Practical Optimization: A Gentle Introduction

John W. Chinneck

Systems and Computer Engineering

Carleton University

Ottawa, Ontario K1S 5B6

Canada

http://www.sce.carleton.ca/faculty/chinneck/po.html

See the Swedish translation of this page by Catherine Desroches.

The chapters appearing below are draft chapters of an introductory textbook on optimization. The ultimate objective is the creation of a complete, yet compact, introductory survey text on the major topics in optimization. The material is derived from the lecture notes used by the author in engineering courses at Carleton University, and reflects the design considerations for those courses:

Students need to have a solid intuitive understanding of how and why optimization methods work. This enables them to recognize when things have gone wrong, and to diagnose the source of the difficulty and take appropriate action. It also permits students to see how methods can be combined or modified to solve non-standard problems.

Explanation and diagrams are more effective in transmitting a real understanding than a total reliance on mathematics.

There should be some exposure to how things are done in practice, which can be significantly different than how they are usually done in textbooks.

The goal is for students to be able to recognize when problems that they encounter in their jobs or in thesis work can be successfully tackled by optimization methods. Students should be able to abstract a useful formulation of a problem from a messy real world description.

The material is written at the introductory level, assuming no more knowledge than high school algebra. Most concepts are developed from scratch. I intend this to be a gentle introduction.

You will need an Adobe Acrobat reader to read the files. A free reader is downloadable from www.adobe.com

Comments and suggestions are actively sought. Please contact the author at chinneck@sce.carleton.ca. This is a draft, so some diagrams are hand-drawn, and there may be typos, etc. Further chapters will be added as time permits. Please let me know of any corrections that are needed.

Browse the algorithm animations. These provide animated illustrations of many of the key concepts. The student developers of these animations were funded through an IBM Faculty Award to Prof. Chinneck. Last revision: August 28, 2007.

Printing out the whole volume? Then add the front matter for a better look. Last revision: June 23, 2015.

Chapter 1: Introduction. An introduction to the process of optimization and an overview of the major topics covered in the course. Last revision: December 12, 2010.

Chapter 2: Introduction to Linear Programming. The basic notions of linear programming and the simplex method. The simplex method is the easiest way to provide a beginner with a solid understanding of linear programming. Last revision: September 19, 2007.

Chapter 3: Towards the Simplex Method for Efficient Solution of Linear Programs. Cornerpoints and bases. Moving to improved solutions. Last revision: August 18, 2006.

See animation LP2

Chapter 4: The Mechanics of the Simplex Method. The tableau representation as a way of illustrating the process of the simplex method. Special cases such as degeneracy and unboundedness. Last revision: September 26, 2017.

See animation LP3

Chapter 5: Solving General Linear Programs. Solving non-standard linear programs. Phase 1 LPs. Feasibility and infeasibility. Unrestricted variables. Last revision: August 18, 2006.

See animations LP4 and LP5

Chapter 6: Sensitivity Analysis. Simple computer-based sensitivity analysis. Last revision: August 18, 2006.

See animation LP6

Chapter 7: Linear Programming in Practice. Mention of other solution methods such as revised simplex method and interior point methods. Mention of advanced techniques used in practice such as advanced and crash start methods, infeasibility analysis, and modelling systems. Last revision: August 18, 2006.

Chapter 8: An Introduction to Networks. Some basic network concepts. The shortest route problem. The minimum spanning tree problem. Last revision: September 10, 2010.

See animations Networks1 and Networks2

Chapter 9: Maximum Flow and Minimum Cut. The maximum flow and minimum cut problems in networks. Last revision: August 18, 2006.

See animation Networks3

Chapter 10: Network Flow Programming. A surprising range of problems can be solved using minimum cost network flow programming, including shortest route, maximum flow and minimum cut, etc. Variations such as generalized and processing networks are also briefly introduced. Last revision: October 12, 2017.

Chapter 11: PERT for Project Planning and Scheduling. PERT is a network-based aid for project planning and scheduling. Many optimization problems involve some aspect of the timing of activities that may run sequentially or in parallel, or the timing of resource use. PERT diagrams help you to understand and formulate such problems. Last revision: November 3, 2016.

See animation Networks4

Chapter 12: Integer/Discrete Programming via Branch and Bound. Branch and bound is the basic workhorse method for solving many kinds of problems that have variables that can take on only integer, binary, or discrete values. Last revision: October 23, 2012.

See animation IP1

Chapter 13: Binary and Mixed-Integer Programming. These are specialized versions of branch and bound. A binary program has only binary variables (0 or 1 only). A mixed-integer program looks like a linear program, except that some or all of the variables are integer-valued (or binary-valued), while others might be real-valued. Last revision: November 22, 2016.

See animations IP2 and IP3

Chapter 14: Heuristics for Discrete Search: Genetic Algorithms and Simulated Annealing. Some problems are just too big for branch and bound, in which case you must abandon the guarantee of finding the optimum solution and instead opt for heuristic methods which can only guarantee to do fairly well most of the time. Genetic Algorithms and Simulated Annealing are two popular heuristic methods for use on very large problems. Last revision: August 22, 2006.

See animations Heuristics1 and Heuristics2

Chapter 15: Dynamic Programming. This optimization technique builds towards a solution by first solving a small part of the whole problem, and then gradually incrementing the size in a series of stages until the whole problem is solved. Efficiency results from combining the local solution for a stage with the optimum found for a previous stage. We look at the simplest deterministic discrete cases. Last revision: December 16, 2015.

See animation DP1

Chapter 16: Introduction to Nonlinear Programming (NLP). NLP is a lot harder than linear programming. We start by looking at the reasons for this. Next we look at the simplest method for solving the simplest type of NLP: unconstrained problems that consist only of a nonlinear objective function. The method of steepest ascent/descent is described. Last revision: December 16, 2015.

See animations NLP1 and NLP2

Chapter 17: Pattern Search for Unconstrained NLP. What do you do if you don’t have access to gradient information? In that case you can use pattern search techniques (also known as derivative-free, direct search, or black box methods). We look at the classic Hooke and Jeeves pattern search method. Last revision: April 29, 2014.

See animation NLP7

Chapter 18: Constrained Nonlinear Programming. Now that we have some idea of how to solve unconstrained NLPs, how do we deal with constrained NLPs? The first idea is to turn them into unconstrained NLPs of course! This is done by using penalty and barrier methods which replace or modify the original objective function in ways that make feasible points attractive in the resulting unconstrained problem. Last revision: April 29, 2014.

Chapter 19: Handling Equality Constraints in NLP. Equality constraints are the hardest to handle in nonlinear programming. We look at two ways of dealing with them: (i) the method of Lagrange, and (ii) the Generalized Reduced Gradient (GRG) method. And we take a look at making linear approximations to nonlinear functions because we need that for the GRG method. Last revision: December 2, 2015.

Chapter 20: Function and Region Shapes, the Karush-Kuhn-Tucker (KKT) Conditions, and Quadratic Programming. Function and region shapes (convex, nonconvex, etc.) are important in understanding how nonlinear solvers work. The KKT conditions are very useful in deciding whether a solution point is really a local optimum in a nonlinear program, and form the basis for some methods of solving quadratic programs. Last revision: December 21, 2018.

Different nonlinear local solvers can give quite different solution trajectories, i.e. the sequence of intermediate solutions reached before the final solution. You can see this in the next few animations.

· See animations NLP3, NLP4, NLP5, NLP6,

Last update December 21, 2018.