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中文摘要：
运用Banach空间几何学、线性拓扑空间理论、Banach空间中非线性逼近理论、不动点理论等现代分析的理论与方法，研究凸约束优化中的若干非线性方程与不等式问题的理论和算法，即研究两个方面(1)凸约束优化中的若干非线性方程解的存在性与迭代算法；(2)凸约束优化中的若干非线性不等式问题解的存在性与迭代算法．本项目将为凸约束优化中迭代算法的研究提供新方法和新途径，并为现行数学技术提供强有力的工具，也为泛函空间中的算子逼近理论的研究和应用带来动力．本研究课题，是理论性和应用性都很强的边缘性交叉课题．在偏微分方程、力学、最优化与控制、运输与经济平衡、电路与结构分析、以及数学与工程科学中许多别的分支学科等方面都有着广泛应用．
英文摘要：
constrained convex optim.；variational inequality；fixed point problem；existence；algorithms
结论摘要：
自2011年以来，本课题一直致力于研究凸约束优化中一些非线性方程与不等式问题的理论与算法．具体地说，利用广义f-投影算子的性质与熟知的KKM 定理、Kakutani-Fan-Glicksberg定理，建立了自反的光滑Banach空间中广义集值混合(拟-)变分不等式解的一些存在定理；构造了一些具强收敛性的松弛与混合粘性迭代法，用于解Hilbert 空间中变分不等式一般系统、分裂可行问题及严格伪压缩映像不动点问题；建议了具强收敛性的混合粘性迭代法，用于解具一致Gateaux可微范数的一致凸Banach空间中变分不等式一般系统与无限族非扩张映像公共不动点问题；设计了具强收敛性的三步Mann迭代法，用于解一致光滑或具有弱连续对偶映像的一致凸Banach空间中变分不等式一般系统与无限族非扩张映像公共不动点问题；构造了具强收敛性的混合粘性CQ迭代法，用于解Hilbert 空间中变分不等式、变分不等式一般系统及依中间意义一致连续渐近严格伪压缩映像的不动点问题；引入了具强收敛性的两步松弛外梯度法，用于解2-一致光滑的或具有一致Gateaux可微范数的一致凸Banach空间中变分不等式一般系统与无限族非扩张映像公共不动点问题；引入了具正则化的多步隐式迭代法，用于解Hilbert 空间中凸连续的Frechet可微泛函的极小化问题与无限族非扩张映像公共不动点问题，并推导了强收敛；建议了具强收敛性一般迭代法，用于解具弱序列连续的广义对偶映像的q-一致光滑且一致凸的Banach空间中变分不等式系统与无限族非扩张映像公共不动点问题；把混合最速下降法、粘性逼近法及梯度投影算法的平均映像途径结合起来，设计了具强收敛性的迭代法，用于解Hilbert空间中的三重分层变分不等式．因此，对照课题申请书，所取得的成果，表明课题要求已顺利完成．
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期刊论文
Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solu
Relaxed hybrid steepest-descent methods with variable parameters for triple-hierarchical variational
Well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbat
Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed
Metric characterizations of $\alpha$-well-posedness for a system of mixed quasivariational-like ineq
Common fixed point for three pairs of self-maps satisfying common (E,A)-property in generalized metr
Convergence of a new modified Ishikawa type iteration for common fixed points of total asymptoticall
Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and varia
A composite iterative method for a finite family of equilibrium problems and a fixed point problem f
Necessary and sufficient condition of the strong convergence for two finite families of uniformly L-
An iterative algorithm combining the viscosity method with the parallel method for a finite family o
On the convergence of a parallel iterative algorithm for two finite families of uniformly L-Lipschit
Hybrid proximal-type algorithms for generalized equilibrium problems, maximal monotone operators, an
Iterative approaches to find zeros of maximal monotone operators by hybrid approximate proximal poin
Iterative methods for genera- lized equilibrium problems, systems of general generalized equilibrium
Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operato
Strong and weak convergence theorems for an infinite family of nonexpansive mappings and application
Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed poin
Relaxed extragradient-like method for general systems of generalized mixed equilibria and fixed poin
Hybrid method for designing explicit hierarchical fixed point approach to monotone variational inequ
Strong convergence theorems for asymptotically weak -pseudo-contractive non-self-mappings with the g
Strong Convergence of the Iterative Methods for Hierarchical Fixed Point Problems of an Infinite Fam
Modified Relaxed Extragradient Method for a General System of Variational Inequalities and Nonexpans
Strong convergence for asymptotically pseudocontractions with the demiclosedness principle in Banach
Hybrid viscosity CQ method for finding a common solution of a variational inequality, a general syst
Iterative algorithms with regularization for hierarchical variational inequality problems and convex
Hybrid projection method for generalized mixed equilibrium problems, variational inequality problems
Multi-step hybrid iterative method for triple hierarchical variational inequality problem with equil
Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and
Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in
Some iterative methods for finding fixed points and for solving constrained convex minimization prob
The Mann-type extragradient iterative algorithms with regularization for solving variational inequal
Relaxed extragradient methods with regularization for general system of variational inequalities wit
Strong convergence for solving a general system of variational inequalities and fixed point problems
A hybrid extragradient-like approximation method with regularization for solving split feasibility a
Relaxed implicit extragradient-like methods for finding minimum-norm solutions of the split feasibil
Relaxed extragradient method for finding a common element of systems of variational inequalities and
On the relaxed hybrid-extragradient method for solving constrained convex minimization problems in H
Existence theorems for generalized set-valued mixed (quasi-) variational inequalities in Banach spac
A general iteration scheme for variational inequality problem and common fixed point problems of non
Mann type iterative methods for finding a common solution of split feasibility and fixed point probl
Hybrid gradient-projection algorithm for solving constrained convex minimization problems and genera
Algorithms for general systems of generalized resolvent equations with corresponding systems of vari
Hybrid extragradient methods for finding zeros of accretive operators and solving variational inequa
Strong convergence of implicit viscosity approximation methods for pseudocontractive mappings in Ban
Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained op
Implicit relaxed and hybrid methods with regularization for minimization problems and asymptotically
Composite iterative algorithms for variational inequality and and fixed point problems in real smoot
Modified hybrid steepest-descent methods for general systems of variational inequalities with soluti
Implicit and explicit iterative methods for systems of variational inequalities and zeros of accreti
Mann-type extragradient methods for general systems of variational inequalities with multivalued var
Relaxed and hybrid viscosity methods for general system of variational inequalities with split feasi
Viscosity approximation methods of common fixed points for an infinite family of asymptotically none
Strong convergence of parallel iterative algorithm with mean errors for two finite families of ?iri?
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