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Optimality conditions for sparse nonlinear programming

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Science China Mathematics 2017年第5期60卷 759-776页

作者： PAN LiLi XIU NaiHua FAN Jun Department of Mathematics Beijing Jiaotong University Department of Mathematics Shandong University of Technology

The sparse nonlinear programming （SNP） is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Frechet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn- Tucker （KKT） conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.

关键词： sparse nonlinear programming constraint qualification normal cone first-order optimality con-dition second-order optimality condition

Optimality Conditions for Rank-Constrained Matrix Optimization

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Journal of the Operations Research Society of China 2019年第2期7卷 285-301页

作者： Xin-Rong Li Wen Song Nai-Hua Xiu Department of Applied Mathematics Beijing Jiaotong UniversityBeijing 100044China School of Mathematical Sciences Harbin Normal UniversityHarbin 150025China

In this paper,we comprehensively study optimality conditions for rank-constrained matrix optimization(RCMO).By calculating the Clarke tangent and normal cones to a rank-constrained set,along with the given Fréchet,Mordukhovich normal cones,we investigate four kinds of stationary points of the RCMO and analyze the relations between each stationary point and local/global minimizer of the ***,the second-order optimality condition of the RCMO is achieved with the help of the Clarke tangent cone.

关键词： Matrix optimization Rank constraint normal cone First-order optimality condition Second-order optimality condition

Sufcient Conditions for Error Bounds and Linear Regularity in Banach Spaces

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Acta Mathematica Sinica,English Series 2014年第3期30卷 423-436页

作者： Zhou WEI Qing Hai HE Department of Mathematics Yunnan University

In this paper, we study error bounds for lower semicontinuous functions defined on Banach space and linear regularity for finitely many closed subset in Banach spaces. By using Clarke＇s subd- ifferentials and Ekeland variational principle, we establish several sufficient conditions ensuring error bounds and linear regularity in Banach spaces.

关键词： Error bound linear regularity normal cone Clarke subdifferential Ekeland variationalprinciple

EXISTENCE　THEOREMS　OF　SOLUTIONS　FOR　TWO－POINT　BOUNDARY　VALUE　PROBLEM　OF　SECOND　ORDER　ORDINARY　DIFFERENTIAL　EQUATIONS　IN　BANACH　SPACES

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Applied Mathematics and Mechanics(English Edition) 1996年第2期17卷 99-108页

作者：张石生王凡 Sichuan University Chengdu P. R. China Nantong Teacher's College Nantong Jiangsu P. R. China

By using partial order method. some existing theorems of solutions for two-point bouniary value problem of second order ordinary differenlial equations in Banach spaces are given.

关键词： two-point boundary value problem weakly sequentially complete Banach space normal cone

Coupled Extremal Quasi-solutions to Nonlinear Impulsive Fredholm Integral Equations in Banach Spaces

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Northeastern Mathematical Journal 2000年第3期16卷 329-338页

作者：戚仕硕 Shandong University

By means of the method of coupled lower and upper quasisolutio ns, the paper applies, instead of establishing the comparison theorem, a new ite rative technique to nonlinear impulsive Fredholm integral equations in Banach sp aces and proves the existence theorem on their coupled extremal quasisolutions . Finally, an infinite system of nonlinear impulsive integral equations is provi ded to demonstrate the obtained results.

关键词： coupled lower and upper quasi-solutions iterative techn ique impulsive Fredholm integral equation Banach space normal cone

Theory of optimal harvesting for a nonlinear size-structured population in periodic environments

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International Journal of Biomathematics 2014年第4期7卷 201-218页

作者： Ze-Rong He Rong Liu Institute of Operational Research and Cybernetics Hangzhou Dianzi University Hangzhou 310018 P. R. China

This paper investigates the theoretical aspects for an optimal harvesting problem of a nonlinear size-structured population model in a periodic environment. We establish the well-posedness of the state system by means of frozen coefficients and fixed point reasoning. The existence of a unique optimal policy is proved via Ekeland＇s variational principle, and the first-order optimality conditions are derived by a suitable normM cone and a dual system. The results obtained would be beneficial for exploration of renewable

关键词： Body size population model harvesting fixed point normal cone Ekeland'sprinciple.

UNIQUENESS OF SOLUTIONS FOR A CLASS OF NON-LINEAR VOLTERRAINTEGRAL EQUATIONS WITHOUT CONTINUITY

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Applied Mathematics and Mechanics(English Edition) 1997年第12期18卷 0-0,0-0+0-0页

作者：董卫

In this paper. the fixed point theorem of increasing opera for with non-continuityis uilized to discuss foe exislence and uniqueness of positire solution for a class of nonlinear Volterra integral equations. An import... 详细信息

关键词： normal cone increasing operator positive solution for integral equation

On Solutions of Sparsity Constrained Optimizatio

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Journal of the Operations Research Society of China 2015年第4期3卷 421-439页

作者： Li-Li Pan Nai-Hua Xiu Sheng-Long Zhou Department of Applied Mathematics Beijing Jiaotong UniversityBeijing 100044China Department of Mathematics Shandong University of TechnologyZibo 255049ShandongChina School of Mathematics University of SouthamptonSouthampton SO171BJUK

In this paper,we mainly study the existence of solutions to sparsity constrained optimization(SCO).Based on the expressions of tangent cone and normal cone of sparsity constraint,we present and characterize two first-order necessary optimality conditions for SCO:N-stationarity and *** we give the second-order necessary and sufficient optimality conditions for *** last,we extend these results to SCO with nonnegative constraint.

关键词： Sparsity constrained optimization Tangent cone normal cone First-order optimality condition Second-order optimality condition