Kung Fu Ng

Error Bounds for Lower Semicontinuous Functions in Normed Spaces

Without the convexity or analyticity assumption, we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient/necessary conditions on the existence of error bounds in infinite dimensional normed spaces. Some characterizations for a convex inequality system to possess an error bound in a reflexive Banach space are also given. As applications, in dealing with the Hoffman error bound result in normed spaces, we give a computable Lipschitz bound constant, which is better than previous Lipschitz bound constants in some examples; we also consider error bounds for quadratic functions on Rn.

Constraint Qualifications for Convex Inequality Systems with Applications in Constrained Optimization

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 qualification studied by Hiriart-Urrutty and Lemarechal and by Li, Nahak, and Singer. Extensions of known results to more general settings are presented, and applications to particular important problems, such as conic programming and approximation theory, are also studied.

Metric Regularity and Constraint Qualifications for Convex Inequalities on Banach Spaces

We introduce new notions of the extended basic constraint qualification and the strong basic constraint qualification and discuss their relationship with other fundamental concepts such as the basic constraint qualification and the metric regularity; in particular we provide a solution to an open problem of Lewis and Pang on characterizing the metric regularity in terms of normal cones. We present a characterization of error bounds for convex inequalities in terms of the strong basic constraint qualification. As applications, we study the linear regularity for infinite collections of closed convex sets in a Banach space.

Constraint Qualifications for Extended Farkas's Lemmas and Lagrangian Dualities in Convex Infinite Programming

For an inequality system defined by a possibly infinite family of proper functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we obtain characterizations of those reverse-convex inequalities which are a consequence of the constrained system, and we provide necessary and/or sufficient conditions for a stable Farkas lemma to hold. Similarly, we provide characterizations for constrained minimization problems to have the strong or strong stable Lagrangian dualities. Several known results in the conic programming problem are extended and improved.

Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces

This paper concerns a generalized equation defined by a closed multifunction between Banach spaces, and we employ variational analysis techniques to provide sufficient and/or necessary conditions for a generalized equation to have the metric subregularity (i.e., local error bounds for the concerned multifunction) in general Banach spaces. Following the approach of Ioffe [Trans. Amer. Math. Soc., 251 (1979), pp. 61-69] who studied the numerical function case, our conditions are described in terms of coderivatives of the concerned multifunction at points outside the solution set. Motivated by the existing modulus representation and point-based criteria for the metric regularity, we establish the corresponding results for the metric subregularity. In the Asplund space case, sharper results are obtained.

Metric Subregularity and Constraint Qualifications for Convex Generalized Equations in Banach Spaces

Several notions of constraint qualifications are generalized from the setting of convex inequality systems to that of convex generalized equations. This is done and investigated in terms of the coderivatives and the normal cones, and thereby we provide some characterizations for convex generalized equations to have the metric subregularity. As applications, we establish formulas of the modulus of calmness and provide several characterizations of the calmness. Extending the classical concept of extreme boundary, we introduce a notion of recession cores of closed convex sets. Using this concept, we establish global metric subregularity (i.e., error bound) results for generalized equations.

The Fermat rule for multifunctions on Banach spaces

Using variational analysis, we study vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In particular, in terms of the coderivatives, we present Fermat’s rules as necessary conditions for an optimal solution of the above problems. As applications, we also provide some necessary conditions (in terms of Clarke’s normal cones or the limiting normal cones) for Pareto efficient points.

Error Bounds of Generalized D-Gap Functions for Nonsmooth and Nonmonotone Variational Inequality Problems

We present some error bound results of generalized D-gap functions for nonsmooth and nonmonotone variational inequality (VI) problems. Application is given in providing a derivative-free descent method.

Linear Regularity for a Collection of Subsmooth Sets in Banach Spaces

Using variational analysis, we study the linear regularity for a collection of finitely many closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting. Moreover, the sharpest linear regularity constant can also be dually represented under the subsmoothness assumption.

The SECQ, Linear Regularity, and the Strong CHIP for an Infinite System of Closed Convex Sets in Normed Linear Spaces

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 intersections epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP, and Jamesons (