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It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples. [ABSTRACT FROM AUTHOR]

The intersection of an affine subspace with the cone of positive semidefinite matrices is called a spectrahedron. An orthogonal projection thereof is called a spectrahedral shadow or projected spec-trahedron. Spectrahedra and their projections can be seen as a generalization of polyhedra. This article is concerned with the problem of approximating the recession cones of spectrahedra and spectrahedral shadows via polyhedral cones. We present two iterative algorithms to compute outer and inner approximations to within an arbitrary prescribed accuracy. The first algorithm is tailored to spectrahedra and is derived from polyhedral approximation algorithms for compact convex sets and relies on the fact, that an algebraic description of the recession cone is available. The second algorithm is designed for projected spectrahedra and does not require an algebraic description of the recession cone, which is in general more difficult to obtain. We prove correctness and finiteness of both algorithms and provide numerical examples. [ABSTRACT FROM AUTHOR]

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces. [ABSTRACT FROM AUTHOR]

We introduce two notions of vector asymptotic functions for vector functions with their image space ordered by the nonnegative orthant. To do this, a suitable extended-valued framework is used. We establish properties and formulas for these notions. The asymptotic notions are used to study multiobjective optimization problems. We obtain (outer/inner) estimates for the asymptotic and recession cones of level, colevel, and weakly efficient solution sets. With this information, we establish coercivity properties, coercive and noncoercive existence results for solutions of multiobjective optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a new approach to finding the minimal representation of dc-functions in the sense of Zalgaller. We introduce the notion of maximal pair of convex sets and show that support functions of this pair are the minimal representation of a given dc-function. For this purpose, we study the properties of Minkowski addition and subtraction in the family of closed convex sets sharing their recession cone. We show how to find maximal pair equivalent to a given one. [ABSTRACT FROM AUTHOR]

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces.

Let C be a closed convex subset of a real Hilbert space containing the origin, and assume that K is the homogenization cone of C, i.e., the smallest closed convex cone containing C × { 1 } . Homogenization cones play an important role in optimization for the construction of examples and counterexamples. A famous examples is the second-order/Lorentz/“ice cream” cone which is the homogenization cone of the unit ball. In this paper, we discuss the polar cone of K as well as an algorithm for finding the projection onto K provided that the projection onto C is available. Various examples illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the recession cone and recession function in the nonconvex case. By virtue of the recession function, we investigate the monotonicity of a function. Then, we obtain a characterization of the Lipschitz continuity for a function by using the recession function. As applications, we study the monotonicity and Lipschitz continuity properties of the optimal value function for a parametric constrained optimization problem. Moreover, we present a characterization of the nonemptiness and boundedness of the solution set for a constrained optimization problem without convexity assumption. [ABSTRACT FROM AUTHOR]

This paper presents two new theorems on Geoffrion's properly efficient solutions and seven examples illustrating their applications to linear fractional vector optimization problems with unbounded constraint sets. Provided that all the components of the objective function are properly fractional, the first theorem gives sufficient conditions for the efficient solution set to coincide with the Geoffrion properly efficient solution set. Admitting that the objective function can have some affine components, in the second theorem we give sufficient conditions for an efficient solution to be a Geoffrion's properly efficient solution. The recession cone of the constraint set, the derivatives of the scalar objective functions, but no tangent cone to the constraint set at the efficient point, are used in the second theorem. [ABSTRACT FROM AUTHOR]

A nonempty closed convex set in R n , not containing the origin, is called a pseudo-cone if with every x it also contains λx for λ ≥ 1. We consider pseudo-cones with a given recession cone C , called C -pseudo-cones. The family of C -pseudo-cones can, with reasonable justification, be considered as a counterpart to the family of convex bodies containing the origin in the interior. For a C -pseudo-cone one can naturally define a surface area measure and a covolume. Since they are in general infinite, we introduce a weighting, leading to modified versions of surface area and covolume. These are finite and still homogeneous, though of other degrees. Our main result is a Minkowski type existence theorem for C -pseudo-cones with given weighted surface area measure. [ABSTRACT FROM AUTHOR]

In this paper we introduce extended generalized complementarity problems in Hausdorff topological vector spaces in duality and study the existence of their solutions. We use a different method than those in literature on the existence of solutions of complementarity problems, which are usually based on arguments from generalized monotonicity. This leads us to obtain new results and improve many existing results in literature. We also prove some existence results for extended generalized complementarity problems in reflexive Banach spaces by means of a Tikhonov regularization procedure under a copositivity assumption and arguments from the recession analysis. [ABSTRACT FROM AUTHOR]

Let $X$ be a Banach space, $Csubset X$ be a closed convex set included in a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set $C$, so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqemptyset$. The latter is a primary condition for subdifferentiability of the support function $sigma_C$. Eventually, we study Gateaux differentiability of support function $sigma_C$ on two sets, the polar cone of $K$ and ${mathop{rm int}}(mathrm{dom} sigma_C)$.

This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines. [ABSTRACT FROM AUTHOR]

Let X be a Banach space, C be a nonempty closed convex set and σ C be the support function of the set C. In this work, we give some necessary and sufficient conditions for the set C to satisfy int (dom σ C) ≠ ∅ , which enable us to study the Frechet and Gateaux differentiability of support function on the set C. [ABSTRACT FROM AUTHOR]

This paper generalizes the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, the authors define a slack operator associated to the set and its polar according to whether the convex set is full dimensional, whether it is a translated cone and whether it contains lines. The authors strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. The authors show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator. [ABSTRACT FROM AUTHOR]

A characterization of bounded sets in Banach spaces in terms of asymptotic cones and the Hausdorff deviations of sets from them homothetic images is obtained. Similar results for generalizations of the notion of boundedness are presented. Boundedness criteria have previously been known only for recessively compact sets. [ABSTRACT FROM AUTHOR]

The paper provides a new subdifferential characterization for Motzkin decomposable (convex) functions. This characterization leads to diverse stability properties for such a decomposability for operations like addition and composition. [ABSTRACT FROM AUTHOR]

The unboundedness of functions was investigated with the recession cones and recession functions. Firstly, the mean value theorem and recession cones were used to characterize the subdifferentials of convex functions on condition of nondifferentiability. Based on the a-bove results, the necessaiy and sufficient conditions for recession functions under the subdiffer-entiable assumption were given. Secondly, the convexity was generalized to E-convex functions, and the unbounded feature of E- convex functions was studied by means of recession functions under the special sub-linear assumption. Finally, an example was given to indicate that these results can not be extended to quasiconvex functions. [ABSTRACT FROM AUTHOR]

Nearly convex sets play important roles in convex analysis, optimization and theory of monotone operators. We give a systematic study of nearly convex sets, and construct examples of subdifferentials of lower semicontinuous convex functions whose domains or ranges are nonconvex. [ABSTRACT FROM AUTHOR]

In this paper, the robust controllability for linear systems with constrained controls x = Ax+Bu, u ∊ Ω is studied under the assumption that Ω is an arbitrary subset satisfying the only condition 0 ∊ clcoΩ and the system matrices are subjected to structured perturbations. The notion of the structured local and global controllability radii are defined. Based on properties of convex processes, some general formulas for the complex and the real controllability radii, for both local and global controllability concepts, are established with respect to structured affine perturbations and multiperturbations of matrices A,B. As particular cases, the general results are applied to linear systems with bounded controls and single-input linear systems, yielding some explicit and computable formulas of controllability radii. Examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

It is known that, in finite dimensions, the support function of a compact convex set with nonempty interior is differentiable excepting the origin if and only if the set is strictly convex. In this paper, we realize a thorough study of the relations between the differentiability of the support function on the interior of its domain and the convexity of the set, mainly for unbounded sets. Then, we revisit some results related to the differentiability of the cost function associated to a production function. [ABSTRACT FROM PUBLISHER]