Pointed Convex Cone Research Articles

Theorem of Alternative and Scalarization of Optimization Problems in Topological Vector Spaces

In this paper, we introduce the definitions of affinelike, preaffinelike, generalized affinelike, and generalized preaffinelike functions by use of "pointed convex cone", and prove that those definitions of generalized affine functions are all different. We discuss optimization problems in topological vector spaces and obtain a theorem of alternative and a scalarization theorem. Our inequalities are given by partial order relations. Our generalized affineness may be used for many discussions in mathematics or applied mathematics wherever the affineness is a condition.

Controllability Properties of Solar Sails

Trajectory design and stationkeeping of solar sails about a celestial body can be formulated as control problems with positivity constraints. Specifically, when reemitted radiation is neglected and the sail is modeled as a flat surface, which are reasonable assumptions for control purposes, the force generated by the solar radiation pressure is contained in a pointed convex cone of revolution with the axis in the sun–satellite direction. Therefore, classical approaches to infer controllability based on the Lie algebra rank condition do not apply to these problems. This study offers a novel condition to decide on controllability of control systems with positivity constraints. This condition is effective because it can be verified by solving an auxiliary convex optimization problem for which reliable numerical methods are available. A crucial ingredient of this approach is the theory of positive trigonometric polynomials. The practical interest of this condition is the assessment of a minimum requirement on the optical properties of the sail, which may be of use for mission design purposes.

Set-Valued Evenly Convex Functions: Characterizations and C-Conjugacy

In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.

Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

In this note, we study in a finite dimensional Lie algebra {mathfrak g} the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone C_x. Assuming that {mathfrak g} is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying [h,x]=x for which C_x pointed. Given x, we show that such elements h can be constructed in such a way that mathop {mathrm{ad}}nolimits h defines a 5-grading, and characterize the cases where we even get a 3-grading.

A variable and a fixed ordering of intervals and their application in optimization with interval-valued functions

In this study, we introduce and analyze the concepts of a fixed ordering structure and a variable ordering structure on intervals. The fixed ordering structures on intervals are defined with the help of a pointed convex cone of intervals. A variable ordering is defined by a set-valued map whose values are convex cones of intervals. In the sequel, a few properties of a cone of intervals are derived. It is shown that a binary relation, defined by a convex cone of intervals, is a partial order relation on intervals; further, the relation is antisymmetric if the convex cone of intervals is pointed. Several results under which a variable ordering map of intervals satisfies the conditions of a partial ordering relation of intervals are provided. The introduced fixed and variable ordering of intervals are applied to define and characterize optimal elements of an optimization problem with interval-valued functions. Finally, we propose a numerical technique and present its algorithmic implementation to obtain the set of optimal elements of an interval optimization problem. We also provide illustrative examples to support the study.

Stiffness pathologies in discrete granular systems: Bifurcation, neutral equilibrium, and instability in the presence of kinematic constraints

SummaryThe paper develops the stiffness relationship between the movements and forces among a system of discrete interacting grains. The approach is similar to that used in structural analysis, but the stiffness matrix of granular material is inherently nonsymmetric because of the geometrics of particle interactions and of the frictional behavior of the contacts. Internal geometric constraints are imposed by the particles' shapes, in particular, by the surface curvatures of the particles at their points of contact. Moreover, the stiffness relationship is incrementally nonlinear, and even small assemblies require the analysis of multiple stiffness branches, with each branch region being a pointed convex cone in displacement space. These aspects of the particle‐level stiffness relationship give rise to three types of microscale failure: neutral equilibrium, bifurcation and path instability, and instability of equilibrium. These three pathologies are defined in the context of four types of displacement constraints, which can be readily analyzed with certain generalized inverses. That is, instability and nonuniqueness are investigated in the presence of kinematic constraints. Bifurcation paths can be either stable or unstable, as determined with the Hill–Bažant–Petryk criterion. Examples of simple granular systems of three, 16, and 64 disks are analyzed. With each system, multiple contacts were assumed to be at the friction limit. Even with these small systems, microscale failure is expressed in many different forms, with some systems having hundreds of microscale failure modes. The examples suggest that microscale failure is pervasive within granular materials, with particle arrangements being in a nearly continual state of instability.

On the finite‐time capture of a fast moving target

SummaryIn this work, we propose a feedback control law that enforces capture of a moving target by a slower pursuer in finite time. It is well known that if this problem is cast as a pursuit‐evasion differential game, then the moving target can always avoid capture by taking advantage of its speed superiority, provided that both the target and the pursuer are employing feedback strategies in the sense of Isaacs. Thus, in order to have a well‐posed pursuit problem, additional assumptions are required so that the pursuer can enforce capture of the faster target in finite time provided that it emanates from a set of ‘favorable’ initial positions, which constitute its winning set. In particular, we assume that the target's velocity either is constant and perfectly known to the pursuer (perfect information case) or can be decomposed into a dominant component, which is constant and known to the pursuer, and a second component that is uncertain and unknown to the pursuer (imperfect information case). It turns out that in both cases, the winning sets of the pursuer are pointed convex cones which have a common apex and a common axis of symmetry but different opening angles. We subsequently propose continuous feedback laws that enforce finite‐time capture while the pursuer never exits its winning set before capture takes place, for both cases. Copyright © 2016 John Wiley & Sons, Ltd.

Realizations of abstract regular polytopes from a representation theoretic view

Peter McMullen has developed a theory of realizations of abstract regular polytopes, and has shown that the realizations up to congruence form a pointed convex cone which is the direct product of certain irreducible subcones. We show that each of these subcones is isomorphic to a set of positive semi-definite hermitian matrices of dimension $m$ over either the real numbers, the complex numbers or the quaternions. In particular, we correct an erroneous computation of the dimension of these subcones by McMullen and Monson. We show that the automorphism group of an abstract regular polytope can have an irreducible character $\chi$ with $\chi\neq \overline{\chi}$ and with arbitrarily large essential Wythoff dimension. This gives counterexamples to a result of Herman and Monson, which was derived from the erroneous computation mentioned before. We also discuss a relation between cosine vectors of certain pure realizations and the spherical functions appearing in the theory of Gelfand pairs.

A direction based classical method to obtain complete Pareto set of multi-criteria optimization problems

This paper provides a direction based parameter dependent classical method to find entire Pareto set of multi-criteria optimization problems. Proposed method is non-gradient based and parameter of the technique essentially lies on the unit sphere in the criterion space. Sequentially solving a set of problems, corresponding to each particular parameter vector on the unit sphere, entire Pareto set of the problem is captured. It is obtained that the method bears a necessary and sufficient condition for Pareto optimality. It is also shown that a simple modification on the constraint set of the formulated problem can ensure general D-Pareto optimality of the outcome solutions, where D is any pointed convex cone. In specific, proposed method can extract 𝜖-Pareto optimal points. As at any 𝜖-Pareto point, trade-off between any two objectives are bounded, outcome solution set of the studied method can facilitate decision maker for final selection of solution. Proposed technique and its illustrations are supported by several numerical and pictorial representations.

Numerical Procedures in Multiobjective Optimization with Variable Ordering Structures

Multiobjective optimization problems with a variable ordering structure, instead of a partial ordering, have recently gained interest due to several applications. In the previous years, a basic theory has been developed for such problems. The binary relations of a variable ordering structure are defined by a cone-valued map that associates, with each element of the linear space ℝ m , a pointed convex cone of dominated or preferred directions. The difficulty in the study of the variable ordering structures arises from the fact that the binary relations are in general not transitive. In this paper, we propose numerical approaches for solving such optimization problems. For continuous problems a method is presented using scalarization functionals, which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn–Graef–Younes method, known from multiobjective optimization with a partial ordering, is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison.

Set-Valued Return Function and Generalized Solutions for Multiobjective Optimal Control Problems (MOC)

In this paper, we consider a multiobjective optimal control problem where the preference relation in the objective space is defined in terms of a pointed convex cone containing the origin, which defines generalized Pareto optimality. For this problem, we introduce the set-valued return function $V$ and provide a unique characterization for $V$ in terms of contingent derivative and coderivative for set-valued maps, which extends two previously introduced notions of generalized solution to the Hamilton--Jacobi equation for single objective optimal control problems.

Projections onto convex sets on the sphere

In this paper some concepts of convex analysis are extended in an intrinsic way from the Euclidean space to the sphere. In particular, relations between convex sets in the sphere and pointed convex cones are presented. Several characterizations of the usual projection onto a Euclidean convex set are extended to the sphere and an extension of Moreau’s theorem for projection onto a pointed convex cone is exhibited.

A duality between the metric projection onto a convex cone and the metric projection onto its dual

If K and L are mutually dual pointed convex cones in Rn with the metric projections onto them denoted by PK and PL respectively, then the following two assertions are equivalent: (i) PK is isotone with respect to the order induced by K (i.e. v−u∈K implies PKv−PKu∈K); (ii) PL is subadditive with respect to the order induced by L (i.e. PLu+PLv−PL(u+v)∈L for any u,v∈Rn).

A Discrete Dynamic Programming Approximation to the Multiobjective Deterministic Finite Horizon Optimal Control Problem

This paper addresses the problem of finding an approximation to the minimal element set of the objective space for the class of multiobjective deterministic finite horizon optimal control problems. The objective space is assumed to be partially ordered by a pointed convex cone containing the origin. The approximation procedure consists of a two-step discretization in time and state space. Following the first-order time discretization, the dynamic programming principle is used to find the multiobjective discrete dynamic programming equation equivalent to the resulting discrete multiobjective optimal control problem. The multiobjective discrete dynamic programming equation is finally discretized in the state space. The convergence of the approximation for both discretization steps is discussed.

Equilibria of Pairs of Nonlinear Maps Associated with Cones

Let K1, K2 be closed, full, pointed convex cones in finite-dimensional real vector spaces of the same dimension, and let F : K1 → span K2 be a homogeneous, continuous, K2-convex map that satisfies F(∂K1) ∩ int K2=∅ and FK1 ∩ int K2 ≠ ∅. Using an equivalent formulation of the Borsuk-Ulam theorem in algebraic topology, we show that we have \(F(K_1 \setminus\{0\}) \cap (-K_2)=\emptyset\) and \(K_2 \subseteq FK_1.\) We also prove that if, in addition, G : K1 → span K2 is any homogeneous, continuous map which is (K1, K2)-positive and K2-concave, then there exist a unique real scalar ω0 and a (up to scalar multiples) unique nonzero vector x0 ∈ K1 such that Gx0 = ω0Fx0, and moreover we have ω0 > 0 and x0 ∈ int K1 and we also have a characterization of the scalar ω0. Then, we reformulate the above result in the setting when K1 is replaced by a compact convex set and recapture a classical result of Ky Fan on the equilibrium value of a finite system of convex and concave functions.

Second order necessary conditions in set constrained differentiable vector optimization

We state second order necessary optimality conditions for a vector optimization problem with an arbitrary feasible set and an order in the final space given by a pointed convex cone with nonempty interior. We establish, in finite-dimensional spaces, second order optimality conditions in dual form by means of Lagrange multipliers rules when the feasible set is defined by a function constrained to a set with convex tangent cone. To pass from general conditions to Lagrange multipliers rules, a generalized Motzkin alternative theorem is provided. All the involved functions are assumed to be twice Frechet differentiable.

Minimax Inequalities for Vector-Valued Mappings onW-Spaces

The topological KKM theorem of Changet al.is slightly modified. By using our topological KKM theorem, we obtain a generalized section theorem and a generalized fixed point theorem onW-spaces which do not have any linear structure, and establish minimax inequalities for vector-valued mappings in Hausdorff topological vector spaces with closed pointed convex cones.

Stability of majorly efficient points and solutions in multiobjective programming

In this paper, we obtain some other properties of the majorly efficient points and solutions of the multiobjective optimization presented in two previous papers of Hu. By decomposing the major cone, which is non-pointed, non-convex and non-closed into a finite union of disjoint strictly supported pointed convex cones, we discuss the continuous perturbations of the decision space. Several sufficient conditions for the continuity of the sets of majorly efficient points and solutions are given.

Optimization of Set-Valued Functions

Let X, Y, and Z be real topological vector spaces and E⊆c X be a convex set. C⊆ Y, D⊆ Z are to be pointed convex cones. Let F: X → 2 Y be C-convex and G: X → 2 Z be D-convex set-valued functions. We consider the problems [formula] This paper generalizes the Moreau-Rockafellar type theorem and the Farkas-Minkowski type theorem for set-valued functions. When Y= R n and Z= R m , we established the necessary and sufficient conditions for the existence of Geoffrion efficient solution of (P) and the relationship between the proper efficient solutions and Geoffrion efficient solutions of (P). The Mond-Weir type and Wolfe type vector duality theorems are also considered in this paper.

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 (Cź{0})+clCźC, which implies the Sterna-Karwat condition for a convex coneC of a Hausdorff topological vector space.