Convexity theories IV. Klein-Hilbert parts in convex modules

The Klein-Hilbert part relation, which was introduced by Gleason in function algebras and investigated for convex subsets of real vector spaces by Bear and Bauer in [3], [5], [2], is defined for convex modules. It turns out that all results that were proved for convex sets can also be proved for convex modules, which constitute the algebraic theory generated by convex sets and which have a close connection to physics and mathematical economics.     

Robust conjugate duality for convex optimization under uncertainty with application to data classification

In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition. This regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem. More importantly, we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem. As an application, we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets. The support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences.     

A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces

In this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping. This condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior-point regularity conditions given so far in the literature. Moreover, it provides a weak sufficient condition for Fenchel duality regarding convex optimization problems in infinite dimensional spaces. As an application, we discuss the strong conical hull intersection property (CHIP) for a finite family of closed convex sets.     

A branch-and-cut method for 0-1 mixed convex programming

We generalize the disjunctive approach of Balas, Ceria, and Cornuéjols [2] and devevlop a branch-and-cut method for solving 0-1 convex programming problems. We show that cuts can be generated by solving a single convex program. We show how to construct regions similar to those of Sherali and Adams [20] and Lovász and Schrijver [12] for the convex case. Finally, we give some preliminary computational results for our method. Received January 16, 1996 / Revised version received April 23, 1999?Published online June 28, 1999     

Bounded linear regularity of convex sets in Banach spaces and its applications

This paper deals with bounded linear regularity, linear regularity and the strong conical hull intersection property (CHIP) of a collection of finitely many closed convex intersecting sets in Banach spaces. It is shown that, as in finite dimensional space setting (see [6]), the standard constraint qualification implies bounded linear regularity, which in turn yields the strong conical hull intersection property, and that the collection of closed convex sets {C ₁, . . . ,C _n } is bounded linearly regular if and only if the tangent cones of {C ₁, . . . ,C _n } has the CHIP and the normal cones of {C ₁, . . . ,C _n } has the property (G)(uniformly on a neighborhood in the intersection C). As applications, we study the global error bounds for systems of linear and convex inequalities. The work of this author was partially supported by the National Natural Sciences Grant (No. 10471032) and the Excellent Young Teachers Program of MOE, P.R.C The authors thank professor K.F.Ng for his helpful discussion and the referee for their helpful suggestions on improving the first version of this paper     

On switching paths polyhedra

A class of integer polyhedra with totally dual integral (tdi) systems is proposed, which generalizes and unifies the “Switching Paths Polyhedra” of Hoffman (introduced in his generalization of Max Flow-Min Cut) and such polyhedra as the convex hull of (the incidence vectors of) all “path-closed sets” of an acyclic digraph, or the convex hull of all sets partitionable intok path-closed sets. As an application, new min-max theorems concerning the mentioned sets are given. A general lemma on when a tdi system of inequalities is box tdi is also given and used.     

On the symmetry function of a convex set

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}We attempt a broad exploration of properties and connections between the symmetry function of a convex set S and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point , let sym(x,S) denote the symmetry value of x in S: , which essentially measures how symmetric S is about the point x, and define x ^* is called a symmetry point of S if x ^* achieves the above maximum. The set S is a symmetric set if sym (S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym (S) and/or sym (x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym (x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the L?wner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x ^* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x ^* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory. Dedicated to Clovis Gonzaga on the occasion of his 60th birthday.     

Generalization of Primal—Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

We generalize primal—dual interior-point methods for linear programming (LP) problems to the convex optimization problems in conic form. Previously, the most comprehensive theory of symmetric primal—dual interior-point algorithms was given by Nesterov and Todd for feasible regions expressed as the intersection of a symmetric cone with an affine subspace. In our setting, we allow an arbitrary convex cone in place of the symmetric cone. Even though some of the impressive properties attained by Nesterov—Todd algorithms are impossible in this general setting of convex optimization problems, we show that essentially all primal—dual interior-point algorithms for LP can be extended easily to the general setting. We provide three frameworks for primal—dual algorithms, each framework corresponding to a different level of sophistication in the algorithms. As the level of sophistication increases, we demand better formulations of the feasible solution sets. Our algorithms, in return, attain provably better theoretical properties. We also make a very strong connection to quasi-Newton methods by expressing the square of the symmetric primal—dual linear transformation (the so-called scaling) as a quasi-Newton update in the case of the least sophisticated framework. August 25, 1999. Final version received: March 7, 2001.     

A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.Corresponding author. Partially supported by Air Force Office of Scientific Research Grant 91-0008 and National Science Foundation Grant DMS-9201297.     

Certain classes of pluricomplex Green functions on ${\mathbb C}^n$

We consider (pluricomplex) Green functions defined on , with logarithmic poles in a finite set and with logarithmic growth at infinity. For certain sets, we describe all the corresponding Green functions. The set of these functions is large and it carries a certain algebraic structure. We also show that for some sets no such Green functions exist. Our results indicate the fact that the set of poles should have certain algebro-geometric properties in order for these Green functions to exist. Received November 24, 1998; in final form April 19, 1999 / Published online July 3, 2000     

Stability of the intersection of solution sets of semi-infinite systems

Many mathematical programming models arising in practice present a block structure in their constraint systems. Consequently, the feasibility of these problems depends on whether the intersection of the solution sets of each of those blocks is empty or not. The existence theorems allow to decide when the intersection of non-empty sets in the Euclidean space, which are the solution sets of systems of (possibly infinite) inequalities, is empty or not. In those situations where the data (i.e., the constraints) can be affected by some kind of perturbations, the problem consists of determining whether the relative position of the sets is preserved by sufficiently small perturbations or not. This paper focuses on the stability of the non-empty (empty) intersection of the solutions of some given systems, which can be seen as the images of set-valued mappings. We give sufficient conditions for the stability, and necessary ones as well; in particular we consider (semi-infinite) convex systems and also linear systems. In this last case we discuss the distance to ill-posedness.     

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR ^v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.     

Quantum convex support

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C^∗-algebra A, which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A and it is a projection of a spectrahedron.Spectrahedra are increasingly investigated at least since the 1990s boom in semi-definite programming. We recall the geometry of the positive semi-definite cone and of the state space. We write a convex duality for general self-dual convex cones. This restricts to projections of state spaces and connects them to results on spectrahedra.Our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections, using natural isomorphisms. The result encodes the face lattice of the convex support into a set of projections in A and enables the integration of convex geometry with matrix calculus or algebraic techniques.     

Self-Scaled Barrier Functions on Symmetric Cones and Their Classification

Self-scaled barrier functions on self-scaled cones were axiomatically introduced by Nesterov and Todd in 1994 as a tool for the construction of primal—dual long-step interior point algorithms. This paper provides firm foundations for these objects by exhibiting their symmetry properties, their close ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and their algebraic classification theory. In the first part we recall the characterization of the family of self-scaled cones as the set of symmetric cones and develop a primal—dual symmetric viewpoint on self-scaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then proceed to showing that any self-scaled barrier function decomposes, in an essentially unique way, into a direct sum of self-scaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of self-scaled barrier functions using the correspondence between symmetric cones and Euclidean—Jordan algebras. December 5, 1999. Final version received: September 6, 2001.     

Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets

Motivated by the subsmoothness of a closed set introduced by Aussel et al. (2005) [8], we introduce and study the uniform subsmoothness of a collection of infinitely many closed subsets in a Banach space. Under the uniform subsmoothness assumption, we provide an interesting subdifferential formula on distance functions and consider uniform metric regularity for a kind of multifunctions frequently appearing in optimization and variational analysis. Different from the existing works, without the restriction of convexity, we consider several fundamental notions in optimization such as the linear regularity, CHIP, strong CHIP and property (G) for a collection of infinitely many closed sets. We establish relationships among these fundamental notions for an arbitrary collection of uniformly subsmooth closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of closed convex sets to the nonconvex setting.     

Lower subdifferentiability of quadratic functions

In this paper we characterize those quadratic functions whose restrictions to a convex set are boundedly lower subdifferentiable and, for the case of closed hyperbolic convex sets, those which are lower subdifferentiable but not boundedly lower subdifferentiable.Once characterized, we will study the applicability of the cutting plane algorithm of Plastria to problems where the objective function is quadratic and boundedly lower subdifferentiable.Financial support from the Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058, is gratefully acknowledged.     

Fixed point optimization algorithm and its application to network bandwidth allocation

A convex optimization problem for a strictly convex objective function over the fixed point set of a nonexpansive mapping includes a network bandwidth allocation problem, which is one of the central issues in modern communication networks. We devised an iterative algorithm, called a fixed point optimization algorithm, for solving the convex optimization problem and conducted a convergence analysis on the algorithm. The analysis guarantees that the algorithm, with slowly diminishing step-size sequences, weakly converges to a unique solution to the problem. Moreover, we apply the proposed algorithm to a network bandwidth allocation problem and show its effectiveness.     

On Scattered Convex Geometries

A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice Ω(η), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.     

Connectedness of efficient points in convex and convex transformable vector optimization

Given a convex vector optimization problem with respect to a closed ordering cone, we show the connectedness of the efficient and properly efficient sets. The Arrow–Barankin–Blackwell theorem is generalized to nonconvex vector optimization problems, and the connectedness results are extended to convex transformable vector optimization problems. In particular, we show the connectedness of the efficient set if the target function f is continuously transformable, and of the properly efficient set if f is differentiably transformable. Moreover, we show the connectedness of the efficient and properly efficient sets for quadratic quasiconvex multicriteria optimization problems.     

Generalized convex relations with applications to optimization and models of economic dynamics

We examine a notion of generalized convex set-valued mapping, extending the notions of a convex relation and a convex process. Under general conditions, we establish duality results for composite set-valued mappings and for convex programming problems involving convex set-valued mappings. We also present applications to the study of economic dynamical systems, by obtaining the characteristics of optimal paths generated by convex processes, and to optimization problems of a certain class of positively homogeneous increasing functions.