cdot )$$ -Normal Regularity Properties 公次正则性和 $$omega (cdot )$$ - 正则性属性
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-09-04 DOI: 10.1007/s10957-024-02476-5
Florent Nacry, Vo Anh Thuong Nguyen, Juliette Venel
{"title":"Metric Subregularity and $$omega (cdot )$$ -Normal Regularity Properties","authors":"Florent Nacry, Vo Anh Thuong Nguyen, Juliette Venel","doi":"10.1007/s10957-024-02476-5","DOIUrl":"https://doi.org/10.1007/s10957-024-02476-5","url":null,"abstract":"<p>In this paper, we establish through an openness condition the metric subregularity of a multimapping with normal <span>(omega (cdot ))</span>-regularity of either the graph or values. Various preservation results for prox-regular and subsmooth sets are also provided.\u0000</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"67 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Converse Lyapunov-Type Theorem for Control Systems with Regulated Cost 具有调节成本的控制系统的逆 Lyapunov 型定理
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-08-31 DOI: 10.1007/s10957-024-02517-z
Anna Chiara Lai, Monica Motta
{"title":"A Converse Lyapunov-Type Theorem for Control Systems with Regulated Cost","authors":"Anna Chiara Lai, Monica Motta","doi":"10.1007/s10957-024-02517-z","DOIUrl":"https://doi.org/10.1007/s10957-024-02517-z","url":null,"abstract":"<p>Given a nonlinear control system, a target set, a nonnegative integral cost, and a continuous function <i>W</i>, we say that the system is <i>globally asymptotically controllable to the target with</i> <i>W</i>-<i>regulated cost</i>, whenever, starting from any point <i>z</i>, among the strategies that achieve classical asymptotic controllability we can select one that also keeps the cost less than <i>W</i>(<i>z</i>). In this paper, assuming mild regularity hypotheses on the data, we prove that a necessary and sufficient condition for global asymptotic controllability with regulated cost is the existence of a special, continuous Control Lyapunov Function, called a <i>Minimum Restraint Function</i>. The main novelty is the necessity implication, obtained here for the first time. Nevertheless, the sufficiency condition extends previous results based on semiconcavity of the Minimum Restraint Function, while we require mere continuity.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"14 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Incomplete Information Mean-Field Games and Related Riccati Equations 不完全信息均势博弈及相关里卡提方程
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-08-31 DOI: 10.1007/s10957-024-02508-0
Min Li, Tianyang Nie, Shujun Wang, Ke Yan
{"title":"Incomplete Information Mean-Field Games and Related Riccati Equations","authors":"Min Li, Tianyang Nie, Shujun Wang, Ke Yan","doi":"10.1007/s10957-024-02508-0","DOIUrl":"https://doi.org/10.1007/s10957-024-02508-0","url":null,"abstract":"<p>We study a class of mean-field games with incomplete information in this paper. For each agent, the state is given by a linear forward stochastic differential equation with common noise. Moreover, both the state and control variables can enter the diffusion coefficients of the state equation. We deduce the open-loop adapted decentralized strategies and feedback decentralized strategies by a mean-field forward–backward stochastic differential equation and Riccati equations, respectively. The well-posedness of the corresponding consistency condition system is obtained and the limiting state-average turns out to be the solution of a mean-field stochastic differential equation driven by common noise. We also verify the <span>(varepsilon )</span>-Nash equilibrium property of the decentralized strategies. Finally, a network security problem is studied to illustrate our results as an application.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"4 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Monotonicity of the Stopping Boundary for Time-Inhomogeneous Optimal Stopping Problems 论时间非均质最优停止问题的停止边界单调性
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-08-30 DOI: 10.1007/s10957-024-02514-2
Alessandro Milazzo
{"title":"On the Monotonicity of the Stopping Boundary for Time-Inhomogeneous Optimal Stopping Problems","authors":"Alessandro Milazzo","doi":"10.1007/s10957-024-02514-2","DOIUrl":"https://doi.org/10.1007/s10957-024-02514-2","url":null,"abstract":"<p>We consider a class of time-inhomogeneous optimal stopping problems and we provide sufficient conditions on the data of the problem that guarantee monotonicity of the optimal stopping boundary. In our setting, time-inhomogeneity stems not only from the reward function but, in particular, from the time dependence of the drift coefficient of the one-dimensional stochastic differential equation (SDE) which drives the stopping problem. In order to obtain our results, we mostly employ probabilistic arguments: we use a comparison principle between solutions of the SDE computed at different starting times, and martingale methods of optimal stopping theory. We also show a variant of the main theorem, which weakens one of the assumptions and additionally relies on the renowned connection between optimal stopping and free-boundary problems.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"33 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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