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A Method for Multi-Leader–Multi-Follower Games by Smoothing the Followers’ Response Function 通过平滑追随者的响应函数实现多领导者与多追随者博弈的方法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-08-02 DOI: 10.1007/s10957-024-02506-2
Atsushi Hori, Daisuke Tsuyuguchi, Ellen H. Fukuda
{"title":"A Method for Multi-Leader–Multi-Follower Games by Smoothing the Followers’ Response Function","authors":"Atsushi Hori, Daisuke Tsuyuguchi, Ellen H. Fukuda","doi":"10.1007/s10957-024-02506-2","DOIUrl":"https://doi.org/10.1007/s10957-024-02506-2","url":null,"abstract":"<p>The multi-leader–multi-follower game (MLMFG) involves two or more leaders and followers and serves as a generalization of the Stackelberg game and the single-leader–multi-follower game. Although MLMFG covers wide range of real-world applications, its research is still sparse. Notably, fundamental solution methods for this class of problems remain insufficiently established. A prevailing approach is to recast the MLMFG as an equilibrium problem with equilibrium constraints (EPEC) and solve it using a solver. Meanwhile, interpreting the solution to the EPEC in the context of MLMFG may be complex due to shared decision variables among all leaders, followers’ strategies that each leader can unilaterally change, but the variables are essentially controlled by followers. To address this issue, we introduce a response function of followers’ noncooperative game that is a function with leaders’ strategies as a variable. Employing this approach allows the MLMFG to be solved as a single-level differentiable variational inequality using a smoothing scheme for the followers’ response function. We also demonstrate that the sequence of solutions to the smoothed variational inequality converges to a stationary equilibrium of the MLMFG. Finally, we illustrate the behavior of the smoothing method by numerical experiments.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"22 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882506","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
Pontryagin’s Maximum Principle for a State-Constrained System of Douglis-Nirenberg Type 道格里斯-尼伦伯格类型状态受限系统的庞特里亚金最大原则
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-08-01 DOI: 10.1007/s10957-024-02499-y
Alexey S. Matveev, Dmitrii V. Sugak
{"title":"Pontryagin’s Maximum Principle for a State-Constrained System of Douglis-Nirenberg Type","authors":"Alexey S. Matveev, Dmitrii V. Sugak","doi":"10.1007/s10957-024-02499-y","DOIUrl":"https://doi.org/10.1007/s10957-024-02499-y","url":null,"abstract":"<p>This article is concerned with optimal control problems for plants described by systems of high order nonlinear PDE’s (whose linear approximation is elliptic in the sense of Douglis-Nirenberg), with a special attention being given to their particular case: the standard stationary system of non-linear Navier–Stokes equations. The objective is to establish an analog of the Pontryagin’s maximum principle. The major challenge stems from the presence of infinitely many point-wise constraints on the system’s state, which are imposed at any point from a given continuum set of independent variables. Necessary conditions for optimality in the form of an “abstract” maximum principle are first obtained for a general optimal control problem couched in the language of functional analysis. This result is targeted at a wide class of problems, with an idea to absorb, in its proof, a great deal of technical work needed for derivation of optimality conditions so that only an interpretation of the discussed result would be basically needed to handle a particular problem. The applicability of this approach is demonstrated via obtaining the afore-mentioned analog of the Pontryagin’s maximum principle for a state-constrained system of high-order elliptic equations and the Navier–Stokes equations.\u0000</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"4 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882505","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
Distributionally Robust Variational Inequalities: Relaxation, Quantification and Discretization 分布稳健的变分不等式：松弛、量化和离散化
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-07-30 DOI: 10.1007/s10957-024-02497-0
Jie Jiang
{"title":"Distributionally Robust Variational Inequalities: Relaxation, Quantification and Discretization","authors":"Jie Jiang","doi":"10.1007/s10957-024-02497-0","DOIUrl":"https://doi.org/10.1007/s10957-024-02497-0","url":null,"abstract":"<p>In this paper, we use the distributionally robust approach to study stochastic variational inequalities under the ambiguity of the true probability distribution, which is referred to as distributionally robust variational inequalities (DRVIs). First of all, we adopt a relaxed value function approach to relax the DRVI and obtain its relaxation counterpart. This is mainly motivated by the robust requirement in the modeling process as well as the possible calculation error in the numerical process. After that, we investigate qualitative convergence properties as the relaxation parameter tends to zero. Considering the perturbation of ambiguity sets, we further study the quantitative stability of the relaxation DRVI. Finally, when the ambiguity set is given by the general moment information, the discrete approximation of the relaxation DRVI is discussed.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"361 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868262","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
The Synthesis of Optimal Control Laws Using Isaacs’ Method for the Solution of Differential Games 用艾萨克方法合成最优控制法则以解决微分博弈问题
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-07-30 DOI: 10.1007/s10957-024-02490-7
Meir Pachter, Isaac E. Weintraub
{"title":"The Synthesis of Optimal Control Laws Using Isaacs’ Method for the Solution of Differential Games","authors":"Meir Pachter, Isaac E. Weintraub","doi":"10.1007/s10957-024-02490-7","DOIUrl":"https://doi.org/10.1007/s10957-024-02490-7","url":null,"abstract":"<p>In this paper we advocate for Isaacs’ method for the solution of differential games to be applied to the solution of optimal control problems. To make the argument, the vehicle employed is Pontryagin’s canonical optimal control example, which entails a double integrator plant. However, rather than controlling the state to the origin, we require the end state to reach a terminal set that contains the origin in its interior. Indeed, in practice, it is required to control to a prescribed tolerance rather than reach a desired end state; constraining the end state to a terminal manifold of co-dimension n − 1 renders the optimal control problem easier to solve. The global solution of the optimal control problem is obtained and the synthesized optimal control law is in state feedback form. In this respect, two target sets are considered: a smooth circular target and a square target with corners. Closed-loop state-feedback control laws are synthesized that drive the double integrator plant from an arbitrary initial state to the target set in minimum time. This is accomplished using Isaacs’ method for the solution of differential games, which entails dynamic programming (DP), working backward from the usable part of the target set, as opposed to obtaining the optimal trajectories using the necessary conditions for optimality provided by Pontryagin’s Maximum Principle (PMP). In this paper, the case is made for Isaacs’ method for the solution of differential games to be applied to the solution of optimal control problems by way of the juxtaposition of the PMP and DP methods.\u0000</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"212 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868140","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
Optimal Relaxed Control for a Decoupled G-FBSDE 解耦 G-FBSDE 的最优松弛控制
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-07-24 DOI: 10.1007/s10957-024-02495-2
Hafida Bouanani, Omar Kebiri, Carsten Hartmann, Amel Redjil
{"title":"Optimal Relaxed Control for a Decoupled G-FBSDE","authors":"Hafida Bouanani, Omar Kebiri, Carsten Hartmann, Amel Redjil","doi":"10.1007/s10957-024-02495-2","DOIUrl":"https://doi.org/10.1007/s10957-024-02495-2","url":null,"abstract":"<p>In this paper we study a system of decoupled forward-backward stochastic differential equations driven by a <i>G</i>-Brownian motion (<i>G</i>-FBSDEs) with non-degenerate diffusion. Our objective is to establish the existence of a relaxed optimal control for a non-smooth stochastic optimal control problem. The latter is given in terms of a decoupled <i>G</i>-FBSDE. The cost functional is the solution of the backward stochastic differential equation at the initial time. The key idea to establish existence of a relaxed optimal control is to replace the original control problem by a suitably regularised problem with mollified coefficients, prove the existence of a relaxed control, and then pass to the limit.\u0000</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771832","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 Dealing with Minima at the Border of a Simplicial Feasible Area in Simplicial Branch and Bound 论在简化分支与边界中处理简化可行区域边界上的最小值
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-07-22 DOI: 10.1007/s10957-024-02480-9
Boglárka G.-Tóth, Eligius M. T. Hendrix, Leocadio G. Casado, Frédéric Messine
{"title":"On Dealing with Minima at the Border of a Simplicial Feasible Area in Simplicial Branch and Bound","authors":"Boglárka G.-Tóth, Eligius M. T. Hendrix, Leocadio G. Casado, Frédéric Messine","doi":"10.1007/s10957-024-02480-9","DOIUrl":"https://doi.org/10.1007/s10957-024-02480-9","url":null,"abstract":"<p>We consider a simplicial branch and bound Global Optimization algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be reduced due to monotonicity of the objective function. If there is a direction in which the objective function is monotone over a simplex, depending on whether the facets that may contain the minimum are at the border of the search region, we can remove the simplex completely, or reduce it to some of its border facets. Our research question deals with finding monotone directions and labeling facets of a simplex as border after longest edge bisection and reduction due to monotonicity. Experimental results are shown over a set of global optimization problems where the feasible set is defined as a simplex, and a global minimum point is located at a face of the simplicial feasible area.\u0000</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737299","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
The Stability of Robustness for Conic Linear Programs with Uncertain Data 具有不确定数据的圆锥线性规划的稳健性
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-07-21 DOI: 10.1007/s10957-024-02492-5
Miguel A. Goberna, Vaithilingam Jeyakumar, Guoyin Li
{"title":"The Stability of Robustness for Conic Linear Programs with Uncertain Data","authors":"Miguel A. Goberna, Vaithilingam Jeyakumar, Guoyin Li","doi":"10.1007/s10957-024-02492-5","DOIUrl":"https://doi.org/10.1007/s10957-024-02492-5","url":null,"abstract":"<p>The robust counterpart of a given conic linear program with uncertain data in the constraints is defined as the robust conic linear program that arises from replacing the nominal feasible set by the robust feasible set of points that remain feasible for any possible perturbation of the data within an uncertainty set. Any minor changes in the size of the uncertainty set can result in significant changes, for instance, in the robust feasible set, robust optimal value and the robust optimal set. The concept of quantifying the extent of these deviations is referred to as the <i>stability of robustness</i>. This paper establishes conditions for the stability of robustness under which minor changes in the size of the uncertainty sets lead to only minor changes in the robust feasible set of a given linear program with cone constraints and ball uncertainty sets.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"36 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737297","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
An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems 状态受限线性抛物线优化控制问题的增量拉格朗日法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-07-19 DOI: 10.1007/s10957-024-02494-3
Hailing Wang, Changjun Yu, Yongcun Song
{"title":"An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems","authors":"Hailing Wang, Changjun Yu, Yongcun Song","doi":"10.1007/s10957-024-02494-3","DOIUrl":"https://doi.org/10.1007/s10957-024-02494-3","url":null,"abstract":"<p>In this paper, we consider a class of state constrained linear parabolic optimal control problems. Instead of treating the inequality state constraints directly, we reformulate the problem as an equality-constrained optimization problem, and then apply the augmented Lagrangian method (ALM) to solve it. We prove the convergence of the ALM without any existence or regularity assumptions on the corresponding Lagrange multipliers, which is an essential complement to the classical theoretical results for the ALM because restrictive regularity assumptions are usually required to guarantee the existence of the Lagrange multipliers associated with the state constraints. In addition, under an appropriate choice of penalty parameter sequence, we can obtain a super-linear non-ergodic convergence rate for the ALM. Computationally, we apply a semi-smooth Newton (SSN) method to solve the ALM subproblems and design an efficient preconditioned conjugate gradient method for solving the Newton systems. Some numerical results are given to illustrate the effectiveness and efficiency of our algorithm.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"64 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737298","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 Globally Convergent Inertial First-Order Optimization Method for Multidimensional Scaling 用于多维扩展的全局收敛惯性一阶优化方法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-07-14 DOI: 10.1007/s10957-024-02486-3
Noga Ram, Shoham Sabach
{"title":"A Globally Convergent Inertial First-Order Optimization Method for Multidimensional Scaling","authors":"Noga Ram, Shoham Sabach","doi":"10.1007/s10957-024-02486-3","DOIUrl":"https://doi.org/10.1007/s10957-024-02486-3","url":null,"abstract":"<p>Multidimensional scaling (MDS) is a popular tool for dimensionality reduction and data visualization. Given distances between data points and a target low-dimension, the MDS problem seeks to find a configuration of these points in the low-dimensional space, such that the inter-point distances are preserved as well as possible. We focus on the most common approach to formulate the MDS problem, known as <i>stress</i> minimization, which results in a challenging non-smooth and non-convex optimization problem. In this paper, we propose an inertial version of the well-known SMACOF Algorithm, which we call AI-SMACOF. This algorithm is proven to be globally convergent, and to the best of our knowledge this is the first result of this kind for algorithms aiming at solving the stress MDS minimization. In addition to the theoretical findings, numerical experiments provide another evidence for the superiority of the proposed algorithm.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"26 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609533","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
Conic Optimization and Interior Point Methods: Theory, Computations, and Applications 圆锥优化和内点法：理论、计算与应用
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-07-11 DOI: 10.1007/s10957-024-02483-6
Tibor Illés, Florian Jarre, Etienne de Klerk, Goran Lesaja
{"title":"Conic Optimization and Interior Point Methods: Theory, Computations, and Applications","authors":"Tibor Illés, Florian Jarre, Etienne de Klerk, Goran Lesaja","doi":"10.1007/s10957-024-02483-6","DOIUrl":"https://doi.org/10.1007/s10957-024-02483-6","url":null,"abstract":"","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"152 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141586747","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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