INTRO TO APPLIED MATH LINEAR AND INTEGER OPTIMIZATION MA 325, SPRING 2018 DÁVID PAPP

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1 INTRO TO APPLIED MATH LINEAR AND INTEGER OPTIMIZATION MA 325, SPRING 2018 DÁVID PAPP

2 THE FORMALITIES Basic info: Me: Dr. Dávid Papp SAS 3222 (Math dept) Textbook: none. One homework assignment (towards the end). Module website What we covered, slides (when applicable) Homework Office hours 2

3 WHAT IS THIS MODULE ABOUT? You ve heard a bit about optimization in the least squares/gps module. Most of that was unconstrained optimization: find a minimum of a given function. In most practical problems, are choices are limited by a variety of reasons: we have constraints. Linear optimization (to be defined soon) is the simplest (but surprisingly useful) family of constrained optimization model. Integer linear optimization is a conceptually similar, but even much more powerful version. (We ll talk about the pros and cons.) 3

4 4 WHAT IS THIS MODULE ABOUT? Linear and integer optimization are the two most commonly used tools in operations research. Recommended reading: Operation everything (Virginia Postrel for the Boston Globe)

5 WHAT IS THIS MODULE ABOUT? Mathematical optimization ( = mathematical programming ) Fundamental tool in computational math, engineering, operations research, even theoretical computer science (and in some branches of pure mathematics) More applications in more industries than we can list here: healthcare management E-commerce energy and the environment supply chain management marketing and revenue management systems manufacturing financial engineering telecommunication networks transportation networks service systems military defense 5

6 WHAT IS THIS MODULE ABOUT? Optimization ( = mathematical programming ) Mathematically speaking, it s simply trying to find min xx SS ff(xx) Plus, we usually want the minimizer, too, not just the value. And, yes, it should really be an inf. And it could equivalently(!) be a max or a sup. Recall the difference between inf and min minimize xx 2 versus minimize ee xx. The minimum/maximum exists in most practical problems. (Might have seen in Calculus or Analysis: Weierstrass s Extreme Value Theorem aka the min-max theorem.) 6

7 7 THE GENERAL OPTIMIZATION PROBLEM Optimization ( = mathematical programming ) Some terminology: (Could be inf/sup/min/max) The constraints min xx SS ff(xx) The objective function The (decision) variables The feasible region (usually in R nn ), aka. the set of feasible solutions. The minimizer is also called the optimal solution.

8 EVERYTHING IS (OR CAN BE) AN OPTIMIZATION PROBLEM Fermat s Last Theorem No three positive integers aa, bb, and cc satisfy aa nn + bb nn = cc nn for any integer value of nn > 2. Was open for over 350 years; it s only known proof is extremely obtuse. We can write it as a very simple(?) optimization problem: minimize (aa nn + bb nn cc nn ) 2 + sin 2 (aaaa) + sin 2 (bbππ) + sin 2 (ccππ) + sin 2 (nnππ) subject to the constraints aa 1, bb 1, cc 1, nn 3. It is not hard to show that the infimum of the above function on the given set is 0. So proving or disproving the conjecture amounts to just figuring out if this value 0 is attained or not. This should make it clear that general optimization problems are hard. 8

9 THE GENERAL OPTIMIZATION PROBLEM In engineering, finance, etc. The variables xx usually represent decisions about the design of the system or its use Design parameters, schedules, etc. Resource allocation The objective function ff represents our primary goal Maximize profit, minimize time the project takes, etc. The constraints xx SS come from many sources: Physical reality Resource availability Constraints from prior decisions or secondary objectives (e.g., maximize profit but also be green and keep customers happy and abide don t upset the IRS) (e.g., maximize profit while risk stays below some threshold) 9

10 10 EXAMPLE 1: GPS AND LEAST SQUARES A relatively simple optimization model of a physical system. Module 1, won t talk about it again

11 THE GENERAL OPTIMIZATION PROBLEM In biological and physical sciences The variables xx usually represent the state of the system, or the evolution of the state of the system Positions, angles, velocities, pressure, etc. The objective and the constraints come from scientific principles and laws Minimize potential energy, free energy, etc. Achieve equilibrium Balance and conservation laws Laws connecting various quantities, etc. Have you played FoldIt? ( 11

12 EXAMPLE 2: PROTEIN FOLDING A complex optimization problem from nature. At a very high level: The protein has lower energy if it s rolled up in a ball. The closeness of some parts gives you special bonus because of additional bonds forming. Hydrophobic parts are better off surrounded by other parts. Can t tear it apart, and no clashes allowed. 12

13 WHAT TO EXPECT FROM THIS MODULE Optimization means at least 3 things: 1. Optimization models: How to turn a real problem into an optimization model? What is a good model? 2. The math behind the models as mathematical constructs (basically, theorems). 3. The algorithms to compute the optimal solution. We will do a little bit of all of the above. You can t do either step in any meaningful way without the other two. (You ll see why.) 13

14 ANY QUESTIONS? 14