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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...

This paper shows, in the metric space setting, that the p-order metric regularity is preserved under perturbation by locally Holder continuous mappings of order 1/p, extending a classical result which says that 1-order metric regularity is preserved under perturbation by locally Lipschitz mappings (see Theorem 3.3 in (Dontchev et al., Trans. Am. Math. Soc. 355:493–517, 2003) and Theorem 1 in (Dontchev, Optimization 64:41–48, 2015)) by allowing p ∈ (0, 1]. The argument is based on a new result of fixed point theorem type.

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...

For a finite/infinite family of closed convex sets with nonempty intersection in Hilbert space, we consider the (bounded) linear regularity property and the linear convergence property of the projection-based methods for solving the convex feasibility problem. Several sufficient conditions are provided to ensure the bounded linear regularity in terms of the interior-point conditions and some finite codimension assumptions. A unified projection method, called Algorithm B-EMOPP, for solving the convex feasibility problem is proposed, and by using the bounded linear regularity, the linear convergence results for this method are established under a new control strategy introduced here.

Using the convex process theory and the majorizing technique, we study the convergence issues of the iterative sequences generated by the extended Newton method for solving the conic inequality system F ≥ C 0 defined by a cone C and a Frechet differentiable function F satisfying the (extended) weak γ-condition. Convergence criterion of the extended Newton method is presented. As an application, we establish, when F is analytic, an extended α-theory similar to Smale's α-theory for nonlinear analytic equations.

Using the Borwein–Preiss variational principle and in terms of the proximal coderivative, we provide a new type of sufficient conditions for the Holder metric subregularity and Holder error bounds in a class of smooth Banach spaces. As an application, new characterizations for the tilt stability of Holder minimizers are established.

In this paper, we introduce and study the notions of Holder weak sharp minimizers, stable Holder weak sharp minimizers and Holder tilt-stable weak minimizers for a proper lower semicontinuous function f on a Banach space. In terms of the Holder metric subregularity/regularity of ∂ f , we consider optimality conditions for Holder weak sharp minimizers and stable Holder weak sharp minimizers. We prove that x is a stable Holder weak sharp minimizer (resp. a Holder tilt-stable weak minimizer) of f if and only if it is a stable Holder sharp minimizer (resp. a Holder tilt-stable minimizer) of f .

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 $\partial f$.

In this paper we discuss some conditions ensuring the error bound property for a family of functions. By virtual of matrix theory on strongly positive matrices we provide some descent/directional descent conditions ensuring the existence of error bounds for systems of functions. These conditions relax the descent requirement for functions and provide useful methods for establishing error bounds.

In this note, we consider error bound issue for conic inequalities. In terms of subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In particular, our result extends Ioffe’s classical result on local error bounds to the vector-valued function case.

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

Under either linearity or convexity assumption, several authors have studied the stability of error bounds for inequality systems when the concerned data undergo small perturbations. In this paper, we consider the corresponding issue for a more general conic inequality (most of the constraint systems in optimization can be described by an inequality of this type). In terms of coderivatives for vector-valued functions, we study perturbation analysis of error bounds for conic inequalities in the subsmooth setting. The main results of this paper are new even in the convex/smooth case.

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.

In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert spaces. We also establish the results of Robinson–Ursescu theorem type for prox-regular multifunctions.

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 first consider subsmoothness for a function family and provide formulas of the subdifferential of the pointwise supremum of a family of subsmooth functions. Next, we consider subsmooth infinite and semi-infinite optimization problems. In particular, we provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum for such optimization problems.

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 an infinite family of proper convex functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications. Under the new constraint qualifications, we provide necessary and/or sufficient conditions for the KKT rules to hold. Similarly, we provide characterizations for constrained minimization problems to have total Lagrangian dualities. Several known results in the conic programming problem are extended and improved.

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.

In terms of the normal cone and the coderivative, we provide some necessary and/or sufficient conditions of metric subregularity for (not necessarily closed) convex multifunctions in normed spaces. As applications, we present some error bound results for (not necessarily lower semicontinuous) convex functions on normed spaces. These results improve and extend some existing error bound results.

Using variational analysis techniques, we study convex-composite optimization problems. In connection with such a problem, we introduce several new notions as variances of the classical KKT conditions. These notions are shown to be closely related to the notions of sharp or weak sharp solutions. As applications, we extend some results on metric regularity of inequalities from the convex case to the convex-composite case.

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.

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\

We study the weak domination property and weakly efficient solutions in vector optimization problems. In particular scalarization of these problems is obtained by virtue of some suitable merit functions. Some natural conditions to ensure the existence of error bounds for merit functions are also given.

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.

In this paper, we discuss some sufficient conditions for the linear regularity and bounded linear regularity (and their variations) of finitely many closed (not necessarily convex) sets in a normed vector space. The accompanying necessary conditions are also given in the setting of Asplund spaces.

We study the Clarke generalized gradient of the D-gap functions for the variational inequality problem (VIP) defined by a locally Lipschitz, but not necessarily differentiable, function in an Euclidean space. Using these results, we study the relationship between minimizing sequences and stationary sequences of the D-gap function, regardless of the existence of solutions of (VIP).

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 general Banach space setting, we study the minimum time function determined by a closed convex set K and a closed set S (this function is simply the usual Minkowski function of K if S is the singleton consisting of the origin). In particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function.

The existence of strictly feasible points is shown to be equivalent to the boundedness of solution sets of generalized complementarity problems with stably pseudomonotone mappings. This generalizes some known results in the literature established for complementarity problems with monotone mappings.

We study the Clarke–Rockafellar directional derivatives of the regularized gap functions (and of some modified ones) for the variational inequality problem (VIP) defined by a locally Lipschitz but not necessarily differentiable function on a closed convex set in an Euclidean space. As applications we show that, under the strong monotonicity assumption, the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of Armijo type converge to the solution of the (VIP).

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 Ωt is a line-segment or a singleton. The main aim of the present paper is to remove this restriction by virtue of a study of the notion of the strong CHIP for an infinite system of closed convex sets in a continuous complex-valued function space.

We study the perturbation property of best approximation to a set defined by an abstract nonlinear constraint system. We show that, at a normal point, the perturbation property of best approximation is equivalent to an equality expressed in terms of normal cones. This equality is related to the strong conical hull intersection property. Our results generalize many known results in the literature on perturbation property of best approximation established for a set defined by a finite system of linear/nonlinear inequalities. The connection to minimization problem is considered.

We investigate whether some merit functions for variational inequality problems (VIP) provide error bounds for the underlying VIP. Under the condition that the involved mapping F is strongly monotone, but not necessarily Lipschitz continuous, we prove that the so-called regularized gap function provides an error bound for the underlying VIP. We give also an example showing that the so-called D-gap function might not provide error bounds for a strongly monotone VIP.

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.

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.

Let E be a normed space, $$a_1^* ,...,a_m^* \in E^* ,c_1 ,...,c_m \in R$$ and $$S = \left\{ {x \in E\left| {\left\langle {a_i^* ,x} \right\rangle - c_i \leqslant 0,1 \leqslant i \leqslant m} \right.} \right\} \ne \emptyset $$ . Let $$\tau _* = \inf \left\{ {\tau \geqslant 0:dist\left( {x,S} \right) \leqslant \tau \max \left\{ {\left[ {\left\langle {a_i^* ,x} \right\rangle - c_i } \right]_ + :i = 1,...,m} \right\}\forall x \in E} \right\}$$ . We give some exact formulas for 7#x03C4;?.

We give two explicit formulas which express the error bound moduli for conic convex systems, one in terms of directional derivative, and the other in terms of the coderivative. As applications, we study error bounds for systems of infinitely many convex inequalities.

In this paper, we mainly study various notions of regularity for a finite collection {C1,⋯,Cm} of closed convex subsets of a Banach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., ϒ-error bound) if and only if the pair {epi(f),X×{0}} of sets in the product space X×ℝ is linearly regular (resp., regular). Similar results for multifunctions are also established. Next, we prove that {C1,⋯,Cm} is linearly regular if and only if it has the strong CHIP and the collection {NC1(z),⋯,NCm(z)} of normal cones at z has property (G) for each z∈C:=∩i=1mCi. Provided that C1 is a closed convex cone and that C2=Y is a closed vector subspace of X, we show that {C1,Y} is linearly regular if and only if there exists α>0 such that each positive (relative to the order induced by C1) linear functional on Y of norm one can be extended to a positive linear functional on X with norm bounded by α. Similar characterization is given in terms of normal cones.

In terms of various derivatives such as contingent derivative and Dini-derivative, we give a series of characterizations of error bounds for convex multifunctions defined on Banach spaces, extending Lewis and Pang's result on a characterization of the existence of global error bounds for approximate solutions of convex inequalities in Euclidean spaces. Applications are given not only to improve some known results but also to provide new results on the existence of error bounds for not necessarily lower semicontinuous convex functions.