Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.     

Sequential optimality conditions for composed convex optimization problems

Using a general approach which provides sequential optimality conditions for a general convex optimization problem, we derive necessary and sufficient optimality conditions for composed convex optimization problems. Further, we give sequential characterizations for a subgradient of the precomposition of a K-increasing lower semicontinuous convex function with a K-convex and K-epi-closed (continuous) function, where K is a nonempty convex cone. We prove that several results from the literature dealing with sequential characterizations of subgradients are obtained as particular cases of our results. We also improve the above mentioned statements.     

Employing different loss functions for the classification of images via supervised learning

Supervised learning methods are powerful techniques to learn a function from a given set of labeled data, the so-called training data. In this paper the support vector machines approach is applied to an image classification task. Starting with the corresponding Tikhonov regularization problem, reformulated as a convex optimization problem, we introduce a conjugate dual problem to it and prove that, whenever strong duality holds, the function to be learned can be expressed via the dual optimal solutions. Corresponding dual problems are then derived for different loss functions. The theoretical results are applied by numerically solving a classification task using high dimensional real-world data in order to obtain optimal classifiers. The results demonstrate the excellent performance of support vector classification for this particular problem.     

Duality for location problems with unbounded unit balls

Given an optimization problem with a composite of a convex and componentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sufficient optimality conditions are derived using strong duality. Furthermore, as special case of this problem, we consider a location problem, where the “distances” are measured by gauges of closed convex sets. We prove that the geometric characterization of the set of optimal solutions for this location problem given by Hinojosa and Puerto in a recently published paper can be obtained via the presented dual problem. Finally, the Weber and the minmax location problems with gauges are given as applications.     

Comparison between different duals in multiobjective fractional programming

The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow us to give dual characterizations for the efficient solutions of the initial fractional problem. The aim of this paper is to compare the intermediate dual problem with other similar dual problems known from the literature. We completely establish the inclusion relations between the image sets of the duals as well as between the sets of maximal elements of the image sets.      

Evaluation of the Potential Release Tendency of Metals and Metalloids from the Estuarine Sediments: Case Study of Raša Bay

Assessing the environmental quality of coastal systems is important not only for the management and protection of such areas, but also for improving the quality of water resources. Since sediment itself can often be a source of certain toxic elements, in addition to information on the distribution of metals in the water column and in the sediment itself, it is useful to determine the bioavailable forms of individual elements, particularly toxic ones. In this study, water and sediment geochemical data were supplemented with oxyanion mobility in sediments estimated by diffusion gradients in thin film (DGTs). The data obtained indicate that the chemical composition of the water in the Raša River estuary primarily reflects the high input of suspended sediment from the catchment, the mixing of freshwater and seawater, and to a lesser extent the effects of anthropogenic activities. Although sediment composition is primarily determined by geological and hydrodynamic conditions in the catchment, it also indicates moderate enrichment in Co, Cr, Mo and Ni. In contrast, the distribution of oxyanions in sediment pore water indicates the influence of sediment as a source of some elements in the bottom water; e.g., sediment contributes to 40% of the arsenic bottom water budget. The obtained depth profiles of the oxyanion distribution in the sediment pore water indicate an early onset of suboxic to anoxic conditions in Raša Bay, which is prone to rapid sedimentation. All this demonstrates the need to consider the bioavailable forms of elements when assessing environmental quality, as the lack of such information can lead to an incomplete assessment, especially in dynamic coastal systems such as estuaries.     

Multiobjective Control Approximation Problems: Duality and Optimality

A general convex multiobjective control approximation problem is considered with respect to duality. The single objectives contain linear functionals and powers of norms as parts, measuring the distance between linear mappings of the control variable and the state variables. Moreover, linear inequality constraints are included. A dual problem is established, and weak and strong duality properties as well as necessary and sufficient optimality conditions are derived. Point-objective location problems and linear vector optimization problems turn out to be special cases of the problem investigated. Therefore, well-known duality results for linear vector optimization are obtained as special cases.     

An alternative formulation for a new closed cone constraint qualification

We give an alternative formulation for the so-called closed cone constraint qualification (CCCQ) related to a convex optimization problem in Banach spaces recently introduced in the literature. This new formulation allows to prove in a simple way that (CCCQ) is weaker than some generalized interior-point constraint qualifications given in the past. By means of some insights from the theory of conjugate duality we also show that strong duality still holds under some weaker hypotheses than the ones considered so far in the literature.     

Characterization of the Fermi surface of the organic superconductor - by measurements of Shubnikov-de Haas and angle-dependent magnetoresistance oscillations and by electronic band-structure calculations

The electronic structure of the quasi two-dimensional (2D) organic superconductor -(ET)₂SF₅CH₂CF₂SO₃ was examined by measuring Shubnikov-de Haas (SdH) and angle-dependent magnetoresistance (AMRO) oscillations and by comparing with electronic band-structure calculations. The SdH oscillation frequencies follow the angular dependence expected for a 2D Fermi surface (FS), and the observed fundamental frequency shows that the 2D FS is 5% of the first Brillouin zone in size. The AMRO data indicate that the shape of the 2D FS is significantly non-circular. The calculated electronic structure has a 2D FS in general agreement with experiment. From the temperature and angular dependence of the SdH amplitude, the cyclotron and band effective masses were estimated to be and ,where g is the conduction electron g factor and the free electron mass. The band effective mass is estimated to be from the calculated electronic band structure. Received: 3 March 1997 / Revised: 5 May 1997 / Received in final form: 5 November 1997 / Accepted: 10 November 1997