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Primal–Dual Stability in Local Optimality 局部最优的原始-双重稳定性
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-24 DOI: 10.1007/s10957-024-02467-6
Matúš Benko, R. Tyrrell Rockafellar
{"title":"Primal–Dual Stability in Local Optimality","authors":"Matúš Benko, R. Tyrrell Rockafellar","doi":"10.1007/s10957-024-02467-6","DOIUrl":"https://doi.org/10.1007/s10957-024-02467-6","url":null,"abstract":"<p>Much is known about when a locally optimal solution depends in a single-valued Lipschitz continuous way on the problem’s parameters, including tilt perturbations. Much less is known, however, about when that solution and a uniquely determined multiplier vector associated with it exhibit that dependence as a primal–dual pair. In classical nonlinear programming, such advantageous behavior is tied to the combination of the standard strong second-order sufficient condition (SSOC) for local optimality and the linear independent gradient condition (LIGC) on the active constraint gradients. But although second-order sufficient conditons have successfully been extended far beyond nonlinear programming, insights into what should replace constraint gradient independence as the extended dual counterpart have been lacking. The exact answer is provided here for a wide range of optimization problems in finite dimensions. Behind it are advances in how coderivatives and strict graphical derivatives can be deployed. New results about strong metric regularity in solving variational inequalities and generalized equations are obtained from that as well.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502011","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 Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems 多目标离散优化问题的加权 p 准则权集分解
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-24 DOI: 10.1007/s10957-024-02481-8
Stephan Helfrich, Kathrin Prinz, Stefan Ruzika
{"title":"The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems","authors":"Stephan Helfrich, Kathrin Prinz, Stefan Ruzika","doi":"10.1007/s10957-024-02481-8","DOIUrl":"https://doi.org/10.1007/s10957-024-02481-8","url":null,"abstract":"<p>Many solution algorithms for multiobjective optimization problems are based on scalarization methods that transform the multiobjective problem into a scalar-valued optimization problem. In this article, we study the theory of weighted <span>(p)</span>-norm scalarizations. These methods minimize the distance induced by a weighted <span>(p)</span>-norm between the image of a feasible solution and a given reference point. We provide a comprehensive theory of the set of eligible weights and, in particular, analyze the topological structure of the normalized weight set. This set is composed of connected subsets, called weight set components which are in a one-to-one relation with the set of optimal images of the corresponding weighted <span>(p)</span>-norm scalarization. Our work generalizes and complements existing results for the weighted sum and the weighted Tchebycheff scalarization and provides new insights into the structure of the set of all Pareto optimal solutions.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502010","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
Efficient Use of Quantum Linear System Algorithms in Inexact Infeasible IPMs for Linear Optimization 在线性优化的不精确不可行 IPM 中有效利用量子线性系统算法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-23 DOI: 10.1007/s10957-024-02452-z
Mohammadhossein Mohammadisiahroudi, Ramin Fakhimi, Tamás Terlaky
{"title":"Efficient Use of Quantum Linear System Algorithms in Inexact Infeasible IPMs for Linear Optimization","authors":"Mohammadhossein Mohammadisiahroudi, Ramin Fakhimi, Tamás Terlaky","doi":"10.1007/s10957-024-02452-z","DOIUrl":"https://doi.org/10.1007/s10957-024-02452-z","url":null,"abstract":"<p>Quantum computing has attracted significant interest in the optimization community because it potentially can solve classes of optimization problems faster than conventional supercomputers. Several researchers proposed quantum computing methods, especially quantum interior point methods (QIPMs), to solve convex conic optimization problems. Most of them have applied a quantum linear system algorithm at each iteration to compute a Newton step. However, using quantum linear solvers in QIPMs comes with many challenges, such as having ill-conditioned systems and the considerable error of quantum solvers. This paper investigates in detail the use of quantum linear solvers in QIPMs. Accordingly, an Inexact Infeasible Quantum Interior Point (II-QIPM) is developed to solve linear optimization problems. We also discuss how we can get an exact solution by iterative refinement (IR) without excessive time of quantum solvers. The proposed IR-II-QIPM shows exponential speed-up with respect to precision compared to previous II-QIPMs. Additionally, we present a quantum-inspired classical variant of the proposed IR-II-QIPM where QLSAs are replaced by conjugate gradient methods. This classic IR-II-IPM has some advantages compared to its quantum counterpart, as well as previous classic inexact infeasible IPMs. Finally, computational results with a QISKIT implementation of our QIPM using quantum simulators are presented and analyzed.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502012","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 Risk Evaluation and Control of Distributed Multi-agent Systems 论分布式多代理系统的风险评估与控制
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-21 DOI: 10.1007/s10957-024-02464-9
Aray Almen, Darinka Dentcheva
{"title":"On Risk Evaluation and Control of Distributed Multi-agent Systems","authors":"Aray Almen, Darinka Dentcheva","doi":"10.1007/s10957-024-02464-9","DOIUrl":"https://doi.org/10.1007/s10957-024-02464-9","url":null,"abstract":"<p>In this paper, we deal with risk evaluation and risk-averse optimization of complex distributed systems with general risk functionals. We postulate a novel set of axioms for the functionals evaluating the total risk of the system. We derive a dual representation for the systemic risk measures and propose new ways to construct families of systemic risk measures using either a collection of linear scalarizations or non-linear risk aggregation. The proposed framework facilitates risk-averse sequential decision-making by distributed methods. The new approach is compared theoretically and numerically to other systemic risk measurements from the existing literature. We formulate a two-stage decision problem for a distributed system using a systemic measure of risk. The structure accommodates distributed systems arising in energy networks, robotics, and other practical situations. A distributed decomposition method for solving the two-stage problem is proposed and applied to a problem arising in communication networks. We have used this problem to compare the methods of systemic risk evaluation. We show that the risk evaluation via linear scalarizations of outcomes leads to less conservative risk evaluation and results in a substantially better solution to the problem at hand than aggregating the risk of individual agents.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526729","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
Numerical Method for a Controlled Sweeping Process with Nonsmooth Sweeping Set 具有非光滑扫频集的受控扫频过程的数值方法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-19 DOI: 10.1007/s10957-024-02470-x
Chadi Nour, Vera Zeidan
{"title":"Numerical Method for a Controlled Sweeping Process with Nonsmooth Sweeping Set","authors":"Chadi Nour, Vera Zeidan","doi":"10.1007/s10957-024-02470-x","DOIUrl":"https://doi.org/10.1007/s10957-024-02470-x","url":null,"abstract":"<p>The numerical method developed in Nour and Zeidan (IEEE Control Syst. Lett. 6:1190-1195, 2022) via the exponential penalization technique for optimal control problems involving sweeping processes with sweeping set <i>C</i> generated by <i>one</i> <i>smooth</i> function, is generalized in this paper to the case where <i>C</i> is <i>nonsmooth</i>. That is, <i>C</i> is the intersection of a <i>finite number</i> (greater than one) of sublevel sets of smooth functions. The change from one to greater than one generating smooth functions is quite challenging. Indeed, while in the latter case <i>C</i> could be reformulated as being generated by one function, however, this function is only <i>Lipschitz</i>, and hence, the method established for one generating smooth function is not applicable to this framework. Therefore, this general setting requires a new approach, which represents the novelty of this paper.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526730","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
Transfer Principles, Fenchel Conjugate, and Subdifferential Formulas in Fan-Theobald-von Neumann Systems 范-特奥伯德-冯-诺依曼系统中的转移原理、芬切尔共轭和次微分公式
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-18 DOI: 10.1007/s10957-024-02474-7
Juyoung Jeong, M. Seetharama Gowda
{"title":"Transfer Principles, Fenchel Conjugate, and Subdifferential Formulas in Fan-Theobald-von Neumann Systems","authors":"Juyoung Jeong, M. Seetharama Gowda","doi":"10.1007/s10957-024-02474-7","DOIUrl":"https://doi.org/10.1007/s10957-024-02474-7","url":null,"abstract":"<p>A Fan-Theobald-von Neumann system [7] is a triple <span>((mathcal {V},mathcal {W},lambda ))</span>, where <span>(mathcal {V})</span> and <span>(mathcal {W})</span> are real inner product spaces and <span>(lambda :mathcal {V}rightarrow mathcal {W})</span> is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). The present article is a continuation of [9] where the concepts of commutativity, automorphisms, majorization, and reduction were introduced and elaborated. Here, we describe some transfer principles and present Fenchel conjugate and subdifferential formulas.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502013","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
Representations for Maximal Monotone Operators of Type (D) in Banach Spaces 巴拿赫空间中 (D) 型最大单调算子的表示法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-04 DOI: 10.1007/s10957-024-02457-8
Bao T. Nguyen, Tran N. Nguyen, Huynh M. Hien
{"title":"Representations for Maximal Monotone Operators of Type (D) in Banach Spaces","authors":"Bao T. Nguyen, Tran N. Nguyen, Huynh M. Hien","doi":"10.1007/s10957-024-02457-8","DOIUrl":"https://doi.org/10.1007/s10957-024-02457-8","url":null,"abstract":"<p>The present paper deals with a maximal monotone operator <i>A</i> of type (D) in a Banach space whose dual space is strictly convex. We establish some representations for the value <i>Ax</i> at a given point <i>x</i> via its values at nearby points of <i>x</i>. We show that the faces of <i>Ax</i> are contained in the set of all weak<span>(^*)</span> convergent limits of bounded nets of the operator at nearby points of <i>x</i>, then we obtain a representation for <i>Ax</i> by use of this set. In addition, representations for the support function of <i>Ax</i> based on the minimal-norm selection of the operator in certain Banach spaces are given.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252855","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
Generalized Sequential Normal Compactness and Weak Differentiabilities 广义序列法向紧凑性和弱微分性
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-03 DOI: 10.1007/s10957-024-02463-w
Bingwu Wang, Xinmin Yang, Pujun Long
{"title":"Generalized Sequential Normal Compactness and Weak Differentiabilities","authors":"Bingwu Wang, Xinmin Yang, Pujun Long","doi":"10.1007/s10957-024-02463-w","DOIUrl":"https://doi.org/10.1007/s10957-024-02463-w","url":null,"abstract":"<p>We study the generalized sequential normal compactness in variational analysis and establish characterizations of the property of graphs of weakly differentiable mappings between Banach spaces, as well as calculus rules involving such functions.\u0000</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253013","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
Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators 两个广义单调算子之和的三阶动态系统
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-06-03 DOI: 10.1007/s10957-024-02437-y
Pham Viet Hai, Phan Tu Vuong
{"title":"Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators","authors":"Pham Viet Hai, Phan Tu Vuong","doi":"10.1007/s10957-024-02437-y","DOIUrl":"https://doi.org/10.1007/s10957-024-02437-y","url":null,"abstract":"<p>In this paper, we propose and analyze a third-order dynamical system for finding zeros of the sum of two generalized operators in a Hilbert space <span>(mathcal {H})</span>. We establish the existence and uniqueness of the trajectories generated by the system under appropriate continuity conditions, and prove exponential convergence to the unique zero when the sum of the operators is strongly monotone. Additionally, we derive an explicit discretization of the dynamical system, which results in a forward–backward algorithm with double inertial effects and larger range of stepsize. We establish the linear convergence of the iterates to the unique solution using this algorithm. Furthermore, we provide convergence analysis for the class of strongly pseudo-monotone variational inequalities. We illustrate the effectiveness of our approach by applying it to structured optimization and pseudo-convex optimization problems.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253348","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
Primal Subgradient Methods with Predefined Step Sizes 具有预定步长的原始次梯度方法
IF 1.9 3区 数学
Journal of Optimization Theory and Applications Pub Date : 2024-05-31 DOI: 10.1007/s10957-024-02456-9
Yurii Nesterov
{"title":"Primal Subgradient Methods with Predefined Step Sizes","authors":"Yurii Nesterov","doi":"10.1007/s10957-024-02456-9","DOIUrl":"https://doi.org/10.1007/s10957-024-02456-9","url":null,"abstract":"<p>In this paper, we suggest a new framework for analyzing primal subgradient methods for nonsmooth convex optimization problems. We show that the classical step-size rules, based on normalization of subgradient, or on knowledge of the optimal value of the objective function, need corrections when they are applied to optimization problems with constraints. Their proper modifications allow a significant acceleration of these schemes when the objective function has favorable properties (smoothness, strong convexity). We show how the new methods can be used for solving optimization problems with functional constraints with a possibility to approximate the optimal Lagrange multipliers. One of our primal-dual methods works also for unbounded feasible set.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141196910","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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