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Optimality conditions for weak efficiency to vector optimization problems with composed convex functions

100%
Boţ R., Hodrea I., Wanka G.
Open Mathematics
|
2008
|
tom 6
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nr 3
453-468
EN
We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.
2
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Latent Semantic Indexing for patent documents

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Moldovan A., Boţ R. I., Wanka G.
International Journal of Applied Mathematics and Computer Science
|
2005
|
tom 15
|
nr 4
551-560
EN
Since the huge database of patent documents is continuously increasing, the issue of classifying, updating and retrieving patent documents turned into an acute necessity. Therefore, we investigate the efficiency of applying Latent Semantic Indexing, an automatic indexing method of information retrieval, to some classes of patent documents from the United States Patent Classification System. We present some experiments that provide the optimal number of dimensions for the Latent Semantic Space and we compare the performance of Latent Semantic Indexing (LSI) to the Vector Space Model (VSM) technique applied to real life text documents, namely, patent documents. However, we do not strongly recommend the LSI as an improved alternative method to the VSM, since the results are not significantly better.
3
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Comparison between different duals in multiobjective fractional programming

100%
Boţ R., Chares R., Wanka G.
Open Mathematics
|
2007
|
tom 5
|
nr 3
452-469
EN
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.
4
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Employing different loss functions for the classification of images via supervised learning

100%
Boţ R., Heinrich A., Wanka G.
Open Mathematics
|
2014
|
tom 12
|
nr 2
381-394
EN
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.
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