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Fin500J: Mathematical Foundations in Finance Topic 5: Numerical Methods for Optimization Philip H. Dybvig Reference: Optimization Toolbox User’s Guide in Matlab, 2008 by the MathWorks, Inc. Slides designed by Yajun Wang. Recall, with optimization, we are seeking f '(x) = 0. f '(x) = 0

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Fin500J: Mathematical Foundations in Finance

Topic 5: Numerical Methods for Optimization

Philip H. Dybvig

Reference: Optimization Toolbox User’s Guide in Matlab, 2008 by the MathWorks, Inc.

Slides designed by Yajun Wang

Fall 2010 Olin Business School

slide2

Recall, with optimization, we are seeking f '(x) = 0

f '(x) = 0

f "(x)< 0

f '(x) = 0

f "(x)>0

Fall 2010 Olin Business School

one dimension unconstrained optimization example
One Dimension Unconstrained Optimization (Example)

Find the maximum of

To solve the root problem for

and the second condition is satisfied at the root

Fall 2010 Olin Business School

one dimension unconstrained optimization example4
One Dimension Unconstrained Optimization (Example)
  • We can solve f '(x) =0 by Bisection using initial interval [1,2], Newton’s with initial point 1.2 or Secant method with initial points 1.2 and 2 presented in Topic 3.
  • We can also solve it in Matlab.
  • >> f=@(x) 2*cos(x)-1/5*x;
  • >> fzero(f,[1,2])
  • ans =1.4276

Fall 2010 Olin Business School

objectives using optimization toolbox in matlab to
Objectives : Using Optimization Toolbox in Matlab to
  • Solve unconstrained optimization with multiple variables
  • Solve linear programming problem
  • Solve quadratic programming problem (for example: optimal portfolio)
  • Solve nonlinear optimization with constraints
  • Mostly, we will focus on minimization in this topic, max f(x) is equivalent to min –f(x)

Fall 2010 Olin Business School

linear programming quadratic programming nonlinear programming
Linear Programming/Quadratic Programming/Nonlinear Programming
  • If f(x) and the constraints are linear, we have linear programming
  • If f(x) is quadratic, and the constraints are linear, we have quadratic programming
  • If f(x) in not linear or quadratic, and/or the constraints are nonlinear, we have nonlinear programming

Fall 2010 Olin Business School

recall the optimality conditions for multiple variables
Recall the Optimality Conditions for Multiple Variables
  • Unconstrained Minimization Problem:
  • min f(x1, x2,…xn)
  • Optimality Condition: x* is a local minimum if
  • Example:
  • What values of x make

Fall 2010 Olin Business School

unconstrained optimization with multiple variables in matlab
Unconstrained Optimization with Multiple Variables in Matlab
  • Step 1: Write an M-file objfun.m and save under the work path of matlab
  • function f=objfun(x)
  • f=exp(x(1)+x(2)-1)+exp(x(1)-x(2)-1)+exp(-x(1)-1);
  • Step 2: >>optimtool in the commend window to open the optimization toolbox

Fall 2010 Olin Business School

unconstrained optimization with multiple variables in matlab cont
Unconstrained Optimization with Multiple Variables in Matlab (cont.)
  • We use the function fminunc to solve unconstrained optimization problem “objfun”

Fall 2010 Olin Business School

quasi newton method is an algorithm used in function fminunc
Quasi-Newton Method is an Algorithm used in function fminunc

Fall 2010 Olin Business School

recall the algorithm of newton method
Recall the Algorithm of Newton Method
  • Newton’s method may fail if Hessian is not positive
  • definite

Fall 2010 Olin Business School

quasi newton methods replace the hessian with some positive definite matrix h
Quasi-Newton Methods Replace the Hessian with some Positive Definite Matrix H
  • The function “fminunc” uses BFGS (Broyden, Fletcher, Goldfarb and Shanno) Hessian Update in the Quasi-Newton algorithm. The formula given by BFGS is

Fall 2010 Olin Business School

linear programming
Linear Programming
  • Both the objective function and the constraints are linear
  • Example: maximizing profit or minimizing cost
  • Objective function
  • Max or Min Z = c1x1 +c2x2 +…..cnxn
  • where cj = payoff of each unit of the jth activity
  • xj = magnitude of the jth activity
  • The constraints can be represented by
  • ai1x1 +ai2x2+…..ainxn  bi
  • where aij = amount of the ith resource that is consumed for each
  • unit of the jth activity, bi = amount of the ith resource available
  • Finally, we add the constraint that all activities have a positive value, xi 0

Fall 2010 Olin Business School

example
Example

x1 = amount of regular and

x2 = amount of premium

Total Profit = 150 x1 + 175 x2

Maximize Z = 150 x1 + 175 x2

Objective function

7x1 + 11x2 77 (material constraint)

10x1 + 8x2  120 (time constraint)

x1  9 (storage constraint)

x2  6 (storage constraint)

x1,x2  0 (positivity constraint)

Fall 2010 Olin Business School

graphical solution
Graphical Solution

(1) 7x1 + 11x2 77

→x2  -7/11 x1 +7

(2) 10x1 + 8x2  120

→ x2  -5/4x1 + 15

(3) x1  9

(4) x2  6

(5) x1,x2  0

Fall 2010 Olin Business School

graphical solution16
Graphical Solution

Now we need to add the objective function to the plot. Start with

Z = 0 (0=150x1 + 175x2)

and

Z = 500 (500=150x1 + 175x2)

Z=1200

Z=1550

Still in feasible region

x1*= 9

x2*  1.5

Fall 2010 Olin Business School

linear programming in matlab
Linear Programming in Matlab

Example:

  • Step 1: >>optimtool in the commend window to open the optimization
  • toolbox
  • Step 2: Define matrices A, Aeq and the vectors f, b, lb, ub

Fall 2010 Olin Business School

linear programming in matlab example
Linear Programming in Matlab (Example)
  • File->export to workspace
  • can export the results including lambda,etc.

Fall 2010 Olin Business School

quadratic programming in matlab
Quadratic Programming in Matlab
  • Step 1: >>optimtool in the commend window to open the optimization
  • toolbox
  • Step 2: Define matrices H,A and the vectors f, b

Fall 2010 Olin Business School

quadratic programming in matlab example portfolio optimization
Quadratic Programming in Matlab (Example: Portfolio Optimization)

Fall 2010 Olin Business School

quadratic programming in matlab quadprog
Quadratic Programming in Matlab (quadprog)

H=[0.017087987 0.003298885 0.001224849; 0.003298885 0.005900944 0.004488271; 0.001224849 0.004488271 0.063000818]

f=[0; 0; 0]

A=[1 1 1; -0.026 -0.008 -0.074; -1 0 0; 0 -1 0; 0 0 -1]

b=[1000; -50; 0; 0; 0]

The function ‘quadprog ’ uses an active set strategy. The first phase involves the calculation of a feasible point. The second phase involves the generation of an iterative sequece of feasible points that converge to the solution.

Fall 2010 Olin Business School

nonlinear programming in matlab constrained nonlinear optimization
Nonlinear Programming in Matlab( Constrained Nonlinear Optimization)

Formulation

Fall 2010 Olin Business School

nonlinear programming in matlab example
Nonlinear Programming in Matlab (Example)

Find x that solves

  • Step 1: Write an M-file objfunc.m for the objective function.
  • function f=objfunc(x)
  • f=exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1);
  • Step 2: Write an M-file confun.m for the constraints.
  • function [c, ceq]=confun(x)
  • %Nonlinear inequality constraints
  • c=[1.5+x(1)*x(2)-x(1)-x(2); -x(1)*x(2)-10];
  • %Nonlinear equality constraints
  • ceq=[];
  • Step 3: >>optimtool to open the optimization toolbox

Fall 2010 Olin Business School

nonlinear programming in matlab example24
Nonlinear Programming in Matlab (Example)

Fall 2010 Olin Business School

sequential quadratic programming is an algorithm used in function fmincon basic idea
Sequential Quadratic Programming is an Algorithm Used in Function ‘fmincon’ (Basic Idea)
  • The basic idea is analogous to Newton’s method for unconstrained optimization.
  • In unconstrained optimization, only the objective function must be approximated, in the NLP, both the objective and the constraint must be modeled.
  • An sequential quadratic programming method uses a quadratic for the objective and a linear model of the constraint ( i.e., a quadratic program at each iteration)

Fall 2010 Olin Business School