A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and $k$-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems

Multi-Objective Optimization Programs and their Application to Amine Absorption Process Design for Natural Gas Sweetening

This chapter presents three MS Excel programs, namely, EMOO (Excel based Multi-Objective Optimization), NDS (Non-Dominated Sorting) and PM (Performance Metrics) useful for Multi-Objective Optimization (MOO) studies. The EMOO program is for finding non-dominated solutions of a given MOO problem. It has both binary-coded and realcoded NSGA-II (Elitist Non-Dominated Sorting Genetic Algorithm), and two termination criteria based on chi-squared test and steady state detection. The known/true Pareto-optimal front for the application problems is not available unlike that for benchmark problems. Hence, a procedure for obtaining known/true Pareto-optimal front is described in this chapter. The NDS program is for non-dominated sorting and crowding distance calculations of the non-dominated solutions obtained from several optimization runs using same or different MOO programs. The PM program can be used to calculate the values of performance metrics between the non-dominated solutions obtained using a MOO program and the true/known Pareto optimal front. It is useful for comparing the performance of MOO programs to find the non-dominated solutions. Finally, use of EMOO, NDS and PM programs is demonstrated on MOO of amine absorption process for natural gas sweetening

Accelerating Convergence of Leapfrogging Optimization - Applications to Nonlinear Process Modeling and Nonlinear Model Predictive Control

Conventionally used optimization methods in chemical engineering applications such as linear programming (LP), Levenberg-Marquardt and sequential quadratic programming (SQP) handle nonlinear objective function (OF) surfaces by linearizing or assuming quadratic behavior of the surfaces [1]. Process modeling and nonlinear model predictive control (NMPC) applications, however, present OF surfaces with surface aberrations such as steep slopes, discontinuities, and hard constraints which require a robust and efficient optimization method. Therefore, an optimization method that can handle surface aberrations is required.Leapfrogging (LF) is a recently developed direct search optimization method, potentially best-in-class, which can handle surface aberrations. LF starts with a set of players (trial solutions), randomly placed in the decision variable (DV) space. The worst player (player with the worst OF value) leaps over the best player into a reflected hypervolume [2]. The leapovers continue until all the players converge. LF is robust and efficient - with minimal computation effort (compared to conventional optimization methods), it can handle the challenges posed by nonlinear OF surfaces. LF was demonstrated on over 40 test functions and several modeling and NMPC applications. Rigorous fundamental analysis of LF is required - for a finer understanding of the method, exploring opportunities for improvement and scaling LF applications to large scale systems.This work is focused on exploring and analyzing methods to accelerate convergence of LF, demonstrating application credibility on nonlinear process modeling of steady state binary distillation and NMPC of a binary distillation column. Accelerating convergence opens the doors for using LF in large scale problems that have several hundred variables such as real time optimization and refinery planning where computational effort and time are of essence. Distillation modeling is constrained, nonlinear, and has optimum confined to a narrow region; distillation control is multivariable, interacting, nonlinear and has severe disturbances.Completion of this work will provide new fundamental understanding of LF which is critical for creating opportunities for algorithm improvement. Demonstrating application to nonlinear process modeling and NMPC will create application credibility, reveal practicality and serve as proof of concept that LF can be an optimizer of choice for use in the process control community.Chemical Engineerin