Optimization

	What is Optimization?
		Finding the best solution under a given set of conditions
		Finding the highest or lowest values of a function
	Example:  Minimize the weight of a part
	Example:  Minimize the cost of a part
	Example:  Maximize the force transmission of a device
	Example:  Find the shortest route to a destination

How are these kinds of problems set up mathematically?

	We need some kind of function to describe what we are trying to achieve
		Minimize weight
			f = Volume*r
			density is a material property—if the material is selected, then r will not be changed
			Volume = f(L,h,d), Length, height, depth
		Certainly we could make L, h, and d very small to make the weight small, or minimum

Objective function

	The function which represents what we are trying to optimize is called the objective function
		F(x)= L*h*w*r
		x is a vector
			x1 = L
			x2 = h
			x3 = w
		x1, x2, and x3 (L,h,w) are called design variables

Why not make L,h,and w very small?

	We could make L, h, and w, very small to reduce weight—this would be called the trivial solution to the problem
	Most applications would require a “window” of values on parameters
		For example, L must not exceed Lmax and must not be less than Lmin,
			Lmin < L < Lmax

Constraints

	The window of values we just placed on L are called constraints
	Most likely, h, and w would have constraints imposed as well
	There are equality constraints, and inequality constraints.

Constraints

	Constraints bound the solution domain
	Within this domain there may be many optima, local optima, or there may be one solution that is the absolute best —the global solution
	Depending on what kind of technique we use to find the optimum value of our function, we may or may not find the global solution

Let’s look at the cable design

	This problem is called a “single variable” problem, because there is only one design variable.
		we found Tension as function of distance along the pole
	We know from calculus that we if we find the place in a domain where the first derivative of a function is zero, and the second derivative is positive there is a minimum of the function

The objective function

	Minimize T =

Constraints

	Subject to
		x<8.0
		x>0

The first derivative

	dT/dx =

The second derivative

	d2T/dx2=

How do we handle multivariable problems?
(no constraints)

	The mathematical necessity of the “first derivative” being zero becomes that the gradient of the function must be zero
	How is a gradient determined and what is its interpretation?

Gradient

What is the gradient

	The gradient is a “line” that points us to an optimum

What is the analogy in n-D to the 2^nd derivative condition?

	It is the Hessian matrix and whether or not the determinant of the Hessian matrix is positive or negative

What about multivariable problems with constraints?

	The necessary condition for identifying an optimal value becomes:
		The gradient of the “Lagrangian” function must become = 0
		L(x,l) = f(x) + l*g(x)
		l is called the Lagrangian multiplier
		f(x) is the function we are optimizing
		g(x) are the constraint equations

Gradient of the Lagrangian

	L(x1,x2,l) = f(x1,x2)+l*g(x1,x2)