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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.

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 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.

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.

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.

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.

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.

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.

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.

Let f be a regular, locally Lipschitz real-valued function defined on an open convex subset of a normed space. We show that at any unit direction u, the upper second-order derivative D+2f(·; u, 0) (in the sense of Dem'yanov and Pevnyi [Dem'yanov, V. F., A. B. Pevnyi. 1974. Expansion with respect to a parameter of the extremal values of game problems. USSR Computational Math. and Math. Phys. 14 33–45.]; Ben-Tal and Zowe [Ben-Tal, A., J. Zowe. 1982. Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems. Math. Programming 24 70–91.]) has the same lower bounds as the lower second-order derivatives D−2f(·; u, 0). Consequently, one can characterize the convexity of f in terms of these derivatives. We also obtain the corresponding results in terms of Chaney's second-order directional derivatives.

We give several general implicit function and closed graph theorems for ser-valued functions. Let Z be a normed space,X, Ymetric spaces withXcomplete. Letf: X ⇉ Z,F: X × Y⇉ Z be multifunctions withz0∈ f(x0) ∩ F(x0, y0) such thatfis open at (x0, y0) andf‘approximates’Fin an appropriate sense. Suppose thatf−1(z) is closed,F(x, y)is compact for eachx, yandzand suppose thatF(x0, ·) is lower semi-continuous aty0. ThenF(·,y) is of closed graph ‘locally’, is open atx0, and there exists a functionx(·) withx(y)→x0fory→y0such thatz0∈F(x(y), (y))for allyneary0. A more general form dealing with the non-linear rate situation is also established.

For a given Banach space ( B , ∥·∥) ordered by a normal cone B + , we introduce an equivalent norm on B : ∥ a ∥ N = inf{ λ ⩾ 0: − λv ⩽ a ⩽ λu for some ∥ u ∥, ∥ v ∥⩽ 1}, and give various characterizations of positive C 0 - or C 0 ∗ -semigroups of operators { S t } t ⩾ 0 on B that satisfy the growth ∥ S t ∥ N ⩽ Me γt . The theorem is an analog of the classical Feller-Miyadera-Phillips theorem and is an extension of recent results of D. W. Robinson.

We consider metric regularity of composite multifucntions and establish an inequality on the moduli of metric regularity. Refining the original proofs of the Robinson-Ursescu theorem and Lyusternik-Graves theorem, we give an unified analysis for these two theorems, applicable to the composite of a closed convex multifunction and a continuous function.

The notion of quasi-regularity, defined for optimization problems in Rn, is extended to the Banach space setting. Examples are given to show that our definition of quasi-regularity is more natural than several other possibilities in the general situation. An infinite dimensional version of the Lagrange multiplier rule is established.

Let K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

Let f be a continuous function, and u a continuous linear function, from a Banach space into an ordered Banach space, such that f − u satisfies a Lipschitz condition and u satisfies an inequality implicit-function condition. Then f also satisfles an inequality implicit-function condition. This extends some results of Flett, Craven and S. M. Robinson.

We prove a version of the Feller-Miyadera-Phillips theorem characterizing the infinitesimal generators of positive C0-semigroups on ordered Banach spaces with normal cones. This is done in terms of N(A) as well as the canonical half-norms of Arendt Chernoff and Kato defined by N(a) = inf{‖b‖ |b ≥ a}, where N(A) = sup{N(Aa) |N(a) ≶ 1} for operator A. A corresponding result on –semigroups is also given.

Using the convex process theory we study the convergence issues of the iterative sequences generated by the Gauss–Newton method for the convex inclusion problem defined by a cone C and a smooth function F (the derivative is denoted by F ′ ). The restriction in our consideration is minimal and, even in the classical case (the initial point x 0 is assumed to satisfy the following two conditions: F ′ is Lipschitz around x 0 and the convex process T x 0 , defined by T x 0 ⋅ = F ′ ( x 0 ) ⋅ − C , is surjective), our results are new in giving sufficient conditions (which are weaker than the known ones and have a remarkable property being affine-invariant) ensuring the convergence of the iterative sequence with initial point x 0 . The same study is also made for the so-called convex-composite optimization problem (with objective function given as the composite of a convex function with a smooth map).

Let f be a lower semi-continuous and bounded below function from a Banach space X into (−∞, +∞] where X is assumed to admit a Lipschitz smooth “bump-function.” Generalizing results of Chaney, we study optimality conditions for[formula]to be a local minimum point of f . These conditions are described in terms of generalized Chaney's subdifferentials and second-order derivatives.

Based on a study of a minimization problem, we present the following results applicable to possibly nonconvex sets in a Banach space: an approximate projection result, an extended extremal principle, a nonconvex separation theorem, a generalized Bishop–Phelps theorem and a separable point result. The classical result of Dieudonne (on separation of two convex sets in a finite-dimensional space) is also extended to a nonconvex setting.