Chong Li

• PhD Student at University of Macau

We study the issue of numerically solving the nonnegative inverse eigenvalue problem (NIEP). At first, we reformulate the NIEP as a convex composite optimization problem on Riemannian manifolds. Then we develop a scheme of the Riemannian linearized proximal algorithm (R-LPA) to solve the NIEP. Under some mild conditions, the local and global conver...

Rotation group $\mathcal{SO}(d)$ synchronization is an important inverse problem and has attracted intense attention from numerous application fields such as graph realization, computer vision, and robotics. In this paper, we focus on the least-squares estimator of rotation group synchronization with general additive noise models, which is a noncon...

We propose an inexact linearized proximal algorithm with an adaptive stepsize, together with its globalized version based on the backtracking line-search, to solve a convex composite optimization problem. Under the assumptions of local weak sharp minima of order p(p≥1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepacka...

The lower-order penalty optimization methods, including the \(\ell _q\) minimization method and the \(\ell _q\) regularization method \((0<q\le 1)\), have been widely applied to find sparse solutions of linear regression problems and gained successful applications in various mathematics and applied science fields. In this paper, we aim to investiga...

Using the result of the error estimate of the simple extended Newton method established in the present paper for solving abstract inequality systems, we study the error bound property of approximate solutions of abstract inequality systems on Banach spaces with the involved function F being Fréchet differentiable and its derivative F′ satisfying th...

The $\ell_p$ regularization problem with $0< p< 1$ has been widely studied for finding sparse solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. The proximal gradient algorithm is one of the most popular algorithms for solving the $\ell_p$ regularisation problem. In the present...

We study the convergence issue for inexact descent algorithm (employing general step sizes) for multiobjective optimizations on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity, local/global convergence results are established. On the other hand, without the assumption of the...

In this paper, we discuss the statistical properties of the $\ell_q$ optimization methods $(0<q\leq 1)$, including the $\ell_q$ minimization method and the $\ell_q$ regularization method, for estimating a sparse parameter from noisy observations in high-dimensional linear regression with either a deterministic or random design. For this purpose, we...

We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp. weak sharp minima), local/global convergence (resp. linear convergence) results are established. As an...

In this paper, we study the convergence of the Newton-type methods for solving the square inverse singular value problem with possible multiple and zero singular values. Comparing with other known results, positivity assumption of the given singular values is removed. Under the nonsingularity assumption in terms of the (relative) generalized Jacobi...

An interesting problem was raised in Vong et al. (SIAM J. Matrix Anal. Appl. 32:412–429, 2011): whether the Ulm-like method and its convergence result can be extended to the cases of multiple and zero singular values. In this paper, we study the convergence of a Ulm-like method for solving the square inverse singular value problem with multiple and...

For a bounded closed convex subset C of a Banach space X with the origin as an interior point, we study the convexity issue of proximinal sets G in Banach spaces in the sense of generalized best approximation determined by the Minkowski functional generated by C. Some sufficient conditions such as the smoothness and compactly locally uniform convex...

We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on thi...

In the present paper, we study the characterization issue of the weak sharp minimaproperties for convex infinite optimization problems in normed linear spaces. We develop a new approach to establish several complete geometric characterizations for the global/bounded/local weaksharp minima property, which extend/improve the corresponding ones in thi...

The $\ell_p$ regularization problem with $0< p< 1$ has been widely studied for finding sparse solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. The proximal gradient algorithm is one of the most popular algorithms for solving the $\ell_p$ regularisation problem. In the present...

The iterative soft thresholding algorithm (ISTA) is one of the most popular optimization algorithms for solving the regularized least squares problem, and its linear convergence has been investigated under the assumption of finite basis injectivity property or strict sparsity pattern. In this paper, we consider the regularized least squares proble...

In the present paper, we consider the varying stepsize CQ algorithm for solving the split feasibility problem in Hilbert spaces, investigate the linear convergence issue and explore an application in systems biology. In particular, we introduce a notion of bounded linear regularity property for the split feasibility problem, and use it to establish...

Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper, we prove that these conditions are mutually equivalent and they hold, if and only if the Hadamard manifold is isometric to the Euclidean space. In...

We study some basic properties of the function $f_0:M\rightarrow\IR$ on Hadamard manifolds defined by $$ f_0(x):=\langle u_0,\exp_{x_0}^{-1}x\rangle\quad\mbox{for any $x\in M$}. $$ A characterization for the function to be linear affine is given and a counterexample on Poincar\'{e} plane is provided, which in particular, shows that assertions (i) a...

We propose several approximate Gauss-Newton methods, i.e., the truncated, perturbed, and truncated-perturbed GN methods, for solving underdetermined nonlinear least squares problems. Under the assumption that the Fréchet derivatives are Lipschitz continuous and of full row rank, Kantorovich-type convergence criteria of the truncated GN method are e...

We consider the convergence problem of some Newton-type methods for solving the inverse singular value problem with multiple and positive singular values. Under the nonsingularity assumption of the relative generalized Jacobian matrices at the solution c⁎, a convergence analysis for the multiple and positive case is provided and the superlinear or...

In the present paper, we study Newton’s method on the Heisenberg group for solving the equation , where is a mapping from Heisenberg group to its Lie algebra. Under certain generalized Lipschitz condition, we obtain the convergence radius of Newton’s method and the estimation of the uniqueness ball of the zero point of . Some applications to specia...

In the present paper, we consider inexact proximal point algorithms for finding singular points of multivalued vector fields on Hadamard manifolds. The rate of convergence is shown to be linear under the mild assumption of metric subregularity. Furthermore, if the sequence of parameters associated with the iterative scheme converges to 0, then the...

We study some basic properties of the function $f_0:M\rightarrow\IR$ on Hadamard manifolds defined by $$ f_0(x):=\langle u_0,\exp_{x_0}^{-1}x\rangle\quad\mbox{for any $x\in M$}. $$ A characterization for the function to be linear affine is given and a counterexample on Poincar\'e plane is provided, which in particular, shows that assertions (i) and...

We establish the porosity results on the existence and the topological structure of fixed points for nonexpansive set-valued maps in hyperbolic spaces, which in particular extend and/or improve some known results in this direction.

Let (S,Σ, µ) be a complete positive σ-finite measure space and let X be a Banach space. We consider the simultaneous proximinality problem in L p (S,Σ,X) for 1 ⩽ p < +∞. We establish some N-simultaneous proximinality results of L p (S,Σ0, Y) in L p (S,Σ,X) without the Radon-Nikodým property (RNP) assumptions on the space \(\overline {spanY}\) and i...

We study the convergence issue of the subgradient algorithm for solving the convex feasibility problems in Riemannian manifolds, which was first proposed and analyzed by Bento and Melo [J. Optim. Theory Appl., 152(2012), pp. 773-785]. The linear convergence property about the subgradient algorithm for solving the convex feasibility problems with th...

This paper is devoted to study the Fréchet and proximal regularities at points outside the target set of perturbed distance functions dSJ(⋅) determined by a closed subset SS and a Lipschitz function JJ. Also, we provide some important results on the Fréchet, proximal, and Clarke subdifferentials of dSJ(⋅) at those points in Banach spaces.

For an indexed collection of convex sets in a Banach space with index set I of cardinality vertical bar I vertical bar of I, which may be finite or infinite, we investigate the interrelationship between various qualification notions including the linear regularity, the normal property, the dual normal conical hull intersection property, and Jameson...

We study the existence problems of fixed points and the topological structure of fixed point sets for nonexpansive set-valued maps in Banach space. We establish some porosity results on the existence of fixed points and the topological structure of fixed point sets for nonexpansive set-valued maps with values in an admissible family. The results ob...

The celebrated Farkas lemma originated in Farkas (1902) provides us an attractively simple and extremely useful characterization for a linear inequality to be a consequence of a linear inequality system on the Euclidean space \(\mathbb {R}^n\). This lemma has been extensively studied and extended in many directions, including conic systems (linear,...

Inexact proximal point methods are extended to find singular points for multivalued vector fields on Hadamard manifolds. Convergence criteria are established under some mild conditions. In particular, in the case of proximal point algorithm, that is, \(\varepsilon _n=0\) for each \(n\) , our results improve sharply the corresponding results in Li e...

This paper is devoted to study the Fréchet and proximal Regularity at points in the target set of perturbed distance functions dJs(·) determined by a closed subset 5 and a Lipschitz function J(·). Also, we provide some important results on the Clarke subdifferential of dJs(·) at those points in arbitrary Banach spaces.

We consider conic inequality systems of the type F(x) ≥K 0, with approximate solution x0 associated to a parameter τ, where F is a twice Fréchet differentiable function between Hilbert spaces X and Y, and ≥K is the partial order in Y defined by a nonempty convex (not necessarily closed) cone K ⊆ Y. We prove that, under the suitable conditions, the...

A notion of quasi-regularity is extended for the inclusion problem \({F(p)\in C}\) , where F is a differentiable mapping from a Riemannian manifold M to \({\mathbb R^n}\) . When C is the set of minimum points of a convex real-valued function h on \({\mathbb R^n}\) and DF satisfies the L-average Lipschitz condition, we use the majorizing function te...

We propose a Ulm-like Cayley transform method for solving inverse eigenvalue problems, which avoids solving approximate Jacobian equations comparing with other known methods. A conver- gence analysis of this method is provided and the quadratic convergence property is proved under the assumption of the distinction of the given eigenvalues. Numerica...

We explore in arbitrary Banach spaces the Fréchet type εε-subdifferentials and the limiting subdifferentials for the perturbed distance function dSJ(⋅) determined by a closed subset SS and a lower semicontinuous function JJ defined on SS. In particular, upper and lower estimates for the Fréchet type εε-subdifferentials and for the limiting subdiffe...

We consider variational inequality problems for set-valued vector fields on general Riemannian manifolds. The existence results of the solution, convexity of the solution set, and the convergence property of the proximal point algorithm for the variational inequality problems for set-valued mappings on Riemannian manifolds are established. Applicat...

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and establish their complete characterizations in...

Firmly nonexpansive mappings are introduced in Hadamard manifolds, a particular class of Riemannian manifolds with nonpositive sectional curvature. The resolvent of a set-valued vector field is defined in this setting and by means of this concept, a strong relationship between monotone vector fields and firmly nonexpansive mappings is established....

Let (S,@S,@m) be a complete positive @s-finite measure space and let X be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in L"p(S,@S,X). Let @S"0 be a sub-@s-algebra of @S and Y a nonempty locally weakly compact convex subset of X such that spanY@? and its dual have the Rado...

Extending and improving some recent results of Hantoute, López, Zălinescu, and others, we provide characterization conditions for subdifferential formulas to hold for the supremum function of a family of convex functions on a real locally convex space.

The notions of the τ C  -Kolmogorov condition, the τ C  -sun and the τ C  -regular point are introduced, and the relationships between them and the best τ C  -approximation are explored. As a consequence, characterizations of best τ C  -approximations from some kind of subsets (not necessarily convex) are obtained. As an application, a characteriza...

The present paper is concerned with the convergence problem of inexact Newton methods. Assuming that the nonlinear operator satisfies the γ-condition, a convergence criterion for inexact Newton methods is established which includes Smale's type convergence criterion. The concept of an approximate zero for inexact Newton methods is proposed in this...

Let A be a nonempty closed subset (resp. nonempty bounded closed subset) of a metric space and x∈X∖A. The nearest point problem (resp. the farthest point problem) w.r.t. x considered here is to find a point a0∈A such that (resp. ). We study the well posedness of nearest point problems and farthest point problems in geodesic spaces. We show that if...

We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator ΠCg and the relative projection operator PCg in terms of the D-approximate (weak) compactness for a nonempty closed set C in a Banach space X. We next present certain sufficient conditions as well as equivalent conditions fo...

The notions of Lipschitz conditions with L average are introduced to the study of convergence analysis of Gauss-Newton's method for singular systems of equations. Unified convergence criteria ensuring the convergence of Gauss-Newton's method for one kind of singular systems of equations with constant rank derivatives are established and unified est...

In general Banach space setting, we study the perturbed distance function ${d_S^J(\cdot)}$ determined by a closed subset S and a lower semicontinuous function J (·). In particular, we show that the Fréchet subdifferential and the proximal subdifferential of a perturbed distance function are representable by virtue of corresponding normal cones of...

Two iterative algorithms for nonexpansive mappings on Hadamard manifolds, which are extensions of the well-known Halpern's and Mann's algorithms in Euclidean spaces, are proposed and proved to be convergent to a fixed point of the mapping. Some numerical examples are provided.

We establish the existence and uniqueness results for variational inequality problems on Riemannian manifolds and solve completely the open problem proposed in [S.Z. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal. 52 (2003) 1491–1498]. Also the relationships between the constrained optimization problem and the variational in...

We first present some sufficient conditions for the upper semicontinuity and/or the continuity of the Bregman farthest-point map Q C g and the relative farthest-point map S C g for a nonempty D-maximally approximately compact subset C of a Banach space X. We next present certain sufficient conditions as well as equivalent conditions for a Klee set...

Assuming that the first derivative of an operator satisfies the Lipschitz condition, a Kantorovich-type convergence criterion for inexact Newton methods is established, which includes the well-known Kantorovich's theorem as a special case. Comparisons and a numerical example are presented to illustrate that our results obtained in the present paper...

The maximal monotonicity notion in Banach spaces is extended to Riemannian manifolds of nonpositive sectional curvature, Hadamard manifolds, and proved to be equivalent to the upper semicontinuity. We consider the problem of finding a singularity of a multivalued vector field in a Hadamard manifold and present a general proximal point method to sol...

Robinson's generalized Newton's method for nonlinear functions with values in a cone is extended to mappings on Riemannian manifolds with values in a cone. When Df satisfies the L-average Lipschitz condition, we use the majorizing function technique to establish the semi-local quadratic convergence of the sequences generated by the extended Newton'...

We introduce the notion of the (one-parameter subgroup) γ-condition for a map f from a Lie group to its Lie algebra and establish α-theory and γ-theory for Newton’s method for a map f satisfying this condition. Applications to analytic maps are provided, and Smale’s α-theory and γ-theory are extended and developed. Examples arising from initial val...

A convergence criterion of the family of Euler-Halley type methods for the vector fields on Riemannian manifolds whose covariant derivatives satisfy the γ-condition is established. The corresponding results due to (12) are extended. An application to analytic vector fields is provided.

With the classical assumptions on f, a convergence criterion of Newton's method (independent of affine connections) to find zeros of a mapping f from a Lie group to its Lie algebra is established, and estimates of the convergence domains of Newton's method are obtained, which improve the corresponding results in Owren & Welfert (2000, BIT Numer. Ma...

We consider the optimization problem (PA) inf x2X{f(x) + g(Ax)} where f and g are proper convex functions defined on locally convex Hausdor topological vector spaces X and Y respectively, and A is a linear operator from X to Y. By using the properties of the epigraph of the conjugated functions, some sucient and necessary conditions for the strong...

We study the convergence properties for some inexact Newton-like methods including the inexact Newton methods for solving nonlinear operator equations on Banach spaces. A new type of residual control is presented. Under the assumption that the derivative of the operator satisfies the Hölder condition, the radius of convergence ball of the inexact N...

This paper is concerned with the problem of nonlinear best simul- taneous approximations in conditional complete lattice Banach spaces with a strong unit. Characterization results of the best simultaneous approximation from simultaneous suns and suns are established. A counterexample, to which the characterization theorem for convex sets due to Moh...

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p⩾1. This paper is concerned with the perturbed optimization problem of finding z0∈Z such that ‖x−z0‖p+J(z0)=infz∈Z{‖x−z‖p+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the K...

Convergence criterion of the inexact methods is established for operators with holder continuous first derivatives. An application to a special nonlinear Hammerstein integral equation of the second kind is provided.

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth.

The famous Newton–Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton's method to a solution of an equation. Here we present a "Kantorovich type" convergence analysis for the Gauss–Newton's method which improves the result in [W.M. Häußler, A Kantorovich-type convergence analysis for the Gaus...

The present paper is concerned with the convergence problem of Newton's method to solve singular systems of equations with constant rank derivatives. Under the hypothesis that the derivatives satisfy a type of weak Lipschitz condition, a convergence criterion based on the information around the initial point is established for Newton's method for s...

One kind of the L-average Lipschitz condition is introduced to covariant derivatives of sections on Rie- mannian manifolds. A convergence criterion of Newton's method and the radii of the uniqueness balls of the singular points for sections on Riemannian manifolds, which is independent of the curvatures, are established under the assumption that th...

The present work is concerned with the uniqueness problem of best simultaneous approximation. An n-dimensional l1- or l∞-simultaneous unicity space is characterized in terms of Property A.

The present paper is concerned with the problem of weighted best simultaneous approximations in Banach spaces. The weighted best simultaneous approximations to sequences from S- and BS-suns in the Banach space are characterized in view of the Kolmogorov conditions. Applications are provided for weighted best simultaneous approximations from RS-sets...

For an inequality system defined by an infinite family of proper convex functions, we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions and study relationships between these new constraint qualifications and other well-known constraint qualifications including the basic constraint...

The convergence properties of Newton's method for systems of equations with constant rank derivatives are studied under the hypothesis that the derivatives satisfy some weak Lipschitz conditions. The unified convergence results, which include Kantorovich type theorems and Smale's point estimate theorems as special cases, are obtained.

This paper is concerned with the problem of the best restricted range approximations of complex-valued continuous functions. Several properties for the approximating set PW\mathcal{P}_\Omega such that the classical characterization results and/or the uniqueness results of the best approximations hold are introduced. Under the very mild conditions...

We introduce a notion of quasi-regularity for points with respect to the inclusion F(x) 2 C where F is a nonlinear Frechet dierentiable function from Rv to Rm. When C is the set of minimum points of a convex real-valued function h on Rm and F0 satisfies the L-average Lipschitz condition of Wang, we use the majorizing function technique to establish...

The present paper is concerned with the convergence problem of the variants of the Chebyshev–Halley iteration family with parameters for solving nonlinear operator equations in Banach spaces. Under the assumption that the first derivative of the operator satisfies the Hölder condition of order p, a convergence criterion of order 1 + p for the itera...

The convergence criterion of Newton's method for underdetermined system of equations under the γ-condition is established and the radius of the convergence ball is obtained. Applications to analytic operator are provided and some results due to Shub and Smale (SIAM J. Numer. Anal. 1996, 33:128–148) are extended and improved.

The convergence criterion of Newton’s method to find the zeros of a map f from a Lie group to its corresponding Lie algebra is established under the assumption that f satisfies the classical Lipschitz condition, and that the radius of convergence ball is also obtained. Furthermore, the radii of the uniqueness balls of the zeros of f are estimated....

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi–continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applicatio...

The convergence of the King-Werner method for finding zeros of nonlinear operators is analyzed. Under the hypothesis that the derivative of f satisfies the radius Lipschitz condition with L-average, the convergence criterion and the convergence ball for the King-Werner method are given. Applying the results to some particular functions L(u), we get...

Let X be a Banach space and Z a nonempty closed subset of X. Let J:Z→R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem supz∈Z{J(z)+‖x−z‖}supz∈Z{J(z)+‖x−z‖}, which is denoted by (x,J)(x,J)-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly...

We consider a (finite or infinite) family of closed convex sets with nonempty intersection in a normed space. A property relating their epigraphs with their intersection's epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP, and Jameson's ($G$)-property) are established. With sui...

Let X be a Banach space and Z a nonempty closed subset of X. Let J:Z→R be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem infz∈Z{J(z)+‖x−z‖}, denoted by (x,J)-inf for x∈X. In the case when X is compactly fully 2-convex, it is proved in the present paper that the set of all points x...

The convergence problem of the family of the deformed Euler–Halley iterations with parameters for solving nonlinear operator equations in Banach spaces is studied. Under the assumption that the second derivative of the operator satisfies the Hölder condition, a convergence criterion of the order 2+p of the iteration family is established. An applic...

Let G be a nonempty closed (resp. bounded closed) subset in a strongly convex Banach space X. Let B(X) denote the space of all nonempty bounded closed subsets of X endowed with the Hausdorff distance and let B-G(X) denote the closure of the set {A is an element of B(X) : A boolean AND G = phi}. We prove that E-o(G) (resp. E-o(G)), the set of all A...

This paper is concerned with the problem of the uniqueness of singular points of vector fields on Riemannian manifolds. The radii of the uniqueness balls of the singular points of vector fields are estimated under the assumption that the vector fields satisfy the gamma-condition, and the results due to Wang and Han in [Criterion a and Newton's meth...

Let ℬ (resp. \({\fancyscript K}\) , ℬ\({\fancyscript C}\), \({\fancyscript K}\) \({\fancyscript C}\)) denote the set of all nonempty bounded (resp. compact, bounded convex, compact convex) closed subsets of the Banach space X, endowed with the Hausdorff metric, and let G be a nonempty relatively weakly compact closed subset of X. Let ℬostand for th...

The γ -conditions for vector fields on Riemannian manifolds are introduced. The γ -theory and the α -theory for Newton's method on Riemannian manifolds are established under the γ -conditions. Applications to analytic vector fields are provided and the results due to Dedieu et al. (2003, IMA J. Numer. Anal. , 23 , 395–419) are improved.

The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vector fields satisfy some kind of general Lipschitz conditions. Some classical results such as the Kantorovich’s type theorem and the S...

Abstract To provide a Kolmogorov-type condition for characterizing a best approximation,in a continuous complex-valued function space, it is usually assumed that the family of closed convex sets in the complex plane used to restrict the range satisfies a strong interior-point condition, and this excludes the interesting case when,some,t is a line-s...

Let G be a strict RS-set (resp. an RS-set) in X and let F be a bounded (resp. totally bounded) subset of X satisfying rG(F)>rX(F)rG(F)>rX(F), where rG(F)rG(F) is the restricted Chebyshev radius of F with respect to G. It is shown that the restricted Chebyshev center of F with respect to G is strongly unique in the case when X is a real Banach space...

Let p be a continuous seminorm on a real locally convex space X. The current paper is concerned with the problem of restricted p-centers in X. Characterizations of restricted p-centers and strongly unique restricted p-centers with respect to a “sun” are provided, and then some recent results due to Laurent and Pai [99. P. J. Laurent and D. Pai (...

The concept of an RS–set in a complex Banach space is introduced and the problem of best approximation from an RS–set in a complex space is investigated. Results consisting of characterizations, uniqueness and strong uniqueness are established.

For a general (possibly infinite) system of closed convex sets in a normed linear space we provide several sufficient conditions for ensuring the strong conical hull intersection property. One set of sufficient conditions is given in terms of the finite subsystems while the other sets are in terms of the relaxed interior-point conditions together w...

For a general infinite system of convex inequalities in a Banach space, we study the basic constraint qualification and its relationship with other fundamental concepts, including various versions of conditions of Slater type, the Mangasarian-Fromovitz constraint qualification, as well as the Pshenichnyi-Levin-Valadier property introduced by Li, Na...

Let X be a real separable strictly convex Banach space and G a nonempty closed subset of X. Let (resp. ) denote the family of all nonempty boundedly compact (resp. compact) convex subsets of X endowed with the Hρ-topology (resp. the Hausdorff distance), (resp. ) the closure of the set (resp. ), and (resp. ) the family of (resp. ) such that the mini...

The present paper is concerned with problems of the strong uniqueness of the best approximation and the characterization of a uniqueness element in operator spaces. Some results on the strong uniqueness of the best approximation operator from RS-sets are proved and the uniqueness element of a sun in the compact operator space from C0 to C0 is chara...