Algorithms for DC Programming via Polyhedral Approximations of Convex Functions

There is an existing exact algorithm that solves DC programming problems if one component of the DC function is polyhedral convex (Loehne, Wagner, 2017). Motivated by this, first, we consider two cutting-plane algorithms for generating an $\epsilon$-polyhedral underestimator of a convex function g. The algorithms start with a polyhedral underestimator of g and the epigraph of the current underestimator is intersected with either a single halfspace (Algorithm 1) or with possibly multiple halfspaces (Algorithm 2) in each iteration to obtain a better approximation. We prove the correctness and finiteness of both algorithms, establish the convergence rate of Algorithm 1, and show that after obtaining an $\epsilon$-polyhedral underestimator of the first component of a DC function, the algorithm from (Loehne, Wagner, 2017) can be applied to compute an $\epsilon$ solution of the DC programming problem without further computational effort. We then propose an algorithm (Algorithm 3) for solving DC programming problems by iteratively generating a (not necessarily $\epsilon$-) polyhedral underestimator of g. We prove that Algorithm 3 stops after finitely many iterations and it returns an $\epsilon$-solution to the DC programming problem. Moreover, the sequence $\{x_k\}_{k\geq 0} outputted by Algorithm 3 converges to a global minimizer of the DC problem when$\epsilon$ is set to zero. Computational results based on some test instances from the literature are provided

Barnum, Wheeler, Allen: the Harput trio's relationship with other communities

Ankara : İhsan Doğramacı Bilkent Üniversitesi İktisadi, İdari ve Sosyal Bilimler Fakültesi, Tarih Bölümü, 2018.This work is a student project of the Department of History, Faculty of Economics, Administrative and Social Sciences, İhsan Doğramacı Bilkent University.The History of Turkey course (HIST200) is a requirement for all Bilkent undergraduates. It is designed to encourage students to work in groups on projects concerning any topic of their choice that relates to the history of Turkey. It is designed as an interactive course with an emphasis on research and the objective of investigating events, chronologically short historical periods, as well as historic representations. Students from all departments prepare and present final projects for examination by a committee, with 10 projects chosen to receive awards.Includes bibliographical references (page 11)by Burcu Feyzullahoğlu