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Fin500J: Mathematical Foundations in Finance Topic 5: Numerical Methods for Optimization Philip H. Dybvig

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

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. Unconstrained Optimization with Multiple Variables in Matlab (cont.) • We use the function fminunc to solve unconstrained optimization problem “objfun” Fall 2010 Olin Business School

  10. Quasi-Newton Method is an Algorithm used in function fminunc Fall 2010 Olin Business School

  11. Recall the Algorithm of Newton Method • Newton’s method may fail if Hessian is not positive • definite Fall 2010 Olin Business School

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. Linear Programming in Matlab (Example) • File->export to workspace • can export the results including lambda,etc. Fall 2010 Olin Business School

  19. 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

  20. Quadratic Programming in Matlab (Example: Portfolio Optimization) Fall 2010 Olin Business School

  21. 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

  22. Nonlinear Programming in Matlab( Constrained Nonlinear Optimization) Formulation Fall 2010 Olin Business School

  23. 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

  24. Nonlinear Programming in Matlab (Example) Fall 2010 Olin Business School

  25. 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

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