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

This thesis is devoted to the development of algorithms for solving nonsmooth nonconvex problems. Some of these algorithms are derivative free methods.Doctor of Philosoph

On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence

We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a highly efficient evaluation of the proximal mapping. The key to this efficiency is a general proximal calculus in the new metric. By using duality, formulas are derived that relate the proximal mapping in a rank-$r$ modified metric to the original metric. We also describe efficient implementations of the proximity calculation for a large class of functions; the implementations exploit the piece-wise linear nature of the dual problem. Then, we apply these results to acceleration of composite convex minimization problems, which leads to elegant quasi-Newton methods for which we prove convergence. The algorithm is tested on several numerical examples and compared to a comprehensive list of alternatives in the literature. Our quasi-Newton splitting algorithm with the prescribed metric compares favorably against state-of-the-art. The algorithm has extensive applications including signal processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115

Extending the solvability of equations using secant-type methods in Banach space

We extend the solvability of equations dened on a Banach space using numerically ecient secant-type methods. The convergence domain of these methods is enlarged using our new idea of restricted convergence region. By using this approach, we obtain a more precise location where the iterates lie than in earlier studies leading to tighter Lipschitz constants. This way the semi-local convergence produces weaker sucient convergence criteria and tighter error bounds than in earlier works. These improvements are also obtained under the same computational eort, since the new Lipschitz constants are special cases of the old ones

Time and Location Aware Mobile Data Pricing

Mobile users' correlated mobility and data consumption patterns often lead to severe cellular network congestion in peak hours and hot spots. This paper presents an optimal design of time and location aware mobile data pricing, which incentivizes users to smooth traffic and reduce network congestion. We derive the optimal pricing scheme through analyzing a two-stage decision process, where the operator determines the time and location aware prices by minimizing his total cost in Stage I, and each mobile user schedules his mobile traffic by maximizing his payoff (i.e., utility minus payment) in Stage II. We formulate the two-stage decision problem as a bilevel optimization problem, and propose a derivative-free algorithm to solve the problem for any increasing concave user utility functions. We further develop low complexity algorithms for the commonly used logarithmic and linear utility functions. The optimal pricing scheme ensures a win-win situation for the operator and users. Simulations show that the operator can reduce the cost by up to 97.52% in the logarithmic utility case and 98.70% in the linear utility case, and users can increase their payoff by up to 79.69% and 106.10% for the two types of utilities, respectively, comparing with a time and location independent pricing benchmark. Our study suggests that the operator should provide price discounts at less crowded time slots and locations, and the discounts need to be significant when the operator's cost of provisioning excessive traffic is high or users' willingness to delay traffic is low.Comment: This manuscript serves as the online technical report of the article accepted by IEEE Transactions on Mobile Computin

Nonsmooth and derivative-free optimization based hybrid methods and applications

"In this thesis, we develop hybrid methods for solving global and in particular, nonsmooth optimization problems. Hybrid methods are becoming more popular in global optimization since they allow to apply powerful smooth optimization techniques to solve global optimization problems. Such methods are able to efficiently solve global optimization problems with large number of variables. To date global search algorithms have been mainly applied to improve global search properties of the local search methods (including smooth optimization algorithms). In this thesis we apply rather different strategy to design hybrid methods. We use local search algorithms to improve the efficiency of global search methods. The thesis consists of two parts. In the first part we describe hybrid algorithms and in the second part we consider their various applications." -- taken from Abstract.Operational Research and Cybernetic

Bibliography on Nondifferentiable Optimization

This is a research bibliography with all the advantages and shortcomings that this implies. The author has used it as a bibliographical data base when writing papers, and it is therefore largely a reflection of his own personal research interests. However, it is hoped that this bibliography will nevertheless be of use to others interested in nondifferentiable optimization