Methods and Algorithms

In the last few years, Algorithms for Convex Optimization have revolutionized algorithm design, both for discrete and continuous optimization problems. For problems like maximum flow, maximum matching, and submodular function minimization, the fastest algorithms involve essential methods such as gradient descent, mirror descent, interior point methods, and ellipsoid methods. The goal of this self-contained book is to enable researchers and professionals in computer science, data science, and machine learning to gain an in-depth understanding of these algorithms. The text emphasizes how to derive key algorithms for convex optimization from first principles and how to establish precise running time bounds. This modern text explains the success of these algorithms in problems of discrete optimization, as well as how these methods have significantly pushed the state of the art of convex optimization itself.

In the last few years, Algorithms for Convex Optimization have revolutionized algorithm design, both for discrete and continuous optimization problems. For problems like maximum flow, maximum matching, and submodular function minimization, the fastest algorithms involve essential methods such as gradient descent, mirror descent, interior point methods, and ellipsoid methods. The goal of this self-contained book is to enable researchers and professionals in computer science, data science, and machine learning to gain an in-depth understanding of these algorithms. The text emphasizes how to derive key algorithms for convex optimization from first principles and how to establish precise running time bounds. This modern text explains the success of these algorithms in problems of discrete optimization, as well as how these methods have significantly pushed the state of the art of convex optimization itself.

Model checking is one of the most successful verification techniques and has been widely adopted in traditional computing and communication hardware and software industries. This book provides the first systematic introduction to model checking techniques applicable to quantum systems, with broad potential applications in the emerging industry of quantum computing and quantum communication as well as quantum physics. Suitable for use as a course textbook and for self-study, graduate and senior undergraduate students will appreciate the step-by-step explanations and the exercises included. Researchers and engineers in the related fields can further develop these techniques in their own work, with the final chapter outlining potential future applications.

This innovative book models pollution mitigation as a negative externality whilst also providing desirable and useful solutions, such as establishing the triangular equivalence relationship among the Lindahl equilibrium without transfers, the Nash bargaining solution with the payoffs of the Cournot-Nash equilibrium as the status quo point, and the social optimum under the Lindahl weights. By introducing programming algorithms to validate these relationships numerically, Zili Yang bridges the gap between analytical results and empirical modelling, ultimately solving the Lindahl equilibrium and hybrid Nash equilibria in the influential RICE model. This text demonstrates the complexity and variety of environment externality problems, ranging from mixed externality to correlated externalities to environmental externality under IRS and policy applications. Integrating theory, algorithms and applications in a comprehensive framework, The Environment and Externality will benefit scholars and

This innovative book models pollution mitigation as a negative externality whilst also providing desirable and useful solutions, such as establishing the triangular equivalence relationship among the Lindahl equilibrium without transfers, the Nash bargaining solution with the payoffs of the Cournot-Nash equilibrium as the status quo point, and the social optimum under the Lindahl weights. By introducing programming algorithms to validate these relationships numerically, Zili Yang bridges the gap between analytical results and empirical modelling, ultimately solving the Lindahl equilibrium and hybrid Nash equilibria in the influential RICE model. This text demonstrates the complexity and variety of environment externality problems, ranging from mixed externality to correlated externalities to environmental externality under IRS and policy applications. Integrating theory, algorithms and applications in a comprehensive framework, The Environment and Externality will benefit scholars and