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Considering a closed multifunction $\Psi$ between two Banach spaces, it is known that metric regularity and strong metric subregularity of $\Psi$ are, respectively, stable with respect to “small Li...

For an inequality system defined by an infinite family of proper lower semicontinuous convex functions in normed linear space, we consider the Farkas--Minkowski (FM for short) type qualification an...

When the objective function undergoes both a tilt perturbation and a general parameter perturbation, this paper considers the notions of a fully stable Holder minimizer, a uniform Holder growth con...

Using the error bound results established in the present paper for approximate solutions, we study the stability issues when perturbed by possibly nonaffine smooth maps $E$ for the abstract inequal...

Using techniques of variational analysis and dual techniques for smooth conjugate functions, for a local minimizer of a proper lower semicontinuous function $f$ on a Banach space, $p\in (0,\;+\infty)$ and $q=\frac{1+p}{p}$, we prove that the following two properties are always equivalent: (i) $\bar x$ is a stable $q$-order minimizer of $f$ and (ii) $\bar x$ is a tilt-stable $p$-order minimizer of $f$. We also consider their relationships in conjunction with the $p$-order strong metric regularity of the subdifferential mapping $\partial f$.

For an indexed collection of convex sets in a Banach space with index set $I$ of cardinality $|I|$ 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's property $(G)$, together with their quantitative versions, and thereby we extend the well-known results of Bakan, Deutsch, and Li [Trans. Amer. Math. Soc., 357 (2005), pp. 3831--3863], and Bauschke, Borwein, and Li [Math. Program. Ser. A, 86 (1999), pp. 135--160] from the finite case to the more general case that allows $|I|$ to be infinite.

In this paper, we consider the metric subregularity property for a piecewise linear multifunction (with respect to a piecewise linear constraint) as well as the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem. Of these properties we pay special attention to the global ones. We first provide a result on a certain relationship between two nonnegative piecewise linear numerical functions for which the kernel of one of them is contained in the kernel of the other. Using this result, we establish the bounded/global metric subregularity results for a piecewise linear multifunction with respect to a piecewise linear set. As applications, we study the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem.

Unlike the Frechet, Penot, Clarke, and other “order one” notions of normal cones, the notion of proximal normal cone describes variational behavior of “order two,” and most of results on proximal normal cone are established in the Hilbert space framework. A possible reason is that in a general Banach space (such as in the classical Banach spaces $l^p$ and $L^p$ with $1 \leq p < 2$), the proximal normal cone of a “smooth” subset $A$ at every point of $A$ may consist of the zero element only. Nevertheless, on the other hand, we show in this paper that for a fairly big class of smooth Banach spaces (including the classical Banach spaces $l^p$ and $L^p$ with $2 \leq p

We consider conic inequality systems of the type $F(x)\ge_K 0$, with approximate solution $x_0$ associated to a parameter $\tau$, where $F$ is a twice Frechet differentiable function between Hilbert spaces $X$ and $Y$, and $\ge_K$ is the partial order in $Y$ defined by a nonempty convex (not necessarily closed) cone $K\subseteq Y$. We prove that, under the suitable conditions, the system $F(x)\ge_K 0$ is solvable, and the ratio of the distance from $x_0$ to the solution set $S$ over the distance from $F(x_0)$ to the cone $K$ has an upper bound given explicitly in terms of $\tau$ and $x_0$. We show that the upper bound is sharp. Applications to analytic function inequality/equality systems on Euclidean spaces are given, and the corresponding results of Dedieu [SIAM J. Optim., 11 (2000), pp. 411--425] are extended and significantly improved.

Using the technique of variational analysis and in terms of normal cones, we establish unified separation results for finitely many closed (not necessarily convex) sets in Banach spaces, which not only cover the existing nonconvex separation results and a classical convex separation theorem, but also recapture the approximate projection theorem. With help of the separation result for closed sets, we provide necessary and sufficient conditions for approximate Pareto solutions of constrained vector optimization problems. In particular, we extend some basic optimality results for approximate solutions of numerical optimization problems to the vector optimization setting.

Extending and improving some recent results of Hantoute, Lopez, and Zalinescu 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.

We establish several set-valued function versions of Ekeland's variational principle and hence provide some sufficient conditions ensuring the existence of error bounds for inequality systems defined by finitely many lower semicontinuous functions.

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.

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.

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.

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.

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.

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.

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 suitable continuity assumption we show how this property can be ensured from the corresponding property of some of its finite subfamilies.

We study general constrained multiobjective optimization problems with objectives being closed multifunctions in Banach spaces. In terms of the coderivatives and normal cones, we provide generalized Lagrange multiplier rules as necessary optimality conditions of the above problems. In an Asplund space setting, sharper results are presented.

In this paper we study error bounds for function $\varphi(x)={\rm dist}(Ax+b,K)$, where $A$ is a continuous linear operator from a Banach space $X$ into another Banach space $Y$, $b\in Y$, and $K\subset Y$ is a closed set. Assuming that $A$ is of closed range ${\rm Im}(A)$, $\varphi$ is shown to have a Lipschitz error bound ($\Upsilon$-error bound) if and only if $\{{\rm Im}(A)+b,K\}$ is linearly regular (regular). Next we identify exactly when the convex function $g=\sqrt{\zeta}-l$ has error bounds (of various types), where $\zeta$ is a quadratic function satisfying $\zeta\geq 0$ and $l$ is an affine function on $\mathbb{R}^{n}$. As an application, we study the existence of various error bounds for the function $\varphi:x\mapsto {\rm dist}(Ax+b,\mathcal{K})$ to supplement the results reported in an earlier paper [K. F. Ng and W. H. Yang, SIAM J. Optim., 13 (2002), pp. 24--43] for the standard second-order cone $\mathcal{K}$ in $\mathbb{R}^{n}$.

For a general (constrained or unconstrained) system of conic inequalities, we provide sufficient conditions to ensure the existence of a common error bound for each of the systems obtained from the original one when the given data undergoes a small perturbation. Moreover, we also give some dual characterizations on the existence of error bounds for systems of conic inequalities.

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.

Several fundamental concepts such as the basic constraint qualification (BCQ), the strong conical hull intersection property (CHIP), and the perturbations for convex systems of inequalities in Banach spaces (over $\mathbb{R}$ or $\mathbb{C}$) are extended and studied; here the systems are not necessarily finite. Their relationships with each other in connection with the best approximations are investigated. As applications, we establish results on the unconstrained reformulation of best approximations with infinitely many constraints in Hilbert spaces; also we give several characterizations of best restricted range approximations in $C(Q)$ under quite general constraints.

We consider a proper lower semicontinuous function $f$ on a Banach space X with $\lambda=\inf\{f(x):\;x\in X\}>-\infty$. Let $\alpha\geq\lambda$ and $S_\alpha=\{x\in X:\; f(x)\leq\alpha\}$. We define the lower derivative of f at the set $S_\alpha$ by $$\underline{D}(f,S_\alpha)=\liminf_{x\rightarrow S_\alpha}\frac{f(x)-\alpha}{dist(x,S_\alpha)},$$ where $x\rightarrow S_\alpha$ can be interpreted in various ways. We show that, when f is convex and $\alpha = \lambda$, it is equal to the largest weak sharp minima constant. In terms of these derivatives and subdifferentials, we present several characterizations for convex f to have global weak sharp minima. Some of these results are also shown to be valid for nonconvex f. As applications, we give error bound results for abstract linear inequality systems.

In this paper we study error bounds of the abstract linear inequality system (A,C,b): $Ax \leqslant b$, where A is a bounded linear operator from a Banach space X to a Banach space Y partially ordered by a closed convex cone C. We also give some general results on the existence of error bounds for a convex function F; we show in particular that F has an error bound if and only if the directional derivative of the distance function (to the solution set S) at each boundary point of S along any nontangential direction is bounded by the derivative of F. As an application, we prove that if C is a polyhedral cone, then the system (A,C,b) has an error bound. When Y is a Hilbert space, our results can be expressed in terms of the angles between Ax-b-P-C(Ax-b) and Im(A), or in terms of the angles between Im(A) and the nonvertex supporting hyperplanes of C. In the case in which $X=\mathbb{R}^{n}$ and C is an "ice-cream" cone, we identify exactly when (A,C,b) has an error bound.

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.

Let f be a bounded below, lower semicontinuous function from a Banach space into $R\cup \{+\infty \}.$ We study the relationships between minimizing and critical sequences of f, where the criticality condition is given in terms of some subdifferential $\partial.$ Here the objective function f is not supposed to be convex or smooth. Our work extends that of Auslender and Crouzeix and that of Chou, Ng, and Pang.

We study some fundamental asymptotic properties associated with the well-posedness of constrained optimization problems. Emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic Karush--Kuhn--Tucker (KKT) optimality conditions. Unlike the subdifferential approach used by Auslender, Cominetti, and Crouzeix, we use a residual function approach that is closely tied to the theory of error bounds; this approach handles constraints explicitly and allows the effective treatment of infeasible, stationary sequences. The asymptotic KKT conditions provide an asymptotic optimality certificate for inequality constrained programs. Specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with Holderian minima are discussed. An application of the results to the convergence of a family of Newton-type iterative descent methods involving singular quasi-Newton matrices for solving a constrained minimization problem is also presented.