Computation of convex conjugates in linear time using graph-matrix calculus

Computational Convex Analysis focuses on computing the convex operators which are used very often in convex analysis. The Fenchel conjugate is one of the most frequently used convex operator. The objective of this thesis is to develop an algorithm for computing the conjugate of a bivariate Piecewise Linear-Quadratic (PLQ) function. Although some algorithms already exist for computing the conjugate of a bivariate PLQ function, their worst-case time complexity is not linear. Our challenge is to improve the worst case time complexity to linear. We use a planar graph to represent the entities of a PLQ function. Each node of the graph contains a GPH matrix which represents an entity of the PLQ function . In addition, we store the adjacency information and type of all entities. We traverse the graph using breadth first search and compute the conjugate of each entity. Moreover we store the information of visited entities using a binary flag. So we never need loop through all entities to check whether it is already visited. As a result we get linear computation time in the worst case. We perform numerical experiment which confirms that our algorithm is running in linear-time. We provide a comparison of the performance of this algorithm with the previous algorithms. Finally we explained how to extend this algorithm in higher dimensions while keeping the worst-case time complexity linear.Arts and Sciences, Irving K. Barber School of (Okanagan)Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)Graduat