Description:
Motivation: The increasing use of DNA microarray-based tumor gene expression profiles for cancer diagnosis requires mathematical methods with high accuracy for solving clustering, feature selection and classification problems of gene expression data. Results: New algorithms are developed for solving clustering, feature selection and classification problems of gene expression data. The clustering algorithm is based on optimization techniques and allows the calculation of clusters step-by-step. This approach allows us to find as many clusters as a data set contains with respect to some tolerance. Feature selection is crucial for a gene expression database. Our feature selection algorithm is based on calculating overlaps of different genes. The database used, contains over 16 000 genes and this number is considerably reduced by feature selection. We propose a classification algorithm where each tissue sample is considered as the center of a cluster which is a ball. The results of numerical experiments confirm that the classification algorithm in combination with the feature selection algorithm perform slightly better than the published results for multi-class classifiers based on support vector machines for this data set.
Description:
Motivation: The increasing use of DNA microarray-based tumor gene expression profiles for cancer diagnosis requires mathematical methods with high accuracy for solving clustering, feature selection and classification problems of gene expression data. Results: New algorithms are developed for solving clustering, feature selection and classification problems of gene expression data. The clustering algorithm is based on optimization techniques and allows the calculation of clusters step-by-step. This approach allows us to find as many clusters as a data set contains with respect to some tolerance. Feature selection is crucial for a gene expression database. Our feature selection algorithm is based on calculating overlaps of different genes. The database used, contains over 16 000 genes and this number is considerably reduced by feature selection. We propose a classification algorithm where each tissue sample is considered as the center of a cluster which is a ball. The results of numerical experiments confirm that the classification algorithm in combination with the feature selection algorithm perform slightly better than the published results for multi-class classifiers based on support vector machines for this data set.
Description:
We consider the problem of discriminating two finite point sets in the n-dimensional space by a finite number of hyperplanes generating a piecewise linear function. If the intersection of these sets is empty, then they can be strictly separated by a max-min of linear functions. An error function is introduced. This function is nonconvex piecewise linear. We discuss an algorithm for its minimization. The results of numerical experiments using some real-world datasets are presented, which show the effectiveness of the proposed approach.
Description:
We consider the problem of discriminating two finite point sets in the n-dimensional space by a finite number of hyperplanes generating a piecewise linear function. If the intersection of these sets is empty, then they can be strictly separated by a max-min of linear functions. An error function is introduced. This function is nonconvex piecewise linear. We discuss an algorithm for its minimization. The results of numerical experiments using some real-world datasets are presented, which show the effectiveness of the proposed approach.
Description:
We use a primal-dual scheme to devise a new update rule for a penalty function method applicable to general optimization problems, including nonsmooth and nonconvex ones. The update rule we introduce uses dual information in a simple way. Numerical test problems show that our update rule has certain advantages over the classical one. We study the relationship between exact penalty parameters and dual solutions. Under the differentiability of the dual function at the least exact penalty parameter, we establish convergence of the minimizers of the sequential penalty functions to a solution of the original problem. Numerical experiments are then used to illustrate some of the theoretical results.
Description:
We use a primal-dual scheme to devise a new update rule for a penalty function method applicable to general optimization problems, including nonsmooth and nonconvex ones. The update rule we introduce uses dual information in a simple way. Numerical test problems show that our update rule has certain advantages over the classical one. We study the relationship between exact penalty parameters and dual solutions. Under the differentiability of the dual function at the least exact penalty parameter, we establish convergence of the minimizers of the sequential penalty functions to a solution of the original problem. Numerical experiments are then used to illustrate some of the theoretical results.
Description:
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.
Description:
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.
Description:
The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented.
Description:
The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented.