Xi Yin Zheng

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.

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.

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.

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.

Calmness for L-Subsmooth Multifunctions in Banach Spaces

Using variational analysis techniques, we study subsmooth multifunctions in Banach spaces. In terms of the normal cones and coderivatives, we provide some characterizations for such multifunctions to be calm. Sharper results are obtained for Asplund spaces. We also present some exact formulas of the modulus of the calmness. As applications, we provide some error bound results on nonconvex inequalities, which improve and generalize the existing error bound results.

Lagrange Multipliers in Nonsmooth Semi-Infinite Optimization Problems

Using the variational analysis technique, in terms of the epi-coderivative, we provide Lagrange multiplier rules for a class of semi-infinite optimization problems where all functions are lower semicontinuous or locally Lipschitz.

Global error bounds with fractional exponents

Abstract.Using the partial order induced by a proper weakly lower semicontinuous function on a reflexive Banach space X we give a sufficient condition for f to have error bounds with fractional exponents. Application is given to identify the set of such exponents for quadratic functions.

Weak Sharp Minima for Semi-infinite Optimization Problems with Applications

We study local weak sharp minima and sharp minima for smooth semi-infinite optimization problems SIP. We provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum of SIP. As applications, we present several sufficient and necessary conditions of calmness for infinitely many smooth inequalities. In particular, we improve some calmness results in [R. Henrion and J. Outrata, Math. Program., 104 (2005), pp. 437-464].