Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C ², in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.      

Approximation and decomposition properties of some classes of locally D.C. functions

We study the connexion between local and global decompositions of some important subclasses of locally d.c. functions (functions which locally split as a difference of two convex functions). Then we tackle the problem of regularizing such functions by the Moreau-Yosida process and prove in particular that the class of lower-C ² functions fits well this approximation procedure.     

Second-Order Epi-Derivatives of Composite Functionals

We compute two-sided second-order epi-derivatives for certain composite functionals f=gF where F is a C ¹ mapping between two Banach spaces X and Y, and g is a convex extended real-valued function on Y. These functionals include most essential objectives associated with smooth constrained minimization problems on Banach spaces. Our proof relies on our development of a formula for the second-order upper epi-derivative that mirrors a formula for a second-order lower epi-derivative from [7], and the two-sided results we obtain promise to support a more precise sensitivity analysis of parameterized optimization problems than has been previously possible.     

Spectral invariance of smooth crossed products, and rapid decay locally compact groups

We show that all rapid-decay locally compact groups are unimodular and that the set of rapid-decay functions on a locally compact rapidly decaying group forms a dense and spectral invariant Fréchet *-subalgebra of the reduced group C ^*-algebra. In general, the set of rapid-decay functions on a locally compact strongly rapid-decay group with values in a commutative C ^*-algebra forms a dense and spectral invariant Fréchet *-subalgebra of the twisted crossed product C ^*-algebra. The spectral invariance property implies that the K-theories of both algebras are naturally isomorphic under inclusion.This project is supported in part by the National Science Foundation Grant #DMS 92-04005.     

Generalized second-order directional derivatives and optimization with C1,1 functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C¹,¹ functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gãteaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C^1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.     

Tight convex underestimators for {{\mathcal C}^2}-continuous problems: I. univariate functions

A novel method for the convex underestimation of univariate functions is presented in this paper. The method is based on a piecewise application of the well-known αBB underestimator, which produces an overall underestimator that is piecewise convex. Subsequently, two algorithms are used to identify the linear segments needed for the construction of its -continuous convex envelope, which is itself a valid convex underestimator of the original function. The resulting convex underestimators are very tight, and their tightness benefits from finer partitioning of the initial domain. It is theoretically proven that there is always some finite level of partitioning for which the method yields the convex envelope of the function of interest. The method was applied on a set of univariate test functions previously presented in the literature, and the results indicate that the method produces convex underestimators of high quality in terms of both lower bound and tightness over the whole domain under consideration.     

Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.     

The second order optimality conditions for nonlinear mathematical programming with C 1,1 data

To find nonlinear minimization problems are considered and standard C ²-regularity assumptions on the criterion function and constrained functions are reduced to C ^1,1-regularity. With the aid of the generalized second order directional derivative for C ^1,1 real-valued functions, a new second order necessary optimality condition and a new second order sufficient optimality condition for these problems are derived.     

Representable K-Theory for σ-C*-algebras

We define a version of K-theory on the category of -C ^*-algebras (countable inverse limits of C ^*-algebras). Our theory is homotopy invariant, has long exact sequences and a Milnor sequence, and satisfies Bott periodicity. On C ^*-algebras it gives the ordinary K-theory, and on the space of continuous functions on a countable direct limit X of compact Hausdorff spaces, it gives the representable K-theory of X. (We do not claim that our theory is in general a representable functor.) We also define an equivariant version, and discuss several related groups.Partially supported by a National Science Foundation Postdoctoral Fellowship.     

Extremal K -Theory and Index for C*-Algebras

We develop the general theory for a new functor K _e on the category of C ^*-algebras. The extremal K-set, K _e (A), of a C ^*-algebra A is defined by means of homotopy classes of extreme partial isometries. It contains K ₁ (A) and admits a partially defined addition extending the addition in K ₁ (A), so that we have an action of K ₁ (A) on K _e (A). We show how this functor relates to K ₀ and K ₁, and how it can be used as a carrier of information relating the various K-groups of ideals and quotients of A. The extremal K-set is then used to extend the classical theory of index for Fredholm and semi-Fredholm operators.     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

Extrema Constrained by a Family of Curves and Local Extrema

This paper considers the connections between the local extrema of a function f:DR and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C ¹ or C ²-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C ¹ or C ²-curves. Surprisingly, what is true for C ¹-curves fails to be true in part for C ²-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Construction of ‘Wachspress type’ rational basis functions over rectangles

In the present paper, we have constructed rational basis functions ofC ⁰ class over rectangular elements with wider choice of denominator function. This construction yields additional number of interior nodes. Hence, extra nodal points and the flexibility of denominator function suggest better approximation.     

Two types of minimax theorems for vector-valued functions

It seems that minimax theorems for vector-valued functions found in recent papers have something in common. Taking note of this, we improve several results in the author's recent works and state two types of minimax theorems for vector-valued functions. One theorem refers to functions with some special convexity properties; the other theorem refers to separated functions of the typef(x, y)=u(x)+v(y). The proofs are based on the existence of weak cone saddle points off and on a condition about a pointed convex cone which induces a partial ordering in the image space off. We need the condition (C{0})+clC–C, which implies the Sterna-Karwat condition for a convex coneC of a Hausdorff topological vector space.The author thanks the referees for their valuable suggestions on the original draft. Also, he is grateful to S. Yoshiara for his useful suggestions on the English presentation.     

Approximate convexity and submonotonicity

It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C¹ functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C¹ functions in infinite dimensions. The following characterization is established: a multivalued operator is maximal cyclically submonotone if, and only if, it coincides with the Clarke subdifferential of a locally Lipschitz directionally approximately convex function, which is unique up to a constant. Furthermore, it is shown that in Asplund spaces, every regular function is generically approximately convex.     

Regular interval Cantor sets of <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> and minimality

It is known that not every Cantor set of S ¹ is C ¹-minimal. In this work we prove that every member of a subfamily of what we here call regular interval Cantor set is not C ¹-minimal. We also prove that no member of a class of Cantor sets that includes this subfamily is C ^1+∈-minimal, for any ∈ > 0. Partially supported by CNPq-Brasil and PEDECIBA-Uruguay.     

Classification of extensions of classifiable -algebras

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-algebras.     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.