A Collection of Challenging Optimization Problems in Science, Engineering and Economics

Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE Congress on Evolutionary Computation (at Sendai International Center, Sendai, Japan, from 25th to 28th May, 2015.

Control of quantum phenomena: Past, present, and future

Quantum control is concerned with active manipulation of physical and chemical processes on the atomic and molecular scale. This work presents a perspective of progress in the field of control over quantum phenomena, tracing the evolution of theoretical concepts and experimental methods from early developments to the most recent advances. The current experimental successes would be impossible without the development of intense femtosecond laser sources and pulse shapers. The two most critical theoretical insights were (1) realizing that ultrafast atomic and molecular dynamics can be controlled via manipulation of quantum interferences and (2) understanding that optimally shaped ultrafast laser pulses are the most effective means for producing the desired quantum interference patterns in the controlled system. Finally, these theoretical and experimental advances were brought together by the crucial concept of adaptive feedback control, which is a laboratory procedure employing measurement-driven, closed-loop optimization to identify the best shapes of femtosecond laser control pulses for steering quantum dynamics towards the desired objective. Optimization in adaptive feedback control experiments is guided by a learning algorithm, with stochastic methods proving to be especially effective. Adaptive feedback control of quantum phenomena has found numerous applications in many areas of the physical and chemical sciences, and this paper reviews the extensive experiments. Other subjects discussed include quantum optimal control theory, quantum control landscapes, the role of theoretical control designs in experimental realizations, and real-time quantum feedback control. The paper concludes with a prospective of open research directions that are likely to attract significant attention in the future.Comment: Review article, final version (significantly updated), 76 pages, accepted for publication in New J. Phys. (Focus issue: Quantum control

Unifying a Geometric Framework of Evolutionary Algorithms and Elementary Landscapes Theory

Evolutionary algorithms (EAs) are randomised general-purpose strategies, inspired by natural evolution, often used for finding (near) optimal solutions to problems in combinatorial optimisation. Over the last 50 years, many theoretical approaches in evolutionary computation have been developed to analyse the performance of EAs, design EAs or measure problem difficulty via fitness landscape analysis. An open challenge is to formally explain why a general class of EAs perform better, or worse, than others on a class of combinatorial problems across representations. However, the lack of a general unified theory of EAs and fitness landscapes, across problems and representations, makes it harder to characterise pairs of general classes of EAs and combinatorial problems where good performance can be guaranteed provably. This thesis explores a unification between a geometric framework of EAs and elementary landscapes theory, not tied to a specific representation nor problem, with complementary strengths in the analysis of population-based EAs and combinatorial landscapes. This unification organises around three essential aspects: search space structure induced by crossovers, search behaviour of population-based EAs and structure of fitness landscapes. First, this thesis builds a crossover classification to systematically compare crossovers in the geometric framework and elementary landscapes theory, revealing a shared general subclass of crossovers: geometric recombination P-structures, which covers well-known crossovers. The crossover classification is then extended to a general framework for axiomatically analysing the population behaviour induced by crossover classes on associated EAs. This shows the shared general class of all EAs using geometric recombination P-structures, but no mutation, always do the same abstract form of convex evolutionary search. Finally, this thesis characterises a class of globally convex combinatorial landscapes shared by the geometric framework and elementary landscapes theory: abstract convex elementary landscapes. It is formally explained why geometric recombination P-structure EAs expectedly can outperform random search on abstract convex elementary landscapes related to low-order graph Laplacian eigenvalues. Altogether, this thesis paves a way towards a general unified theory of EAs and combinatorial fitness landscapes

The Evolution of Dispersal in Random Environments and The Principle of Partial Control

McNamara and Dall (2011) identified novel relationships between the abundance of a species in different environments, the temporal properties of environmental change, and selection for or against dispersal. Here, the mathematics underlying these relationships in their two-environment model are investigated for arbitrary numbers of environments. The effect they described is quantified as the fitness-abundance covariance. The phase in the life cycle where the population is censused is crucial for the implications of the fitness-abundance covariance. These relationships are shown to connect to the population genetics literature on the Reduction Principle for the evolution of genetic systems and migration. Conditions that produce selection for increased unconditional dispersal are found to be new instances of departures from reduction described by the "Principle of Partial Control" proposed for the evolution of modifier genes. According to this principle, variation that only partially controls the processes that transform the transmitted information of organisms may be selected to increase these processes. Mathematical methods of Karlin, Friedland, and Elsner, Johnson, and Neumann, are central in generalizing the analysis. Analysis of the adaptive landscape of the model shows that the evolution of conditional dispersal is very sensitive to the spectrum of genetic variation the population is capable of producing, and suggests that empirical study of particular species will require an evaluation of its variational properties.Comment: Dedicated to the memory of Professor Michael Neumann, one of whose many elegant theorems provides for a result presented here. 28 pages, 1 table, 1 figur

An exploration of evolutionary computation applied to frequency modulation audio synthesis parameter optimisation

With the ever-increasing complexity of sound synthesisers, there is a growing demand for automated parameter estimation and sound space navigation techniques. This thesis explores the potential for evolutionary computation to automatically map known sound qualities onto the parameters of frequency modulation synthesis. Within this exploration are original contributions in the domain of synthesis parameter estimation and, within the developed system, evolutionary computation, in the form of the evolutionary algorithms that drive the underlying optimisation process. Based upon the requirement for the parameter estimation system to deliver multiple search space solutions, existing evolutionary algorithmic architectures are augmented to enable niching, while maintaining the strengths of the original algorithms. Two novel evolutionary algorithms are proposed in which cluster analysis is used to identify and maintain species within the evolving populations. A conventional evolution strategy and cooperative coevolution strategy are defined, with cluster-orientated operators that enable the simultaneous optimisation of multiple search space solutions at distinct optima. A test methodology is developed that enables components of the synthesis matching problem to be identified and isolated, enabling the performance of different optimisation techniques to be compared quantitatively. A system is consequently developed that evolves sound matches using conventional frequency modulation synthesis models, and the effectiveness of different evolutionary algorithms is assessed and compared in application to both static and timevarying sound matching problems. Performance of the system is then evaluated by interview with expert listeners. The thesis is closed with a reflection on the algorithms and systems which have been developed, discussing possibilities for the future of automated synthesis parameter estimation techniques, and how they might be employed