Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Some solutions of karman''s equation describing the transonic flow past a corner point on a profile with a curvilinear generatrix

Transonic flows in the neighbourhood of a corner point on a profile are investigated in a class of self-similar solutions of Karman's equation. The corner point is formed by the intersection of two smooth curves, the tangents to which make a convex angle. The generatrix, which lies in the subsonic part of the flow is assumed to be curvilinear and to vary according to a power law. Values of the self-similarity index are found for which transonic flows are possible either with a free streamline or with a rarefaction wave.     

On generalized trade-off directions in nonconvex multiobjective optimization

Trade-off information related to Pareto optimal solutions is important in multiobjective optimization problems with conflicting objectives. Recently, the concept of trade-off directions has been introduced for convex problems. These trade-offs are characterized with the help of tangent cones. Generalized trade-off directions for nonconvex problems can be defined by replacing convex tangent cones with nonconvex contingent cones. Here we study how the convex concepts and results can be generalized into a nonconvex case. Giving up convexity naturally means that we need local instead of global analysis. Received: December 2000 / Accepted: October 2001?Published online February 14, 2002     

On the Volume Formulas of Cones and Orthogonal Multi-cones in S~n(1) and H~n(-1)

In the study of n-dimensional spherical or hyperbolic geometry, n ≥3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in Sn(l) and Hn(-1).     

Characterization of Solid Cones

The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a general expression of the interior of a solid cone, the author obtains a number of necessary and sufficient conditions for convex cones to be solid in the framework of Banach spaces. In particular, the author gives a dual relationship between solid cones and generalized sharp cones. The related known results are improved and extended.     

Extremality of convex sets with some applications

In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machinery to deriving new calculus results on intersection rules for normal cones to convex sets and on infimal convolutions of support functions.     

Projective and inductive limits in locally convex cones

We define and study the projective and inductive limit notions for locally convex cones. We use convex quasiuniform structure method for this purpose. Also we study the barreledness in the locally convex cones and introduce the notion upper-barreled cones and prove that the inductive limit of upper-barreled cones is upper-barreled.     

Restricted Normal Cones and the Method of Alternating Projections: Applications

The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving normal cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in a three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted normal cone, which is a generalization of the classical Mordukhovich normal cone. Numerous examples are provided to illustrate the theory.     

A convex sets in general position

The notion of convex cones in general position has turned out to be useful in convex programming theory. In this paper we extend the notion to convex sets and give some characterizations which yield a better insight into this concept. We also consider the case of convex sets in S-general position.     

Theory of convex cones in multicriteria decision making

We develop the theory of convex polyhedral cones in the objective-function space of a multicriteria decision problem. The convex cones are obtained from the decision-maker's pairwise judgments of decision alternatives and are applicable to any quasiconcave utility function. Therefore, the cones can be used in any progressively articulated solution procedure that employs pairwise comparisons. The cones represent convex sets of solutions that are inferior to known solutions to a multicriteria problem. Therefore, these convex sets can be eliminated from consideration while solving the problem. We develop the underlying theory and a framework for representing knowledge about the decision-maker's preference structure using convex cones. This framework can be adopted in the interactive solution of any multicriteria problem after taking into account the characteristics of the problem and the solution procedure. Our computational experience with different multicriteria problems shows that this approach is both viable and efficient in solving practical problems of moderate size.     

Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization

The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property. Jameson’s duality for two cones, which relates bounded linear regularity to property (G), is re-derived and refined. For polyhedral cones, a statement dual to Hoffman’s error bound result is obtained. A sharpening of a result on error bounds for convex inequalities by Auslender and Crouzeix is presented. Finally, for two subspaces, property (G) is quantified by the angle between the subspaces. Received October 1, 1997 / Revised version received July 21, 1998? Published online June 11, 1999     

On the wellposedness of constitutive laws involving dissipation potentials

We consider a material with memory whose constitutive law is formulated in terms of internal state variables using convex potentials for the free energy and the dissipation. Given the stress at a material point depending on time, existence of a strain and a set of inner variables satisfying the constitutive law is proved. We require strong coercivity assumptions on the potentials, but none of the potentials need be quadratic.

As a technical tool we generalize the notion of an Orlicz space to a cone ``normed' by a convex functional which is not necessarily balanced. Duality and reflexivity in such cones are investigated.

     

Convex Duality and Calculus: Reduction to Cones

An attempt is made to justify results from Convex Analysis by means of one method. Duality results, such as the Fenchel-Moreau theorem for convex functions, and formulas of convex calculus, such as the Moreau-Rockafellar formula for the subgradient of the sum of sublinear functions, are considered. All duality operators, all duality theorems, all standard binary operations, and all formulas of convex calculus are enumerated. The method consists of three automatic steps: first translation from the given setting to that of convex cones, then application of the standard operations and facts (the calculi) for convex cones, finally translation back to the original setting. The advantage is that the calculi are much simpler for convex cones than for other types of convex objects, such as convex sets, convex functions and sublinear functions. Exclusion of improper convex objects is not necessary, and recession directions are allowed as points of convex objects. The method can also be applied beyond the enumeration of the calculi.     

A cone separation theorem

The notion of separation is extended here to include separation by a cone. It is shown that two closed cones, one of them acute and convex, can be strictly separated by a convex cone, if they have no point in common. As a matter of fact, an infinite number of convex closed acute cones can be constructed so that each of them is a separating cone.The research was done while the author was a visiting professor at the University of British Columbia, Vancouver, British Columbia, Canada.     

Radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations

In this paper, we study the existence of radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations. By applying a well‐known fixed point theorem in cones, we shall establish several new criteria for the existence of nontrivial radially symmetric convex solutions for the systems of Monge‐Ampère equations with or without an eigenvalue parameter. Copyright © 2015 John Wiley & Sons, Ltd.     

A uniform boundedness theorem for locally convex cones

We prove a uniform boundedness theorem for families of linear operators on ordered cones. Using the concept of locally convex cones we introduce the notions of barreled cones and of weak cone-completeness. Our main result, though no straightforward generalization of the classical case, implies the Uniform Boundedness Theorem for Fréchet spaces.

     

Abstract ordered compact convex sets and algebras of the (sub)probabilistic powerdomain monad over ordered compact spaces

The majority of categories used in denotational semantics are topological in nature. One of these is the category of stably compact spaces and continuous maps. Previously, Eilenberg–Moore algebras were studied for the extended probabilistic powerdomain monad over the category of ordered compact spaces X and order-preserving continuous maps in the sense of Nachbin. Appropriate algebras were characterized as compact convex subsets of ordered locally convex topological vector spaces. In so doing, functional analytic tools were involved. The main accomplishments of this paper are as follows: the result mentioned is re-proved and is extended to the subprobabilistic case; topological methods are developed which defy an appeal to functional analysis; a more topological approach might be useful for the stably compact case; algebras of the (sub)probabilistic powerdomain monad inherit barycentric operations that satisfy the same equational laws as those in vector spaces. Also, it is shown that it is convenient first to embed these abstract convex sets in abstract cones, which are simpler to work with. Lastly, we state embedding theorems for abstract ordered locally compact cones and compact convex sets in ordered topological vector spaces.     

Characterization of properly optimal elements with variable ordering structures

In vector optimization with a variable ordering structure, the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications, it was started to develop a comprehensive theory for these vector optimization problems. Thereby, also notions of proper efficiency were generalized to variable ordering structures. In this paper, we study the relation between several types of proper optimality. We give scalarization results based on new functionals defined by elements from the dual cones which allow complete characterizations also in the nonconvex case.     

The area minimizing problem in conformal cones,I

In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Euclidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we establish the existence of bounded minimal graphs in mean convex conformal cones. Moreover those minimal graphs are the solutions to corresponding area minimizing problems. We can solve the area minimizing problem in non-mean convex translating conformal cones if these cones are contained in a larger mean convex conformal cones with the NCM assumption. We give examples to illustrate that this assumption can not be removed for our main results.     

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.