Subadditivity Of Piecewise Linear Functions
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Subadditivity of Piecewise Linear Functions
An optimization problem is trying to minimize or maximize a real objective function such that the input variables satisfies certain constraints. The simplest optimization problem is the linear program (LP), which has been proved to be in the P class and are usually solved by the simplex method or interior point methods. However, imposing integral constraints to an optimization problem can make the problem difficult to solve, and the mixed integer program (MIP) belongs to the NP-hard class. Mixed integer optimization problems have a large number of applications in various fields, such as operations research and machine learning. Although MIP are typically hard to solve, the good news is that researchers are making substantial progress on solving problems more efficiently using better computation powers and more robust algorithms. Cutting plane algorithms, which were first proposed by Gomory and Johnson in the 1960s, are widely used in the state-of-the-art solvers. In the cutting plane algorithms, a family of real functions (so called cut-generating functions) are used to provide valid constraints to the optimization problem so that they can be solved faster. Cut-generating functions are typically piecewise linear functions and satisfy subadditivity. In this dissertation, we study the subadditivity of piecewise linear functions and use a software to verify subadditivity more efficiently. It is important to verify subadditivity of a cut-generating function before applying it to optimization problems. As the structure of the cut-generating function gets complicated, the existing algorithm can take a very long time to verify subadditivity. We develop a spatial branch and bound algorithm to prove or disprove subadditivity of a given piecewise linear function. We use a benchmark work to show that the new algorithm works better for functions with complicated structures. We also address the reproducibility of the benchmark work, and we provide a open sourced repository so that other interested researchers can reproduce the experiment. Dual-feasible functions are in the scope of superadditive duality theory, and they are an important family of functions which have been used in certain combinatorial optimization problems. We provide a new characterization of strong dual-feasible functions and we relate them to cut-generating functions. Inspired by results on cut-generating functions, we discover new results on dual-feasible functions. Software has been used to study properties of dual-feasible functions, based on which new dual-feasible functions can be found.
Integer Programming and Combinatorial Optimization
This book constitutes the refereed proceedings of the 19th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2017, held in Waterloo, IN, Canada, in June 2017. The 36 full papers presented were carefully reviewed and selected from 125 submissions. The conference is a forum for researchers and practitioners working on various aspects of integer programming and combinatorial optimization. The aim is to present recent developments in theory, computation, and applications in these areas. The scope of IPCO is viewed in a broad sense, to include algorithmic and structural results in integer programming and combinatorial optimization as well as revealing computational studies and novel applications of discrete optimization to practical problems.
An Introduction to Computational Risk Management of Equity-Linked Insurance
The quantitative modeling of complex systems of interacting risks is a fairly recent development in the financial and insurance industries. Over the past decades, there has been tremendous innovation and development in the actuarial field. In addition to undertaking mortality and longevity risks in traditional life and annuity products, insurers face unprecedented financial risks since the introduction of equity-linking insurance in 1960s. As the industry moves into the new territory of managing many intertwined financial and insurance risks, non-traditional problems and challenges arise, presenting great opportunities for technology development. Today's computational power and technology make it possible for the life insurance industry to develop highly sophisticated models, which were impossible just a decade ago. Nonetheless, as more industrial practices and regulations move towards dependence on stochastic models, the demand for computational power continues to grow. While the industry continues to rely heavily on hardware innovations, trying to make brute force methods faster and more palatable, we are approaching a crossroads about how to proceed. An Introduction to Computational Risk Management of Equity-Linked Insurance provides a resource for students and entry-level professionals to understand the fundamentals of industrial modeling practice, but also to give a glimpse of software methodologies for modeling and computational efficiency. Features Provides a comprehensive and self-contained introduction to quantitative risk management of equity-linked insurance with exercises and programming samples Includes a collection of mathematical formulations of risk management problems presenting opportunities and challenges to applied mathematicians Summarizes state-of-arts computational techniques for risk management professionals Bridges the gap between the latest developments in finance and actuarial literature and the practice of risk management for investment-combined life insurance Gives a comprehensive review of both Monte Carlo simulation methods and non-simulation numerical methods Runhuan Feng is an Associate Professor of Mathematics and the Director of Actuarial Science at the University of Illinois at Urbana-Champaign. He is a Fellow of the Society of Actuaries and a Chartered Enterprise Risk Analyst. He is a Helen Corley Petit Professorial Scholar and the State Farm Companies Foundation Scholar in Actuarial Science. Runhuan received a Ph.D. degree in Actuarial Science from the University of Waterloo, Canada. Prior to joining Illinois, he held a tenure-track position at the University of Wisconsin-Milwaukee, where he was named a Research Fellow. Runhuan received numerous grants and research contracts from the Actuarial Foundation and the Society of Actuaries in the past. He has published a series of papers on top-tier actuarial and applied probability journals on stochastic analytic approaches in risk theory and quantitative risk management of equity-linked insurance. Over the recent years, he has dedicated his efforts to developing computational methods for managing market innovations in areas of investment combined insurance and retirement planning.