Baire generalized topological spaces, generalized metric spaces and infinite games

Different generalizations of topological Baire spaces to the case of generalized topological spaces are considered and the properties of such spaces are examined. In particular, these considerations are focused on the relationship between Baire generalized topological spaces and semi-continuous real functions and infinite games. The notion of generalized metric spaces corresponding to generalized topological spaces is introduced as an important tool in this discussion.     

Well-posed constrained optimization problems in metric spaces

Relations between different notions of well-posedness of constrained optimization problems are studied. A characterization of the class of metric spaces in which Hadamard, strong, and Levitin-Polyak well-posedness of continuous minimization problems coincide is given. It is shown that the equivalence between the original Tikhonov well-posedness and the ones above provides a new characterization of the so-called Atsuji spaces. Generalized notions of well-posedness, not requiring uniqueness of the solution, are introduced and investigated in the above spirit.     

Inclusion problems in ordered topological vector spaces and applications

In this paper we study the existence of minimal and maximal solutions of inclusion problems in ordered topological vector spaces, and apply the obtained results to operator inclusions in ordered Banach spaces and to differential inclusions.     

Baire spaces and infinite games

It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.     

Conjugate dual problems in constrained set-valued optimization and applications

In this paper, two conjugate dual problems are proposed by considering the different perturbations to a set-valued vector optimization problem with explicit constraints. The weak duality, inclusion relations between the image sets of dual problems, strong duality and stability criteria are investigated. Some applications to so-called variational principles for a generalized vector equilibrium problem are shown.     

A general method for solving constrained optimization problems

In this paper a general problem of constrained minimization is studied. The minima are determined by searching for the asymptotical values of the solutions of a suitable system of ordinary differential equations.For this system, if the initial point is feasible, then any trajectory is always inside the set of constraints and tends towards a set of critical points. Each critical point that is not a relative minimum is unstable. For formulas of one-step numerical integration, an estimate of the step of integration is given, so that the above mentioned qualitative properties of the system of ordinary differential equations are kept.     

Scalarization of set-valued optimization problems and variational inequalities in topological vector spaces

This paper deals with set-valued vector optimization problems and set-valued vector variational inequalities in topological vector spaces, and provides some scalarization approaches for these problems by means of the polar cone and Gerstewitz’s scalarization functions.     

Incorrect problems in topological spaces

Conclusions 15. According to the methods of § 3 and § 4, we are considering the solution on a bicompact set M, taking as approximate solutions the elements of this set, i.e., we restrict X to a bicompact space M.By virtue of the corollary of Theorem 5, this re-establishes the stability of the problem. Thus by using bicompact sets and the related additional information about the solution, it is possible to go over from an incorrect problem to a problem which is correct in the sense of Tikhonov (see [19], p. 4).Translated from Sibirskii Matematicheskii Zhurnal, Vol. 10, No. 5, pp. 1065–1074, September–October, 1969.     

Constraint qualifications for constrained Lipschitz optimization problems and applications to a MPCC

Three constraint qualifications (the weak generalized Robinson constraint qualification, the bounded constraint qualification, and the generalized Abadie constraint qualification), which are weaker than the generalized Robinson constraint qualification (GRCQ) given by Yen (1997) [1], are introduced for constrained Lipschitz optimization problems. Relationships between those constraint qualifications and the calmness of the solution mapping are investigated. It is demonstrated that the weak generalized Robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping, whereas the proposed generalized Abadie constraint qualification, described in terms of graphical derivatives in variational analysis, is weaker than the calmness of the solution mapping. Finally, those constraint qualifications are written for a mathematical program with complementarity constraints (MPCC), and new constraint qualifications ensuring the C-stationary point condition of a MPCC are obtained.     

Duality gap of the conic convex constrained optimization problems in normed spaces

In this paper, motivated by a result due to Champion [Math. Program.99, 2004], we introduce a property for a conic quasi-convex vector-valued function in a general normed space. We prove that this property characterizes the zero duality gap for a class of the conic convex constrained optimization problem in the sense that if this property is satisfied and the objective function f is continuous at some feasible point, then the duality gap is zero, and if this property is not satisfied, then there exists a linear continuous function f such that the duality gap is positive. We also present some sufficient conditions for the property The work of this author was partially supported by the National Natural Sciences Grant (No. 10671050) and the Excellent Young Teachers Program of MOE,  P.R.C.     

Collective fixed points, generalized games and systems of generalized quasi-variational inclusion problems in topological spaces

In this paper, by using a fixed point theorem for expansive set-valued mappings with noncompact and nonconvex domains and ranges in topological spaces due to the author, we first prove a collective fixed point theorem and an existence theorem of equilibrium points for a generalized game. As applications, some new existence theorems of solutions for systems of generalized quasi-variational inclusion problems are established in noncompact topological spaces. Our results are different from known results in the literature.     

Qualification conditions for multivalued functions in Banach spaces with applications to nonsmooth vector optimization problems

In this paper we introduce qualification conditions for multivalued functions in Banach spaces involving the A-approximate subdifferential, and we show that these conditions guarantee metric regularity of multivalued functions. The results are then applied for deriving Lagrange multipliers of Fritz—John type and Kuhn—Tucker type for infinite non-smooth vector optimization problems.     

Certain classes of weakly infinite-dimensional spaces and topological games

In [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52] classes w-m-C of weakly infinite-dimensional spaces, 2?m?∞, were introduced. We prove that all of them coincide with the class wid of all weakly infinite-dimensional spaces in the Alexandroff sense. We show also that transfinite dimensions dimwm, introduced in [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52], coincide with dimension dimw2=dim, where dim is the transfinite dimension invented by Borst [P. Borst, Classification of weakly infinite-dimensional spaces. I. A transfinite extension of the covering dimension, Fund. Math. 130 (1) (1988) 1-25]. Some topological games which are related to countable-dimensional spaces, to C-spaces, and some other subclasses of weakly infinite-dimensional spaces are discussed.     

Invariants and dimension functions in general topological spaces

Summary We give here a general method of defining invariants in topological spaces. As applications we give a new concept of dimension and a theorem on separation of spaces by sub-spaces. Entrata in Redazione il 6 luglio 1968.     

Products of monotonically normal spaces with factors defined by topological games

Recently, it has been proved that orthocompactness implies normality for the products of a monotonically normal space and a compact space. It had been known that normality, collectionwise normality and the shrinking property are equivalent for the same products. We extend these two results for the products replacing the compact factor with a factor defined by topological games. Moreover, we prove the equivalence of orthocompactness and weak suborthocompactness in these products.     

Nonlinear separation in the image space with applications to constrained optimization

In this paper, by means of the image space analysis, we obtain optimality conditions for vector optimization of objective multifunction with multivalued constraints based on disjunction of two suitable subsets of the image space. By the oriented distance function a nonlinear regular separation is introduced and some optimality conditions for the constrained extremum problem are obtained. It is shown that the existence of a nonlinear separation is equivalent to a saddle point condition for the generalized Lagrangian function.