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Application of nonsmooth optimisation to data analysis

**Authors:**Ugon, Julien**Date:**2005**Type:**Text , Thesis , PhD**Full Text:****Description:**The research presented in this thesis is two-fold: on the one hand, major data mining problems are reformulated as mathematical programming problems. These problems should be carefully designed, since from their formulation depends the efficiency, perhaps the existence, of the solvers. On the other hand, optimisation methods are adapted to solve these problems, most of which are nonsmooth and nonconvex. This part is delicate, as the solution is often required to be good and obtained fast. Numerical experiments on real-world datasets are presented and analysed.**Description:**Doctor of Philosophy

**Authors:**Ugon, Julien**Date:**2005**Type:**Text , Thesis , PhD**Full Text:****Description:**The research presented in this thesis is two-fold: on the one hand, major data mining problems are reformulated as mathematical programming problems. These problems should be carefully designed, since from their formulation depends the efficiency, perhaps the existence, of the solvers. On the other hand, optimisation methods are adapted to solve these problems, most of which are nonsmooth and nonconvex. This part is delicate, as the solution is often required to be good and obtained fast. Numerical experiments on real-world datasets are presented and analysed.**Description:**Doctor of Philosophy

Conditions for global minimum through abstract convexity

**Authors:**Sharikov, Evgenii**Date:**2008**Type:**Text , Thesis , PhD**Full Text:****Description:**The theory of abstract convexity generalizes ideas of convex analysis by using the notion of global supports and the global definition of subdifferential. In order to apply this theory to optimization, we need to extend subdifferential calculus and separation properties into the area of abstract convexity.**Description:**Doctor of Philosophy

**Authors:**Sharikov, Evgenii**Date:**2008**Type:**Text , Thesis , PhD**Full Text:****Description:**The theory of abstract convexity generalizes ideas of convex analysis by using the notion of global supports and the global definition of subdifferential. In order to apply this theory to optimization, we need to extend subdifferential calculus and separation properties into the area of abstract convexity.**Description:**Doctor of Philosophy

G-coupling functions and properties of strongly star-shaped cones

**Authors:**Morales-Silva, Daniel**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**The main part of this thesis presents a new approach to the topic of conjugation, with applications to various optimization problems. It does so by introducing (what we call) G-coupling functions.**Description:**Doctor of Philosophy

**Authors:**Morales-Silva, Daniel**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**The main part of this thesis presents a new approach to the topic of conjugation, with applications to various optimization problems. It does so by introducing (what we call) G-coupling functions.**Description:**Doctor of Philosophy

A class of Increasing Positively Homogeneous functions for which global optimization problem is NP-hard

**Authors:**Sultanova, Nargiz**Date:**2009**Type:**Text , Thesis , Masters**Full Text:****Description:**It is well known that global optimization problems are, generally speaking, computationally infeasible, that is solving them would require an unreasonably large amount of time and/or space. In certain cases, for example, when objective functions and constraints are convex, it is possible to construct a feasible algorithm for solving global optimization problem successfully. Convexity, however, is not a phenomenon to be often expected in the applications. Nonconvex problems frequently arise in many industrial and scienti¯c areas. Therefore, it is only natural to try to replace convexity with some other structure at least for some classes of nonconvex optimization problems to render the global optimization problem feasible. A theory of abstract convexity has been developed as a result of the above considerations. Monotonic analysis, a branch of abstract convex analysis, is analogous in many ways to convex analysis, and sometimes is even simpler. It turned out that many problems of nonconvex optimization encountered in applications can be described in terms of monotonic functions. The analogies with convex analysis were considered to aid in solving some classes of nonconvex optimization problems. In this thesis we will focus on one of the elements of monotonic analysis - Increasing Positively Homogeneous functions of degree one or in short IPH functions. The aim of present research is to show that finding the solution and ²-approximation to the solution of the global optimization problem for IPH functions restricted to a unit simplex is an NP-hard problem. These results can be further extended to positively homogeneous functions of degree ´, ´ > 0.**Description:**Master of Mathematical Sciences (Research)

**Authors:**Sultanova, Nargiz**Date:**2009**Type:**Text , Thesis , Masters**Full Text:****Description:**It is well known that global optimization problems are, generally speaking, computationally infeasible, that is solving them would require an unreasonably large amount of time and/or space. In certain cases, for example, when objective functions and constraints are convex, it is possible to construct a feasible algorithm for solving global optimization problem successfully. Convexity, however, is not a phenomenon to be often expected in the applications. Nonconvex problems frequently arise in many industrial and scienti¯c areas. Therefore, it is only natural to try to replace convexity with some other structure at least for some classes of nonconvex optimization problems to render the global optimization problem feasible. A theory of abstract convexity has been developed as a result of the above considerations. Monotonic analysis, a branch of abstract convex analysis, is analogous in many ways to convex analysis, and sometimes is even simpler. It turned out that many problems of nonconvex optimization encountered in applications can be described in terms of monotonic functions. The analogies with convex analysis were considered to aid in solving some classes of nonconvex optimization problems. In this thesis we will focus on one of the elements of monotonic analysis - Increasing Positively Homogeneous functions of degree one or in short IPH functions. The aim of present research is to show that finding the solution and ²-approximation to the solution of the global optimization problem for IPH functions restricted to a unit simplex is an NP-hard problem. These results can be further extended to positively homogeneous functions of degree ´, ´ > 0.**Description:**Master of Mathematical Sciences (Research)

Derivative-free hybrid methods in global optimization and their applications

**Authors:**Zhang, Jiapu**Date:**2005**Type:**Text , Thesis , PhD**Full Text:****Description:**In recent years large-scale global optimization (GO) problems have drawn considerable attention. These problems have many applications, in particular in data mining and biochemistry. Numerical methods for GO are often very time consuming and could not be applied for high-dimensional non-convex and / or non-smooth optimization problems. The thesis explores reasons why we need to develop and study new algorithms for solving large-scale GO problems .... The thesis presents several derivative-free hybrid methods for large scale GO problems. These methods do not guarantee the calculation of a global solution; however, results of numerical experiments presented in this thesis demonstrate that they, as a rule, calculate a solution which is a global one or close to it. Their applications to data mining problems and the protein folding problem are demonstrated.**Description:**Doctor of Philosophy

**Authors:**Zhang, Jiapu**Date:**2005**Type:**Text , Thesis , PhD**Full Text:****Description:**In recent years large-scale global optimization (GO) problems have drawn considerable attention. These problems have many applications, in particular in data mining and biochemistry. Numerical methods for GO are often very time consuming and could not be applied for high-dimensional non-convex and / or non-smooth optimization problems. The thesis explores reasons why we need to develop and study new algorithms for solving large-scale GO problems .... The thesis presents several derivative-free hybrid methods for large scale GO problems. These methods do not guarantee the calculation of a global solution; however, results of numerical experiments presented in this thesis demonstrate that they, as a rule, calculate a solution which is a global one or close to it. Their applications to data mining problems and the protein folding problem are demonstrated.**Description:**Doctor of Philosophy

Derivative free algorithms for nonsmooth and global optimization with application in cluster analysis

**Authors:**Ganjehlou, Asef Nazari**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**This thesis is devoted to the development of algorithms for solving nonsmooth nonconvex problems. Some of these algorithms are derivative free methods.**Description:**Doctor of Philosophy

**Authors:**Ganjehlou, Asef Nazari**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**This thesis is devoted to the development of algorithms for solving nonsmooth nonconvex problems. Some of these algorithms are derivative free methods.**Description:**Doctor of Philosophy

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