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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

Comparison of metaheuristic algorithms for pump operation optimization

- Bagirov, Adil, Ahmed, S. T., Barton, Andrew, Mala-Jetmarova, Helena, Al Nuaimat, Alia, Sultanova, Nargiz

**Authors:**Bagirov, Adil , Ahmed, S. T. , Barton, Andrew , Mala-Jetmarova, Helena , Al Nuaimat, Alia , Sultanova, Nargiz**Date:**2012**Type:**Text , Conference paper**Relation:**14th Water Distribution Systems Analysis Conference 2012, WDSA 2012 Vol. 2; Adelaide, Australia; 24th-27th September 2012; p. 886-896**Relation:**http://purl.org/au-research/grants/arc/LP0990908**Full Text:**false**Reviewed:****Description:**Pumping cost constitutes the main part of the overall operating cost of water distribution systems. There are different optimization formulations of the pumping cost minimization problem including those with application of continuous and integer programming approaches. To date mainly various metaheuristics have been applied to solve this problem. However, the comprehensive comparison of those metaheuristics has not been done. Such a comparison is important to identify strengths and weaknesses of different algorithms which reflects on their performance. In this paper, we present a methodology for comparative analysis of widely used metaheuristics for solving the pumping cost minimization problem. This methodology includes the following comparison criteria: (a) the "optimal solution" obtained; (b) the efficiency; and (c) robustness. Algorithms applied are: particle swarm optimization, artificial bee colony and firefly algorithms. These algorithms were applied to one test problem available in the literature. The results obtained demonstrate that the artificial bee colony is the most robust and the firefly is the most efficient and accurate algorithm for this test problem. Funding :ARC

A novel approach to optimal pump scheduling in water distribution systems

- Bagirov, Adil, Barton, Andrew, Mala-Jetmarova, Helena, Al Nuaimat, Alia, Ahmed, S. T., Sultanova, Nargiz, Yearwood, John

**Authors:**Bagirov, Adil , Barton, Andrew , Mala-Jetmarova, Helena , Al Nuaimat, Alia , Ahmed, S. T. , Sultanova, Nargiz , Yearwood, John**Date:**2012**Type:**Text , Conference paper**Relation:**14th Water Distribution Systems Analysis Conference 2012, WDSA 2012 Vol. 1; Adelaide, Australia; 24th-27th September; p. 618-631**Relation:**http://purl.org/au-research/grants/arc/LP0990908**Full Text:**false**Reviewed:****Description:**The operation of a water distribution system is a complex task which involves scheduling of pumps, regulating water levels of storages, and providing satisfactory water quality to customers at required flow and pressure. Pump scheduling is one of the most important tasks of the operation of a water distribution system as it represents the major part of its operating costs. In this paper, a novel approach for modeling of pump scheduling to minimize energy consumption by pumps is introduced which uses pump's start/end run times as continuous variables. This is different from other approaches where binary integer variables for each hour are typically used which is considered very impractical from an operational perspective. The problem is formulated as a nonlinear programming problem and a new algorithm is developed for its solution. This algorithm is based on the combination of the grid search with the Hooke-Jeeves pattern search method. The performance of the algorithm is evaluated using literature test problems applying the hydraulic simulation model EPANet.**Description:**E1

Solving minimax problems : Local smoothing versus global smoothing

- Bagirov, Adil, Sultanova, Nargiz, Al Nuaimat, Alia, Taheri, Sona

**Authors:**Bagirov, Adil , Sultanova, Nargiz , Al Nuaimat, Alia , Taheri, Sona**Date:**2018**Type:**Text , Conference proceedings**Relation:**4th International Conference on Numerical Analysis and Optimization, NAO-IV 2017; Muscat, Oman; 2nd-5th January 2017; published in Numerical Analysis and Optimization NAO-IV (part of the Springer Proceedings in Mathematics and Statistics book series PROMS, volume 235) Vol. 235, p. 23-43**Full Text:**false**Reviewed:****Description:**The aim of this chapter is to compare different smoothing techniques for solving finite minimax problems. We consider the local smoothing technique which approximates the function in some neighborhood of a point of nondifferentiability and also global smoothing techniques such as the exponential and hyperbolic smoothing which approximate the function in the whole domain. Computational results on the collection of academic test problems are used to compare different smoothing techniques. Results show the superiority of the local smoothing technique for convex problems and global smoothing techniques for nonconvex problems. © 2018, Springer International Publishing AG, part of Springer Nature.**Description:**Springer Proceedings in Mathematics and Statistics

Hyperbolic smoothing function method for minimax problems

- Bagirov, Adil, Al Nuaimat, Alia, Sultanova, Nargiz

**Authors:**Bagirov, Adil , Al Nuaimat, Alia , Sultanova, Nargiz**Date:**2013**Type:**Text , Journal article**Relation:**Optimization Vol. 62, no. 6 (2013), p. 759-782**Full Text:**false**Reviewed:****Description:**In this article, an approach for solving finite minimax problems is proposed. This approach is based on the use of hyperbolic smoothing functions. 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 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 general algebraic modelling system. 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. © 2013 Copyright Taylor and Francis Group, LLC.**Description:**2003011099

- Bagirov, Adil, Barton, Andrew, Mala-Jetmarova, Helena, Al Nuaimat, Alia, Ahmed, S. T., Sultanova, Nargiz, Yearwood, John

**Authors:**Bagirov, Adil , Barton, Andrew , Mala-Jetmarova, Helena , Al Nuaimat, Alia , Ahmed, S. T. , Sultanova, Nargiz , Yearwood, John**Date:**2013**Type:**Text , Journal article**Relation:**Mathematical and Computer Modelling Vol. 57, no. 3-4 (2013), p. 873-886**Relation:**http://purl.org/au-research/grants/arc/LP0990908**Full Text:**false**Reviewed:****Description:**The operation of a water distribution system is a complex task which involves scheduling of pumps, regulating water levels of storages, and providing satisfactory water quality to customers at required flow and pressure. Pump scheduling is one of the most important tasks of the operation of a water distribution system as it represents the major part of its operating costs. In this paper, a novel approach for modeling of explicit pump scheduling to minimize energy consumption by pumps is introduced which uses the pump start/end run times as continuous variables, and binary integer variables to describe the pump status at the beginning of the scheduling period. This is different from other approaches where binary integer variables for each hour are typically used, which is considered very impractical from an operational perspective. The problem is formulated as a mixed integer nonlinear programming problem, and a new algorithm is developed for its solution. This algorithm is based on the combination of the grid search with the Hooke-Jeeves pattern search method. The performance of the algorithm is evaluated using literature test problems applying the hydraulic simulation model EPANet. © 2012 Elsevier Ltd.**Description:**2003010583

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