Description:
The research of the first author was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the second author was supported by the Grant Agency of the Czech Republic, Project 17-04301S and the Australian Research Council, Project 10.13039/501100000923DP160100854.
Description:
The research of the first author was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the second author was supported by the Grant Agency of the Czech Republic, Project 17-04301S and the Australian Research Council, Project 10.13039/501100000923DP160100854.
Description:
We present an algorithm to locally minimize nonsmooth, nonconvex functions. In order to find descent directions, the notion of quasisecants, introduced in this paper, is applied. We prove that the algorithm converges to Clarke stationary points. Numerical results are presented demonstrating the applicability of the proposed algorithm to a wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.
Description:
We introduce and analyse outer approximation schemes for solving variational inequality problems in which the constraint set is as in generalized semi-infinite programming. We call these problems generalized semi-infinite variational inequality problems. First, we establish convergence results of our method under standard boundedness assumptions. Second, we use suitable Tikhonov-like regularizations for establishing convergence in the unbounded case.
Description:
The classical Remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval. In this paper the classical Remez algorithm is generalised to the problem of polynomial spline (piece-wise polynomial) approximation with the spline defect equal to the spline degree. Also, the values of the splines in the end points of the approximation interval may be fixed Polynomial splines combine simplicity of polynomials and flexibility, which allows one to significantly decrease the degree of the corresponding polynomials and oscillations of deviation functions. Therefore polynomial splines are a powerful tool for function and data approximation. The generalisation of the Remez algorithm developed in this research has been tested on several approximation problems. The results of the numerical experiments are presented.
Description:
Clustering is an important task in data mining. It can be formulated as a global optimization problem which is challenging for existing global optimization techniques even in medium size data sets. Various heuristics were developed to solve the clustering problem. The global k-means and modified global k-means are among most efficient heuristics for solving the minimum sum-of-squares clustering problem. However, these algorithms are not always accurate in finding global or near global solutions to the clustering problem. In this paper, we introduce a new algorithm to improve the accuracy of the modified global k-means algorithm in finding global solutions. We use an auxiliary cluster problem to generate a set of initial points and apply the k-means algorithm starting from these points to find the global solution to the clustering problems. Numerical results on 16 real-world data sets clearly demonstrate the superiority of the proposed algorithm over the global and modified global k-means algorithms in finding global solutions to clustering problems.
Description:
Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.
Description:
Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.