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Nonsmooth nonconvex optimization approach to clusterwise linear regression problems

- Bagirov, Adil, Ugon, Julien, Mirzayeva, Hijran

**Authors:**Bagirov, Adil , Ugon, Julien , Mirzayeva, Hijran**Date:**2013**Type:**Text , Journal article**Relation:**European Journal of Operational Research Vol. 229, no. 1 (2013), p. 132-142**Full Text:**false**Reviewed:****Description:**Clusterwise regression consists of finding a number of regression functions each approximating a subset of the data. In this paper, a new approach for solving the clusterwise linear regression problems is proposed based on a nonsmooth nonconvex formulation. We present an algorithm for minimizing this nonsmooth nonconvex function. This algorithm incrementally divides the whole data set into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate a good starting point for solving global optimization problems at each iteration of the incremental algorithm. Such an approach allows one to find global or near global solution to the problem when the data sets are sufficiently dense. The algorithm is compared with the multistart Späth algorithm on several publicly available data sets for regression analysis. © 2013 Elsevier B.V. All rights reserved.**Description:**2003011018

Hyperbolic smoothing in nonsmooth optimization and applications

**Authors:**Al Nuaimat, Alia**Date:**2014**Type:**Text , Thesis , PhD**Full Text:****Description:**Nonsmooth nonconvex optimization problems arise in many applications including economics, business and data mining. In these applications objective functions are not necessarily differentiable or convex. Many algorithms have been proposed over the past three decades to solve such problems. In spite of the significant growth in this field, the development of efficient algorithms for solving this kind of problem is still a challenging task. The subgradient method is one of the simplest methods developed for solving these problems. Its convergence was proved only for convex objective functions. This method does not involve any subproblems, neither for finding search directions nor for computation of step lengths, which are fixed ahead of time. Bundle methods and their various modifications are among the most efficient methods for solving nonsmooth optimization problems. These methods involve a quadratic programming subproblem to find search directions. The size of the subproblem may increase significantly with the number of variables, which makes the bundle-type methods unsuitable for large scale nonsmooth optimization problems. The implementation of bundle-type methods, which require the use of the quadratic programming solvers, is not as easy as the implementation of the subgradient methods. Therefore it is beneficial to develop algorithms for nonsmooth nonconvex optimization which are easy to implement and more efficient than the subgradient methods. In this thesis, we develop two new algorithms for solving nonsmooth nonconvex optimization problems based on the use of the hyperbolic smoothing technique and apply them to solve the pumping cost minimization problem in water distribution. Both algorithms use smoothing techniques. The first algorithm is designed for solving finite minimax problems. In order to apply the hyperbolic smoothing we reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems. We also study the main properties of the hyperbolic smoothing function. Based on these results an algorithm for solving the finite minimax problem is proposed and this algorithm is implemented in GAMS. We present preliminary results of numerical experiments with well-known nonsmooth optimization test problems. We also compare the proposed algorithm with the algorithm that uses the exponential smoothing function as well as with the algorithm based on nonlinear programming reformulation of the finite minimax problem. The second nonsmooth optimization algorithm we developed was used to demonstrate how smooth optimization methods can be applied to solve general nonsmooth (nonconvex) optimization problems. In order to do so we compute subgradients from some neighborhood of the current point and define a system of linear inequalities using these subgradients. Search directions are computed by solving this system. This system is solved by reducing it to the minimization of the convex piecewise linear function over the unit ball. Then the hyperbolic smoothing function is applied to approximate this minimization problem by a sequence of smooth problems which are solved by smooth optimization methods. Such an approach allows one to apply powerful smooth optimization algorithms for solving nonsmooth optimization problems and extend smoothing techniques for solving general nonsmooth nonconvex optimization problems. The convergence of the algorithm based on this approach is studied. The proposed algorithm was implemented in Fortran 95. Preliminary results of numerical experiments are reported and the proposed algorithm is compared with an other five nonsmooth optimization algorithms. We also implement the algorithm in GAMS and compare it with GAMS solvers using results of numerical experiments.**Description:**Doctor of Philosophy

**Authors:**Al Nuaimat, Alia**Date:**2014**Type:**Text , Thesis , PhD**Full Text:****Description:**Nonsmooth nonconvex optimization problems arise in many applications including economics, business and data mining. In these applications objective functions are not necessarily differentiable or convex. Many algorithms have been proposed over the past three decades to solve such problems. In spite of the significant growth in this field, the development of efficient algorithms for solving this kind of problem is still a challenging task. The subgradient method is one of the simplest methods developed for solving these problems. Its convergence was proved only for convex objective functions. This method does not involve any subproblems, neither for finding search directions nor for computation of step lengths, which are fixed ahead of time. Bundle methods and their various modifications are among the most efficient methods for solving nonsmooth optimization problems. These methods involve a quadratic programming subproblem to find search directions. The size of the subproblem may increase significantly with the number of variables, which makes the bundle-type methods unsuitable for large scale nonsmooth optimization problems. The implementation of bundle-type methods, which require the use of the quadratic programming solvers, is not as easy as the implementation of the subgradient methods. Therefore it is beneficial to develop algorithms for nonsmooth nonconvex optimization which are easy to implement and more efficient than the subgradient methods. In this thesis, we develop two new algorithms for solving nonsmooth nonconvex optimization problems based on the use of the hyperbolic smoothing technique and apply them to solve the pumping cost minimization problem in water distribution. Both algorithms use smoothing techniques. The first algorithm is designed for solving finite minimax problems. In order to apply the hyperbolic smoothing we reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems. We also study the main properties of the hyperbolic smoothing function. Based on these results an algorithm for solving the finite minimax problem is proposed and this algorithm is implemented in GAMS. We present preliminary results of numerical experiments with well-known nonsmooth optimization test problems. We also compare the proposed algorithm with the algorithm that uses the exponential smoothing function as well as with the algorithm based on nonlinear programming reformulation of the finite minimax problem. The second nonsmooth optimization algorithm we developed was used to demonstrate how smooth optimization methods can be applied to solve general nonsmooth (nonconvex) optimization problems. In order to do so we compute subgradients from some neighborhood of the current point and define a system of linear inequalities using these subgradients. Search directions are computed by solving this system. This system is solved by reducing it to the minimization of the convex piecewise linear function over the unit ball. Then the hyperbolic smoothing function is applied to approximate this minimization problem by a sequence of smooth problems which are solved by smooth optimization methods. Such an approach allows one to apply powerful smooth optimization algorithms for solving nonsmooth optimization problems and extend smoothing techniques for solving general nonsmooth nonconvex optimization problems. The convergence of the algorithm based on this approach is studied. The proposed algorithm was implemented in Fortran 95. Preliminary results of numerical experiments are reported and the proposed algorithm is compared with an other five nonsmooth optimization algorithms. We also implement the algorithm in GAMS and compare it with GAMS solvers using results of numerical experiments.**Description:**Doctor of Philosophy

An incremental piecewise linear classifier based on polyhedral conic separation

- Ozturk, Gurkan, Bagirov, Adil, Kasimbeyli, Refail

**Authors:**Ozturk, Gurkan , Bagirov, Adil , Kasimbeyli, Refail**Date:**2015**Type:**Text , Journal article**Relation:**Machine Learning Vol. 101, no. 1-3 (2015), p. 397-413**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**In this paper, a piecewise linear classifier based on polyhedral conic separation is developed. This classifier builds nonlinear boundaries between classes using polyhedral conic functions. Since the number of polyhedral conic functions separating classes is not known a priori, an incremental approach is proposed to build separating functions. These functions are found by minimizing an error function which is nonsmooth and nonconvex. A special procedure is proposed to generate starting points to minimize the error function and this procedure is based on the incremental approach. The discrete gradient method, which is a derivative-free method for nonsmooth optimization, is applied to minimize the error function starting from those points. The proposed classifier is applied to solve classification problems on 12 publicly available data sets and compared with some mainstream and piecewise linear classifiers. © 2014, The Author(s).

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