Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
ATZ-Fachkongress

Chassis.tech plus 2024


4 Kongresse in einer Veranstaltung

Treffen Sie in München die Fachleute von Chassis, Lenkung, Bremsen sowie Reifen/Räder.
Erweiterte Suche
Anmelden
nach oben

2006 | Buch

Combined Relaxation Methods for Generalized Monotone Variational Inequalities Kapitel lesen Erstes Kapitel lesen

Generalized Convexity and Related Topics

verfasst von: Professor Igor V. Konnov, Professor Dinh The Luc, Professor Alexander M. Rubinov

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Economics and Mathematical Systems

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Inhaltsverzeichnis

Frontmatter

Invited Papers

Frontmatter
Combined Relaxation Methods for Generalized Monotone Variational Inequalities
Summary
The paper is devoted to the combined relaxation approach to constructing solution methods for variational inequalities. We describe the basic idea of this approach and implementable methods both for single-valued and for multi-valued problems. All the combined relaxation methods are convergent under very mild assumptions. This is the case if there exists a solution to the dual formulation of the variational inequality problem. In general, these methods attain a linear rate of convergence. Several classes of applications are also described.
Igor V. Konnov
Abstract Convexity and the Monge-Kantorovich Duality
Summary
In the present survey, we reveal links between abstract convex analysis and two variants of the Monge-Kantorovich problem (MKP), with given marginals and with a given marginal difference. It includes: (1) the equivalence of the validity of duality theorems for MKP and appropriate abstract convexity of the corresponding cost functions; (2) a characterization of a (maximal) abstract cyclic monotone map F: X → L ⊂ IRX in terms connected with the constraint set
$$ Q_0 (\varphi ): = \{ u \in \mathbb{R}^z :u(z_1 ) - u(z_2 ) \leqslant \varphi (z_1 ,z_2 ){\text{ }}\forall z_1 ,z_1 \in Z = dom{\text{ }}F\} $$
of a particular dual MKP with a given marginal difference and in terms of L-subdifferentials of L-convex functions; (3) optimality criteria for MKP (and Monge problems) in terms of abstract cyclic monotonicity and non-emptiness of the constraint set Q 0(ϕ), where ϕ is a special cost function on X × X determined by the original cost function c on X × Y. The Monge-Kantorovich duality is applied then to several problems of mathematical economics relating to utility theory, demand analysis, generalized dynamics optimization models, and economics of corruption, as well as to a best approximation problem.
Vladimir L. Levin
Optimality Conditions and Duality for Multiobjective Programming Involving (C, α, ρ, d) type-I Functions
Summary
In this chapter, we present a unified formulation of generalized convex functions. Based on these concepts, sufficient optimality conditions for a nondifferentiable multiobjective programming problem are presented. We also introduce a general Mond-Weir type dual problem of the problem and establish weak duality theorem under generalized convexity assumptions. Strong duality result is derived using a constraint qualification for nondifferentiable multiobjective programming problems.
Dehui Yuan, Altannar Chinchuluun, Xiaoling Liu, Panos M. Pardalos

Contributed Papers

Frontmatter
Partitionable Variational Inequalities with Multi-valued Mappings
Summary
We consider multi-valued variational inequalities defined on a Cartesian product of finite-dimensional subspaces. We introduce extensions of order monotonicity concepts for set-valued mappings, which are adjusted to the case where the subspaces need not be real lines. These concepts enable us to establish new existence and uniqueness results for the corresponding partitionable multi-valued variational inequalities. Following a parametric coercivity approach, we obtain convergence of the Tikhonov regularization method without monotonicity conditions.
Elisabetta Allevi, Adriana Gnudi, Igor V. Konnov
Almost Convex Functions: Conjugacy and Duality
Summary
We prove that the formulae of the conjugates of the precomposition with a linear operator, of the sum of finitely many functions and of the sum between a function and the precomposition of another one with a linear operator hold even when the convexity assumptions are replaced by almost convexity or nearly convexity. We also show that the duality statements due to Fenchel hold when the functions involved are taken only almost convex, respectively nearly convex.
Radu Ioan Boţ, Sorin-Mihai Grad, Gert Wanka
Pseudomonotonicity of a Linear Map on the Interior of the Positive Orthant
Summary
In this paper we will establish some necessary and/or sufficient conditions for both a nonsingular and a singular matrix A (interpreted as a linear map) to be pseudomonotone. The given results are in terms of the sign of the determinants of the principal submatrices and of the cofactors of A in the nonsingular case and in terms of the structure of A in the singular case. A complete characterization of pseudomonotonicity in terms of the coefficients of a 3 × 3 matrix is given and a method for constructing a merely pseudomonotone matrix is suggested.
Alberto Cambini, Laura Martein
An Approach to Discrete Convexity and Its Use in an Optimal Fleet Mix Problem
Summary
A notion of convexity for discrete functions is first introduced, with the aim to guarantee both the increasing monotonicity of marginal increments and the convexity of the sum of convex functions. Global optimality of local minima is then studied both for single variable functions and for multi variables ones. Finally, a concrete optimal fleet mix problem is studied, pointing out its discrete convexity properties.
Riccardo Cambini, Rossana Riccardi1, Ümit Yüceer
A Unifying Approach to Solve a Class of Parametrically-Convexifiable Problems
Summary
The aim of this paper is to show how a wide class of generalized quadratic programs can be solved, in a unifying framework, by means of the so-called optimal level solutions method. In other words, the problems are solved by analyzing, explicitly or implicitly, the optimal solutions of particular quadratic strictly convex parametric subproblems. In particular, it is pointed out that some of these problems share the same set of optimal level solutions. A solution algorithm is proposed and fully described. The results achieved are then deepened in the particular case of box constrained problems.
Riccardo Cambini, Claudio Sodini
Mathematical Programming with (Φ, ρ)-invexity
Summary
We introduce new invexity-type properties for differentiable functions, generalizing (F, ρ)-convexity. Optimality conditions for nonlinear programming problems are established under such assumptions, extending previously known results. Wolfe and Mond-Weir duals are also considered, and we obtain direct and converse duality theorems.
Giuseppe Caristi, Massimiliano Ferrara, Anton Stefanescu
Some Classes of Pseudoconvex Fractional Functions via the Charnes-Cooper Transformation
Summary
Using a very recent approach based on the Charnes-Cooper trasformation we characterize the pseudoconvexity of the sum between a quadratic fractional function and a linear one. Furthemore we prove that the ratio between a quadratic fractional function and the cube of an affine one is pseudoconvex if and only if the product between a quadratic fractional function and an affine one is pseudoconvex and we provide a sort of canonical form for this latter class of functions. Benefiting by the new results we are able to characterize the pseudoconvexity of the ratio between a quadratic fractional function and the cube of an affine one.
Laura Carosi, Laura Martein
Equilibrium Problems Via the Palais-Smale Condition
Summary
Inspired by some results from nonsmooth critical point theory, we propose in this paper to study equilibrium problems by means of a general Palais-Smale condition adapted to bifunctions. We introduce the notion of critical points for equilibrium problems and we give some existence results for (EP) with lack of compacity.
Ouayl Chadli, Zaki Chbani, Hassan Riahi
Points of Efficiency in Vector Optimization with Increasing-along-rays Property and Minty Variational Inequalities
Summary
Minty variational inequalities are studied as a tool for vector optimization. Instead of focusing on vector inequalities, we propose an approach through scalarization which allows to construct a proper variational inequality type problem to study any concept of efficiency in vector optimization.
This general scheme gives an easy and consistent extension of scalar results, providing also a notion of increasing along rays vector function. This class of generalized convex functions seems to be intimately related to the existence of solutions to a Minty variational inequality in the scalar case, we now extend this fact to vector case.
Finally, to prove a reversal of the main theorem, generalized quasiconvexity is considered and the notion of *-quasiconvexity plays a crucial role to extend scalar evidences. This class of functions, indeed, guarantees a Minty-type variational inequality is a necessary and sufficient optimality condition for several kind of efficient solution.
Giovanni P. Crespi, Ivan Ginchev, Matteo Rocca
Higher Order Properly Efficient Points in Vector Optimization
Summary
We consider the constrained vector optimization problem minC f(x), g(x) ∈ − K, where f: ℝn → ℝm and g : ℝn → ℝp are given functions and C ∈ ℝm and K ∈ ℝp are closed convex cones. Two type of solutions are important for our considerations, namely i-minimizers (isolated minimizers) of order k and pminimizers (properly efficient points) of order k (see e.g. [11]). Every i-minimizer of order k ≥ 1 is a p-minimizer of order k. For k = 1, conditions under which the reversal of this statement holds have been given in [11]. In this paper we investigate the possible reversal of the implication i-minimizer ⇒ p-minimizer in the case k = 2. To carry on this study, we develop second-order optimality conditions for p-minimizers, expressed by means of Dini derivatives. Together with the optimality conditions obtained in [11] and [12] in the case of i-minimizers, they play a crucial role in the investigation. Further, to get a satisfactory answer to the posed reversal problem, we deal with sense I and sense II solution concepts, as defined in [11] and [5].
Ivan Ginchev, Angelo Guerraggio, Matteo Rocca
Higher-order Pseudoconvex Functions
Summary
In terms of n-th order Dini directional derivative with n positive integer we define n-pseudoconvex functions being a generalization of the usual pseudoconvex functions. Again with the n-th order Dini derivative we define n-stationary points, and prove that a point x 0 is a global minimizer of a n-pseudoconvex function f if and only if x 0 is a n-stationary point of f. Our main result is the following. A radially continuous function f defined on a radially open convex set in a real linear space is n-pseudoconvex if and only if f is quasiconvex function and any n-stationary point is a global minimizer. This statement generalizes the results of Crouzeix, Ferland, Math. Program. 23 (1982), 193–205, and Komlósi, Math. Program. 26 (1983), 232–237. We study also other aspects of the n-pseudoconvex functions, for instance their relations to variational inequalities.
Ivan Ginchev, Vsevolod I. Ivanov
Sufficient Optimality Conditions and Duality in Nonsmooth Multiobjective Optimization Problems under Generalized Convexity
Summary
We consider a multiobjective optimization problem in ℝn with a feasible set defined by inequality and equality constraints and a set constraint. All the involved functions are, at least, directionally differentiable. We provide sufficient optimality conditions for global and local Pareto minimum under several kinds of generalized convexity. Also Wolfe-type and Mond-Weir-type dual problems are considered, and weak and strong duality theorems are proved.
Giorgio Giorgi, Bienvenido Jiménez, Vicente Novo
Optimality Conditions for Tanaka’s Approximate Solutions in Vector Optimization
Summary
In this work, approximate solutions of vector optimization problems in the sense of Tanaka [18] are characterized via scalarization. Necessary and sufficient conditions are obtained using a new order representing property and a new monotonicity concept, respectively. A family of gauge functions defined by generalized Chebyshev norms and verifying both properties is introduced in order to characterize approximate solutions of vector optimization problems via approximate solutions of several scalarizations.
César Gutiérrez, Bienvenido Jiménez, Vicente Novo
On the Work of W. Oettli in Generalized Convexity and Nonconvex Optimization — a Review and Some Perspectives
Joachim Gwinner
Local and Global Consumer Preferences
Summary
Several kinds of continuous (generalized) monotone maps are characterized by partial gradient maps of skew-symmetric real-valued bifunctions displaying corresponding (generalized) concavity-convexity properties. As an economic application, it is shown that two basic approaches explaining consumer choice are behaviorally equivalent.
Reinhard John
Optimality Conditions for Convex Vector Functions by Mollified Derivatives
Summary
Necessary and sufficient optimality conditions for nonsmooth multiobjective optimization problems and some characterizations of convex vector functions are proved by means of mollified derivatives.
Davide La Torre
On Arcwise Connected Convex Multifunctions
Summary
In this paper arcwise connected convex multifunctions are introduced and studied. Optimality conditions involving this type of data are analyzed.
Davide La Torre
A Sequential Method for a Class of Bicriteria Problems
Summary
The aim of the paper is to suggest a sequential method for generating the set E of all efficient points of a bicriteria problem P B where the feasible region is a polytope and whose criteria are a linear function and a concave function which is the sum of a linear and the reciprocal of an affine function. The connectedness of E and some theoretical properties of P B allow to give a finite simplex-like algorithm based on a suitable post-optimality analysis carried on a scalar parametric problem where the linear criteria plays the role of a parametric constraint.
Laura Martein, Valerio Bertolucci
Decomposition of the Measure in the Integral Representation of Piecewise Convex Curves
Summary
The notions of a convex arc and piecewise convex curve in the plane generalize the notion of a convex curve, the latter is usually defined as the boundary of a planar compact convex set with nonempty interior. The integral representation of a piecewise convex curve through a Riemann-Stieltjes integral with a corresponding one-dimensional measure is studied. It is shown that the Minkowski operations known from the convex sets can be generalized to piecewise convex curves. It is shown that the decomposition of the measure in the integral representation of the piecewise convex curve leads to a decomposition of the piecewise convex curve into a sum of corresponding piecewise convex curves. On this base, applying the natural decomposition of the one-dimensional measure into an absolutely continuous function, a jump function, and a singular function, the structure of a piecewise convex curve is investigated. As some curious consequences, the existence of polygons with infinitely many sides and no vertices, and polygons with infinitely many vertices and no sides is shown.
Mariana Nedelcheva
Rambling Through Local Versions of Generalized Convex Functions and Generalized Monotone Operators
Summary
Two classes of functions encompassing the cone of convex functions and the space of strictly differentiable functions are presented and compared. Related properties for sets and multimappings are dealt with.
Huynh Van Ngai, Jean-Paul Penot
Monotonicity and Dualities
Summary
There is a recent surge of interest for the representation of monotone operators by convex functions. It can explained by the success of convex analysis in obtaining the fundamental results about maximal monotone operators. Convex analysis can also be combined with variational analysis to get new convergence results. Here we take another direction and connect such a stream with the concept of duality in a general framework, heavily using order methods.
Jean-Paul Penot
On Variational-like Inequalities with Generalized Monotone Mappings
Summary
We consider two new classes of generalized relaxed α-monotone and semimonotone functions and using the KKM technique we prove the existence of solutions for variational-like inequalities relative to these types of mappings in Banach spaces. Several examples and special cases are also considered.
Vasile Preda, Miruna Beldiman, Anton Bătătorescu
Almost Pure Nash Equilibria in Convex Noncooperative Games
Summary
This paper considers n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. For such games it is shown that there are two-point Nash equilibria in them, that is equilibria in players’ strategies with support consisting of at most two points. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.
Tadeusz Radzik, Wojciech Połowczuk
A Spectral Approach to Solve Box-constrained Multi-objective Optimization Problems
Summary
This paper presents some first and second order conditions necessary for the Pareto optimality of box-constrained multi-objective optimization problems. These necessary conditions are related to the spectrum of a matrix defined via the gradient vectors and the Hessian matrices of the objective functions. These necessary conditions are used to develop two algorithms. The first one is built taking into account the first order necessary conditions and determines some critical points for the multi-objective problems considered. The second one is based on the second order necessary conditions and discards the critical points that do not belong to the local Pareto optimal front. Some numerical results are shown.
Maria Cristina Recchioni
Backmatter
Metadaten
Titel
Generalized Convexity and Related Topics
verfasst von
Professor Igor V. Konnov
Professor Dinh The Luc
Professor Alexander M. Rubinov
Copyright-Jahr
2006
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-37007-9
Print ISBN
978-3-540-37006-2
DOI
https://doi.org/10.1007/978-3-540-37007-9

Premium Partner

    Bildnachweise