Duality in Reverse Convex Optimization

In this paper, we are concerned with the problem of boundedness in the constrained global maximization of a convex function. In particular, we present necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and we establish sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints. We also address the problem of boundedness in the global maximization problem when a feasible region is convex and unbounded.

In this paper, we study complexity results of sparse optimization problems and reverse convex optimization problems. These problems are very important subjects of optimization problems. We prove that the complexity result of the sparsity constraint problem and sparse solution problem are all NP-hard in the strong sense and even testing feasibility of the sparsity constraint is NP-complete in the strong sense. Then the sparse optimization problem is NP-hard in the strong sense. We also prove that the reverse convex problem is NP-hard in the strong sense by transforming the sparsity constraint into a reverse convex constraint.

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example.

One of the standard goals of in nite-dimensional topology is the topological classi cation of linear metric spaces and their convex subsets. The origins of this problem go back to the questions of Fr echet [Fr] and Banach [Ba] whether all in nite-dimensional separable Fr echet (Banach) spaces are homeomorphic (we explain our terminology at the end of the introduction). These questions have been a rmatively answered by the combined results of Anderson and Kadec (see [BP]) which may be stated as follows: every separable Fr echet space is homeomorphic to a Hilbert space. This result was generalized by Toru nczyk [To] for the nonseparable case. Since then the main interest in the eld has turned to topological classi cation of incomplete or nonlocally convex linear metric spaces. Quite recently, Cauty [Ca2] has constructed an example of a separable linear complete metric space X which is not an AR. In particular, X is not homeomorphic to any convex subset of a locally convex linear metric space. The topological classi cation of incomplete locally convex metric linear spaces, or even incomplete pre-Hilbert spaces, is much more complex than for the complete ones. First, there exist pre-Hilbert spaces of an arbitrary Borel class. Moreover, the Borel type invariant does not classify such spaces, and in the class of -compact spaces there exist uncountably many pairwise nonhomeomorphic such spaces [BP], [Ca1]. Until recently, there was hope that, in spite of these di culties, the classi cation problem could be restricted to subspaces of Hilbert spaces. The result of Bessaga and Dobrowolski [BD] stating that every locally convex linear -compact metric space is homeomorphic to a pre-Hilbert space looked like a promising rst step. The main objective of this paper is to show that, in general, it is not possible to restrict ones attention to subspaces of Hilbert spaces only. We show that the classes of pre-Hilbert,

In mathematical programming, duality theorems play a central role. Especially, in convex and quasiconvex programming, Lagrange duality and surrogate duality have been studied extensively. Additionally, constraint qualifications are essential ingredients of the powerful duality theory. The best-known constraint qualifications are the interior point conditions, also known as the Slater-type constraint qualifications. A typical example of mathematical programming is a minimization problem of a real-valued function on a vector space. This types of problems have been studied widely and have been generalized in several directions. Recently, the authors investigate set functions and Fenchel duality. However, duality theorems and its constraint qualifications for mathematical programming with set functions have not been studied yet. It is expected to study set functions and duality theorems. In this paper, we study duality theorems for convex and quasiconvex set functions. We show Lagrange duality theorem for convex set functions and surrogate duality theorem for quasiconvex set functions under the Slater condition. As an application, we investigate an uncertain problem with motion uncertainty.

In this paper, we establish global necessary and sufficient optimality conditions for D.C. vector optimization problems under reverse convex constraints. An application to vector fractional mathematical programming is also given.

The various properties of topological spaces in the classical Baire category theorems which imply the Baire property also imply the stronger property of pseudo-completeness. In contrast to some of these properties and to the Baire property, J. C. Oxtoby has shown that pseudo-completeness is productive. The following main result places pseudo-completeness in the context of linear topological spaces: Let E and F be linear topological spaces and g be a continuous linear mapping of E into F. If E is pseudo-complete, g is almost open, and the completion of g[E] has a continuous metric, then g[E] is complete. The proof of this result uses the difference theorem, but not an open mapping theorem. The hypotheses lead to a discussion of conditions for a linear mapping to be almost open and for a linear topological space to have a continuous metric. An example shows that, although a translation invariant continuous metric on a linear topological space E extends to a translation invariant continuous pseudo-metric on the completion of E, this extension need not be a metric, even if it induces a normed topology on E. Othes examples show that a pseudo-complet e linear topological space need not be complete in its natural uniformity, and that the almost open condition of the main result may not be omitted and is not implied by the combination of the other conditions and the conclusion. Pseudo-complete topological spaces, introduced by Oxtoby [8], successfully bind together the classical Baire category theorems. They have been the subject of recent work; Aarts and Lutzer [1], for example, have shown that a metrizable space is pseudo-comple te if and only if it has a dense subspace which is topologically complete. In the context of linear topological spaces, Todd [13] shows that, under certain conditions, a linear topological space which is pseudocomplete is complete in the natural uniformity. Theorem 3.1 extends this result for the image of a pseudo-complete linear topological space under a continuous linear almost open mapping; in contrast. Example 5.5 gives pseudo-complete locally convex linear topological spaces which are not complete, and hence are not Ptak spaces. From open mapping theorems arise restrictions on continuous images of certain types of spaces. As an application of his open mapping theorem, Banach ([2], p. 38), showed that the continuous linear image of a complete metric linear space in a space of the same

A deterministic global optimization algorithm for generalized geometric programming

It is shown that, in a non-necessarily Hausdorff real topological vector space, if a subset is a countable disjoint union of convex sets closed in the subset, then those convex sets must be its convex components. On the other hand, by means of convex components we extend the notion of extreme point to non-convex sets, which entails a new equivalent reformulation of the Krein--Milman property (involving drops among other objects). Finally, we study the nature of convex functions and provide some results on their support in order to introduce the concept of multi-slice, that is, slices determined by convex functions (instead of by linear functions). Among other things, we prove that the boundary of a closed convex set with non-empty interior can be obtained as the set of support points of a certain lower semi-continuous convex function on that convex set.

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this manuscript we solve the “faceless problem”, that is, the informally posed open problem of characterizing those convex sets free of proper faces. Indeed, we prove that a non-singleton convex subset of a real vector space is free of proper faces if and only if every point of the convex set is an inner point. As a consequence, we prove the following dichotomy theorem: a point of a convex set of a real vector space is either a extreme point or an inner point of some face. We also characterize the non-singleton closed convex subsets of a topological vector space free of proper faces, which turn out to be the linear manifolds. An application of this allows us to show that the only possible minimal faces of a linearly bounded closed convex subset of a topological vector space are the extreme points. As an application of the technique we develop to prove our results, we easily construct dense proper faces of convex sets and non-dense proper faces whose closure is not a face. This fact allows us to equivalently renorm any infinite dimensional Banach space so that its unit ball contains a non-closed face. Even more, we prove that every infinite dimensional Banach space containing an isomorphic copy of $$c_0$$ or $$\ell _p$$ , $$1<p<\infty $$ , can be equivalently renormed so that its unit ball contains a face whose closure is not a face.

The problem of maximizing a concave functional / over a convex set S, subset of a linear space M, is studied. It is related to the problem of .minimizing f(x) over M subject to sup [ILM0001] & where F is a directional derivative of . Consequences are discussed pro¬ves viding characterizations of the optimizing point p. Applications to statistics are demon¬strated including an (additional) derivation of the theorem of WHITTLE

Interest in the study of dichotomy for various classes of probability measures arose after the well-known Kakutani theorem [1] concerning the equivalence and singularity of product measures had appeared. The development and various generalizations of this result can be found in Feldman [2], Hajek [3], Fernique [4], Kanter [5], and Okazaki [6]. In the last paper, Okazaki proved a dichotomy theorem for H-ergodic probability measures on a locally convex Hausdorff space equipped with a cylindrical c-algebra C(E, {x~ }), where {x*},~z C E*; this theorem generalizes all earlier dichotomy results, since in every case one can find an appropriate set H with respect to which the measures are ergodic. The proof of the Okasaki theorem uses topological properties of the original space and the separability of the c-algebra C(E, {x~ }) on which the measure p is defined. In this paper we prove a dichotomy theorem for ergodic measures on linear measure spaces in the most general situation under rather weak requirements. We do not assume that the linear space is equipped with a topological structure, and the measurable space structure need not agree with the linear operations. Let (E, ~) be an arbitrary measure space such that E is a linear set and the a-algebra ~ is invariant under translations by elements z E E. Let p be a probability measure on ~, and let H C E.

It is known that the design of optimal transmit beamforming vectors for cognitive radio multicast transmission can be formulated as indefinite quadratic optimization programs. Given the challenges of such nonconvex problems, the conventional approach in literature is to recast them as convex semidefinite programs (SDPs) together with rank-one constraints. Then, these nonconvex and discontinuous constraints are dropped allowing for the realization of a pool of relaxed candidate solutions, from which various randomization techniques are utilized with the hope to recover the optimal solutions. However, it has been shown that such approach fails to deliver satisfactory outcomes in many practical settings, wherein the determined solutions are found to be unacceptably far from the actual optimality. On the contrary, we in this contribution tackle the aforementioned optimal beamforming problems differently by representing them as SDPs with additional reverse convex (but continuous) constraints. Nonsmooth optimization algorithms are then proposed to locate the optimal solutions of such design problems in an efficient manner. Our thorough numerical examples verify that the proposed algorithms offer almost global optimality whilst requiring relatively low computational load.

In this paper, strongly (α ,T ) -convex functions, i.e., functions f : D → R satisfying the functional inequality f (tx+(1− t)y) t f (x)+(1− t) f (y)− tα(1− t)(x− y)− (1− t)αt(y− x) for x,y ∈ D and t ∈ T ∩ [0,1] are investigated. Here D is a convex set in a linear space, α is a nonnegative function on D−D , and T ⊆ R is a nonempty set. The main results provide various characterizations of strong (α ,T ) -convexity in the case when T is a subfield of R . Mathematics subject classification (2010): Primary 39B62, 26B25.

The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane tok given points in the plane. This problem was generalized by Witzgall ton-dimensional space and to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multi-facility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual.