Search

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]

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]

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]

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]

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]

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]

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]

Copyright of Applied Mathematics & Mechanics (1000-0887) is the property of Applied Mathematics & Mechanics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

In this paper, some characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are studied in finite and infinite dimensional spaces, respectively. By using a new proof method which is different from the one used in Huang et al. (J Optim Theory Appl 162:548-558 ), a sufficient and necessary condition for the nonemptiness and boundedness of solution sets is established. Basing on this result, some new characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are proved. Compared with the known results in Huang et al. (), the key assumption that $$K_\infty \cap (F(K))^{w\circ }_C=\{0\}$$ is not required in finite dimensional spaces. Furthermore, the corresponding result of Huang et al. () is extended to the case of infinite dimensional spaces. Some examples are also given to illustrated the main results. [ABSTRACT FROM AUTHOR]

This article deals with the analysis of the contact complementarity problem for Lagrangian systems subjected to unilateral constraints, and with a singular mass matrix and redundant constraints. Previous results by the authors on existence and uniqueness of solutions of some classes of variational inequalities are used to characterize the well-posedness of the contact problem. Criteria involving conditions on the tangent cone and the constraints gradient are given. It is shown that the proposed criteria easily extend to the case where the system is also subjected to a set of bilateral holonomic constraints, in addition to the unilateral ones. In the second part, it is shown how basic convex analysis may be used to show the equivalence between the Lagrangian and the Hamiltonian formalisms when the mass matrix is singular. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the Lipschitzian/Hölderian-type global error bound for systems of finitely many convex polynomial inequalities over a polyhedral constraint. First, for systems of this type, we show that under a suitable asymptotic qualification condition the Lipschitzian-type global error bound property is equivalent to the Abadie qualification condition; in particular, the Lipschitzian-type global error bound is satisfied under the Slater condition. Second, without regularity conditions, the Hölderian global error bound with an explicit exponent is investigated. [ABSTRACT FROM AUTHOR]

A subset B of a closed convex set A is recession-compatible with respect to A if A can be expressed as the Minkowski sum of B and the recession cone of A. We show that if A contains no line, then there exists a recession-compatible subset of A that is minimal with respect to set inclusion. The proof only uses basic facts of convex analysis and does not depend on Zorn's Lemma. An application of this result to the error bound theory in optimization is presented. [ABSTRACT FROM AUTHOR]

We propose verifiable necessary and sufficient conditions for the solution existence of a convex quadratic program whose constraint set is defined by finitely many convex linear-quadratic inequalities. A related stability analysis of the problem is also considered. [ABSTRACT FROM AUTHOR]

In this paper, error bounds for γ-paraconvex multifunctions are considered. A Robinson-Ursescu type Theorem is given in normed spaces. Some results on the existence of global error bounds are presented. Perturbation error bounds are also studied. [ABSTRACT FROM AUTHOR]

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566-2574, ) and He (J. Math. Anal. Appl. 330:352-363, ). [ABSTRACT FROM AUTHOR]

We give an answer to the Problem 11.6 posed by Stephen Simons in his book “From Hahn-Banach to Monotonicity”: Do there exist a nonzero finite dimensional Banach space and a pair of extended real-valued, proper and convex functions which is totally Fenchel unstable? The answer is negative. [ABSTRACT FROM AUTHOR]

Under a weak compactness assumption on the functions involved, which always holds in finite-dimensional normed linear spaces, this paper extends various characterizations of the nonemptiness and boundedness of weakly efficient solution sets of convex vector optimization problems, obtained previously by the author (Deng in J. Optim. Theory Appl. 96:123–131, ) in the real finite-dimensional normed linear space setting, to those in the real reflexive Banach space setting. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate error bounds for convex polynomials. We prove that for a convex polynomial f in n variables which is not everywhere positive and which is not constant on any affine subspace, either f is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of f is compact. As an application, we deduce various types of error bounds for unconstrained and polyhedral-constrained convex polynomials. [ABSTRACT FROM AUTHOR]

We present a geometrical interpretation of the weighting method for constrained (finite dimensional) vector optimization. This approach is based on rigid movements which separate the image set from the negative of the ordering cone. We study conditions on the existence of such translations in terms of the boundedness of the scalar problems produced by the weighting method. Finally, using recession cones, we obtain the main result of our work: a sufficient condition under which weighting vectors yield solvable scalar problems. [ABSTRACT FROM AUTHOR]

The notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. It has been studied extensively by several authors. This paper is the second of a series on this subject where the basic results on weak sharp minima in Part I are applied to a number of important problems in convex programming. In Part II we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming. We obtain both new results and reproduce several existing results from a fresh perspective. [ABSTRACT FROM AUTHOR]

A variant of Robinson-Ursescu Theorem is given in normed spaces. Several error bound theorems for convex inclusions are proved and in particular a positive answer to Li and Singer's conjecture is given under weaker assumption than the assumption required in their conjecture. Perturbation error bounds are also studied. As applications, we study error bounds for convex inequality systems. [ABSTRACT FROM AUTHOR]

Conditions for the existence of various effective solutions of vector optimization problems with an unbounded convex closed feasible set of solutions are established. The study is based on the use of the properties of recession cones of sets of feasible solutions and cones of perspective directions of optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, various necessary and sufficient conditions are given for the nonemptiness and compactness of the weakly efficient solution set of a convex vector optimization problem. [ABSTRACT FROM AUTHOR]

Copyright of Trabajos de Investigacion Operativa is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Following a few words on multifunctions in the mathematical literature, a very brief recall on dual spaces, some preliminary notations and definitions in the introduction, we give some results on those functions in the second paragraph. In the third paragraph, a duality theory in cone-optimization involving multifunctions is developed with the concept of the strong instead of the weak cone-optimality criterium. The results so obtained account for existing ones on univocal vector-function optimization and they hold in spaces of arbitrary dimension. [ABSTRACT FROM AUTHOR]

In this paper we give a criterion for d-stability of convex programs and a perturbation result which subsume and generalize some recent results in semi-infinite programming and limiting Lagrangians. [ABSTRACT FROM AUTHOR]

This paper presents a study of recession cones of nonconvex sets in infinite dimensional spaces. The results are then applied to investigate efficiency conditions and the domination property in vector optimization. [ABSTRACT FROM AUTHOR]

We use the cover or the closure of the cover in order to determine when two disjoint closed convex sets have parallel faces, one of them being bounded. Moreover, we show that the study of the closure of the cover gives interesting results about the cover itself. [ABSTRACT FROM AUTHOR]

The following result is proved using solvability and optimality criteria for linear programs. The duals to the cones of feasible directions at vertices of a polyhedral set constitute a partition of the dual to the recession cone of the set. [ABSTRACT FROM AUTHOR]