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Introduction to Optimization. Anjela Govan North Carolina State University SAMSI NDHS Undergraduate workshop 2006. What is Optimization?. Optimization is the mathematical discipline which is concerned with finding the maxima and minima of functions, possibly subject to constraints. .

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introduction to optimization

Introduction to Optimization

Anjela Govan

North Carolina State University

SAMSI NDHS Undergraduate workshop 2006

what is optimization
What is Optimization?
  • Optimization is the mathematical discipline which is concerned with finding the maxima and minima of functions, possibly subject to constraints.
where would we use optimization
Where would we use optimization?
  • Architecture
  • Nutrition
  • Electrical circuits
  • Economics
  • Transportation
  • etc.
what do we optimize
What do we optimize?
  • A real function of n variables
  • with or without constrains
lets optimize
Lets Optimize
  • Suppose we want to find the minimum of the function
review max min for r 2
Review max-min for R2
  • What is special about a local max or a local min of a function f (x)?

at local max or local min f’(x)=0

f”(x) > 0 if local min

f”(x) < 0 if local max

review max min for r 310
Review max-min for R3
  • Second Derivative Test
  • Local min, local max, saddle point
  • Gradient of f – vector (df/dx, df/dy, df/dz)

direction of fastest increase of f

  • Global min/max vs. local min/max
gradient descent method examples
Gradient Descent Method Examples
  • Minimize function
  • Minimize function
gradient descent method examples12
Gradient Descent Method Examples
  • Use function gd(alpha,x0)
    • Does gd.m converge to a local min? Is there a difference if  > 0 vs.  < 0?
    • How many iterations does it take to converge to a local min? How do starting points x0 affect number of iterations?
  • Use function gd2(x0)
    • Does gd2.m converge to a local min?
    • How do starting points x0 affect number of iterations and the location of a local minimum?
how good are the optimization methods
How good are the optimization methods?
  • Starting point
  • Convergence to global min/max.
  • Classes of nice optimization problems

Example: f(x,y) = 0.5(x2+y2),  > 0

Every local min is global min.

other optimization methods
Other optimization methods
  • Non smooth, non differentiable surfaces 

can not compute the gradient of f 

can not use Gradient Method

  • Nelder-Mead Method
  • Others
convex hull
Convex Hull
  • A set C is convex if every point on the line segment connecting x and y is in C.
  • The convex hull for a set of points X is the minimal convex set containing X.
simplex
Simplex
  • A simplex or n-simplex is the convex hull of a set of (n +1) . A simplex is an n-dimensional analogue of a triangle.
  • Example:
    • a 1-simplex is a line segment
    • a 2-simplex is a triangle
    • a 3-simplex is a tetrahedron
    • a 4-simplex is a pentatope
nelder mead method
Nelder-Mead Method
  • n = number of variables, n+1 points
  • form simplex using these points; convex hull
  • move in direction away from the worst of these points: reflect, expand, contract, shrink
  • Example:
    • 2 variables  3 points  simplex is triangle
    • 3 variables  4 points  simplex is tetrahedron
fminsearch function
fminsearch function
  • parameters: q =[C,K]
  • cost function:
  • Minimize cost function

[q,cost]=

fninsearch(@cost_beam, q0,[],time,y_tilde)

our optimization problem
Our optimization problem
  • In our problem
    • Our function:
    • cost function “lives” in R3
    • 2 parameters C and K, n=2
    • Simplex is a triangle