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Primal-Dual Stability in Local Optimality

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第2期203卷 1325-1354页

作者： Benko, Matus Rockafellar, R. Tyrrell Johann Radon Inst Computat & Appl Math Linz Austria Univ Washington Dept Math Box 354350 Seattle WA 98195 USA

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.

关键词： Second-order variational analysis Local optimality Primal-dual stability Tilt stability, full stability, metric regularity Kummer's inverse theorem Implicit mapping theorems Graphically Lipschitzian mappings Crypto-continuity Strict graphical derivatives coderivatives Variational sufficiency

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Lipschitz-Like Property for Linear Constraint Systems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第3期199卷 1281-1296页

作者： Yao, Wenfang Yang, Xiaoqi Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China

In this paper, we consider a linear constraint system with a set constraint. We investigate the Lipschitz-like property of such systems with an explicit set constraint under full perturbations (including the matrix perturbation) and derive some sufficient and necessary conditions for this property. We also make use of some other approaches like outer-subdifferentials and error bounds to characterize such a property. We later apply the obtained results to linear portfolio selection problems with different settings and obtain some sufficient conditions for the parametric feasible set mapping to enjoy the Lipschitz-like property with various stock selection constraints.

关键词： Linear systems coderivatives Lipschitz-like property Linear portfolio selection

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On the Application of the SCD Semismooth* Newton Method to Variational Inequalities of the Second Kind

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第4期30卷 1453-1484页

作者： Gfrerer, Helmut Outrata, Jiri V. Valdman, Jan Johannes Kepler Univ Linz Inst Computat Math A-4040 Linz Austria Czech Acad Sci Inst Informat Theory & Automat Prague 18208 Czech Republic Federat Univ Australia Ctr Informat & Appl Optimizat POB 663 Ballarat Vic 3350 Australia Czech Tech Univ Fac Informat Technol Dept Appl Math Thakurova 9 Prague 160006 6 Czech Republic

The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD semismooth* Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD semismooth* Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities.

关键词： Newton method Semismoothness Superlinear convergence Global convergence Generalized equation coderivatives

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Variational Analysis of Paraconvex Multifunctions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2022年第1-3期193卷 180-218页

作者： Huynh Van Ngai Nguyen Huu Tron Nguyen Van Vu Thera, Michel Quy Nhon Univ Dept Math 170 An Duong Vuong Quy Nhon Binh Dinh Vietnam Univ Limoges XLIM UMR CNRS 7252 123 Ave Albert Thomas F-87060 Limoges France Federation Univ Ctr Informat & Appl Optimizat Sch Engn Informat Technol & Phys Sci Ballarat Vic Australia

Our aim in this article is to study the class of so-called rho-paraconv ex multifunctions from a Banach space X into the subsets of another Banach space Y. These multifunctions are defined in relation with a modulus function rho : X -> [0, +infinity) satisfying some suitable conditions. This class of multifunctions generalizes the class of gamma-paraconvex multifunctions with gamma > 1 introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. We establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study.

关键词： Weak convexity Lower C-2 functions Paraconvexity Paramonotonicity Approximate convex function Frechet subdifferential Frechet normal cone coderivatives Fuzzy mean value theorem

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Algebraic core and convex calculus without topology

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OPTIMIZATION 2022年第2期71卷 309-335页

作者： Dang Van Cuong Mordukhovich, Boris S. Nguyen Mau Nam Cartmell, Addison Duy Tan Univ Fac Nat Sci Dept Math Da Nang Vietnam Duy Tan Univ Inst Res & Dev Da Nang Vietnam Wayne State Univ Dept Math Detroit MI 48202 USA Portland State Univ Farlborz Maseeh Dept Math & Stat Portland OR 97207 USA

In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the separation theorem of convex sets in vector spaces allows us to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended-real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. We also obtain a precise formula for computing the subdifferential of optimal value functions associated with convex problems of parametric optimization in vector spaces. Functions of this type play a crucial role in many aspects of convex optimization and its applications.

关键词： Algebraic core convex separation normals coderivatives subgradients

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Sensitivity Analysis of Stochastic Constraint and Variational Systems via Generalized Differentiation

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SET-VALUED AND VARIATIONAL ANALYSIS 2023年第1期31卷 4-4页

作者： Mordukhovich, Boris S. Perez-Aros, Pedro Wayne State Univ Dept Math Detroit MI 48202 USA Univ OHiggins Inst Ciencias Ingn Rancagua Chile

This paper conducts sensitivity analysis of random constraint and variational systems related to stochastic optimization and variational inequalities. We establish efficient conditions for well-posedness, in the sense of robust Lipschitzian stability and/or metric regularity, of such systems by employing and developing coderivative characterizations of well-posedness properties for random multifunctions and efficiently evaluating coderivatives of special classes of random integral set-valued mappings that naturally emerge in stochastic programming and stochastic variational inequalities.

关键词： Variational analysis Set-valued analysis Lipschitzian stability Generalized differentiation coderivatives Stochastic programming Stochastic variational inequalities Primary Secondary

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ON A SEMISMOOTH* NEWTON METHOD FOR SOLVING GENERALIZED EQUATIONS

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SIAM JOURNAL ON OPTIMIZATION 2021年第1期31卷 489-517页

作者： Gfrerer, Helmut Outrata, Jiri, V Johannes Kepler Univ Linz Inst Computat Math A-4040 Linz Austria Czech Acad Sci Inst Informat Theory & Automat Prague 18208 Czech Republic Federat Univ Australia Ctr Informat & Appl Optimizat Ballarat Vic 3350 Australia

In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multivalued part of the considered GE. The method is based on the new notion of semismoothness*, which, together with a suitable regularity condition, ensures the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models.

关键词： Newton method semismoothness superlinear convergence generalized equation coderivatives

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Sensitivity Analysis of Multiobjective Optimization Problems with Parameterized Quasi-Variational Inequalities

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2018年第2期39卷 155-178页

作者： Pang, Li-Ping Meng, Fan-Yun Lv, Jian Tian, Qi Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China Qingdao Technol Univ Comp Engn Inst Qingdao Peoples R China Zhejiang Univ Finance & Econ Dept Finance Hangzhou Zhejiang Peoples R China

This paper mainly establishes the sensitivity analysis of a multiobjective optimization problem with parameterized quasi-variational inequalities (QVIs). Using the (regular) coderivative of the associated epigraphical multifunction, the (regular) subdifferentials of the efficient frontier maps are estimated, which involve the (regular) coderivatives of the solution mapping to the parameterized QVIs. Under the linear independent constraint qualification, the defined auxiliary set-valued mappings in the parameterized QVIs are clam. The detailed formulae of subdifferentials of the efficient frontier maps are obtained and examples are simultaneously provided for analyzing and illustrating the obtained results.

关键词： coderivatives efficient frontier maps multiobjective optimization parameterized quasi-variational inequalities

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Full Stability of General Parametric Variational Systems

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SET-VALUED AND VARIATIONAL ANALYSIS 2018年第4期26卷 911-946页

作者： Mordukhovich, B. S. Nghia, T. T. A. Pham, D. T. Wayne State Univ Dept Math Detroit MI 48202 USA RUDN Univ Moscow 117198 Russia Oakland Univ Dept Math & Stat Rochester MI 48309 USA

The paper introduces and studies the notions of Lipschitzian and Holderian full stability of solutions to three-parametric variational systems described in the generalized equation formalism involving nonsmooth base mappings and partial subgradients of prox-regular functions acting in Hilbert spaces. Employing advanced tools and techniques of second-order variational analysis allows us to establish complete characterizations of, as well as directly verifiable sufficient conditions for, such full stability properties under mild assumptions. Furthermore, we derive exact formulas and effective quantitative estimates for the corresponding moduli. The obtained results are specified for important classes of variational inequalities and variational conditions in both finite and infinite dimensions.

关键词： Variational analysis Parametric variational systems Variational inequalities and variational conditions Lipschitzian and Holderian full stability Prox-regularity Legendre forms Polyhedricity Generalized differentiation Subgradients coderivatives Primary 49J53 Secondary 49J52 90C31

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Normal and mixed partial second-order subdifferentials in variational analysis and optimization

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POSITIVITY 2018年第3期22卷 671-685页

作者： Sadeqi, Ildar Nadi, Somayeh Sahand Univ Technol Dept Math Tabriz Iran

A partial second-order subdifferential is defined here for extended real valued functions of two variables corresponding to its variables through coderivatives of first-order partial subdifferential mappings. In addition, some rules are presented to calculate these second-order structures along with defining some conditions to insure the equality and . Moreover, as an application, some conditions are stated which show the relation between local minimum of a function and positiveness of principal minors of its hessian matrix.

关键词： coderivatives Second-order subdifferentials Stability Strictly differentiable mappings Variational analysis

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