Block 4 Nonlinear Systems Lesson 12 – Nonlinear Optimization Is It Not the Best of All Possible Worlds? An engineer who forgot to optimize

The Goal of this Lesson Goal: To make this “best of all possible nonlinear worlds” - a little better! Right on. "It is demonstrable," said he, "that things cannot be otherwise than as they are; for as all things have been created for some end, they must necessarily be created for the best end.” Candide by Voltaire

What do Operations Researchers (OR) do? OR is concerned with optimal decision-making and modeling of deterministic and probabilistic systems that originate from real life. These applications, which occur in government, business, engineering, economics, and the natural and social sciences are largely characterized by the need to allocate scarce resources. (Hillier & Lieberman)

The General Optimization Problem where f, g 1, …,g m are real-valued functions

The Single Variable Problem open interval: closed interval:

The Real Problem local max local min unbounded x f(x) ab closed interval global min global max

Local Minimum local min: x’ is a local minimum (maximum) if for an arbitrary small neighborhood, N, about x’, f(x’)  (  ) f(x) for all x in N. x f(x) x’ N N

Global Minimum global min: x* is a global minimum if f(x*)  f(x) for all x such that a  x  b. x f(x) a b x*

Global Maximum global max: x* is a global maximum if f(x*)  f(x) for all x such that a  x  b. x f(x) x* a b

x x x f(x) concave convex stationary point stationary point Animated

Our very first nonlinear optimization problem

Global Minimum – Convex Functions If f(x) is a convex function if and only if x f(x)

Global Maximum – Concave Functions If f(x) is a concave function if and only if x f(x)

An Unbounded Function

The Single Variable Problem on the Open Interval necessary condition for global solution: f(x) is bounded and sufficient condition: for all x:

A Bounded Example concave function

A word problem A pipeline from the port in NYC to St. Louis, a distance of 1000 miles, is to be constructed by the Leak E. Oil Company with automatic shutoff values installed every x miles in the event of a leak. Environmentalists have estimated that such a pipeline is likely to have two major leaks during its lifetime. The cost of a valve is $500 and the cost of a cleanup in the event of a leak is $2500 per pipeline mile of oil spilled. How far apart should the valves be placed? f(x) = 2 (2500) x (1000) / x 0  x  1000 Animated

f(x) = 2 (2500) x (1000) / x therefore f(x) is convex and x* is a global minimum Animated A word problem (continued)

The Single Variable Problem on the Closed Interval define a stationary point as any point x’ such that find This looks too easy. There must be more to it.

Our very next example problem I bet that can be factored!

xf(x)f”(x) local/global min local max 4 42local/global min 6 20global max Our very next example problem (continued)

Another example For a particular government 12-year health care program for the elderly, the number of people in thousands receiving direct benefits as a function of the number of years, t, after the start of the program is given by For what value of t does the maximum number receive benefits? My health benefits will expire soon!

The Answer t = 0 (n=0), t= 4 (53/3), t = 8 (n = 42.67), t = 12 (n = 96) local max local min f n

Multi -Variable Optimization i.e. going from one to two

2-Variable Function with a Maximum z = f(x,y)

2-Variable Function with both Maxima and Minima z = f(x,y)

2-Variable Function with a Saddle Point z = f(x,y)

The General Problem necessary conditions: sufficient conditions: f(x 1,x 2,…,x n ) is convex for a minimum f(x 1,x 2,…,x n ) is concave for a maximum

Recall Taylor’s Series Approximation in 2-variables? I sure do!

2-Variable Problem sufficient conditions: and saddle point

A 2-variable example Max f(x,y) = 100 – (x – 4) 2 – 2 (y – 2) 2 necessary conditions: sufficient conditions: concave function

f(x,y) = 2x 3 – 2x 2 – 10x + y 3 – 3y (3x – 5) (x + 1) = 0 x = 5/3, -1 3y (y – 2) = 0 y = 0, 2 Not Another Example? A Cubic no less! …and it has four solutions! (x*,y*) = (5/3,0), (5/3, 2), (-1,0), (-1,2)

xy 5/3016-6saddle pt 5/32166local min local max saddle pt Not Another Example (continued)

A special container must be constructed to transport 40 cubic yards of material. The transportation cost is one dollar per round trip. It costs $10 per square yard to construct the sides, $30 per square yard to construct the bottom of the container and $20 dollars to construct the ends. It has no top and no salvage value. It must be rectangular in shape and only one can be made. Find the dimensions which will minimize the construction and transportation costs. I need a box, quick! A Logistics Design Problem

The Formulation let x = the length, y = the width, and z = the height then volume = xyz and transportation cost = $1 [40 / (xyz)] cost of bottom = $30 xy cost of sides = $10 xz cost of ends = $20 yz

The necessary conditions

Is the function convex? I see, all 9 2 nd partials must be analyzed. They show us how to do that in MSC 523. I am going to sign up today!

Power Plant Location citynbr lines Cincinnati 7 Dayton 4 Columbus 10 Toledo 4 Cleveland 12 Youngstown 3 DPL desires to construct a nuclear power plant in Ohio that will provide electrical power to the cities shown below. Also shown are the number of transmission lines required to meet each city’s demands for additional electricity. The problem is to locate the power plant so that the total transmission loss is minimized.

The Great State of Ohio x y (11,35) (3,9) (6,15) (15,18) (31,30) (24,34)

Euclidean Distances x y (a,b) (x,y) (x – a) (y – b) h h 2 = (x – a) 2 + (y – b) 2 Animated

The Formulation let x = the x-coordinate of power station y = the y-coordinate of power station (x i,y i ) = coordinate of i th city w i = number of transmission lines to i th city Euclidean distance squared

The Solution – necessary conditions

The Solution – sufficient conditions convex function Why it is everywhere convex. Truly you have found the global minimum.

Power Plant Location city nbr lines (w i )locationw i * x i w i * y i Cincinnati 73, Dayton 46, Columbus 1015, Toledo 411, Cleveland 1224, Youngstown 331, Totals x* = 620 / 40 = 15.5y* = 941 / 40 =

The Great State of Ohio x y Waldo, Ohio Route 23 Marion County (15.5,23.5)

Much ado about Waldo Waldo is a village located in Marion County, Ohio. As of the 2000 census, the village had a total population of 332. Waldo is known in the central Ohio region for excellent fried baloney sandwiches from the G&R Tavern. Waldo is also home to several vineyards. According to the United States Census Bureau, the village has a total area of 1.7 km² (0.6 mi²). 1.7 km² (0.6 mi²) of it is land and none of the area is covered with water.