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Optimization Methods in Engineering: Theory and Applications

Explore different optimization criteria and methods such as steepest descent, Newton's method, and Powell's conjugate direction. Understand how to find local minima with examples and applications in engineering problems.

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Optimization Methods in Engineering: Theory and Applications

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  1. Chapter 3 UNCONSTRAINED OPTIMIZATION Prof. S.S. Jang Department of Chemical Engineering National Tsing-Hua Univeristy

  2. Contents • Optimality Criteria • Direct Search Methods • Nelder and Mead (Simplex Search) • Hook and Jeeves(Pattern Search) • Powell’s Method (The Conjugated Direction Search) • Gradient Based Methods • Cauchy’s method (the Steepest Decent) • Newton’s Method • Conjugate Gradient Method (The Fletcher and Reese Method) • Quasi-Newton Method – Variable Metric Method

  3. 3-1 The Optimality Criteria • Given a function (objective function) f(x), where and let • Stationary Condition: • Sufficient minimum criteria positive definite • Sufficient maximum criteria negative definite

  4. Definiteness • A real quadratic for Q=xTCx and its symmetric matrix C=[cjk] are said to be positive definite (p.d.) if Q>0 for all x≠0. A necessary and sufficient condition for p.d. is that all the determinants: are positive, C is negative definitive (n.d.) if –C is p.d.

  5. Example : • x=linspace(-3,3);y=x; • >> for i=1:100 • for j=1:100 • z(i,j)=2*x(i)*x(i)+4*x(i)*y(j)-10*x(i)*y(j)^3+y(j)^2; • end • end • mesh(x,y,z) • contour(x,y,z,100)

  6. Example- Continued (mesh and contour)

  7. Example-Continued At Indefinite, x* is a non-optimum point.

  8. Mesh and Contour – Himmelblau’s function : f(x)=(x12+x2-11)2+(x1+x22-7)2 • Optimality Criteria may find a local minimum in this case • x=linspace(-5,5);y=x; • for i=1:100 • for j=1:100 • z(i,j)=(x(i)^2+y(j)-11)^2+(x(i)+y(j)^2-7)^2; • end • end • mesh(x,y,z) • contour(x,y,z,80)

  9. Mesh and Contour - Continued

  10. Example: f(x)=x12+25x22

  11. Rosenbrock’s Banana Functionf(x,y)=100(y-x2)2+(1-x)2

  12. Example: f(x)=2x13+4x1x23-10x1x2+x23Steepest decent search from the initial point at (2,2); we have d=[6x12+4x23-10x2, 12x1x22-10x1+3x22]’=

  13. Example: Curve Fitting • From the theoretical considerations, it is believe that the dependent variable y is related to variable x via a two-parameter function: The parameters k1 and k2 are to be determined by a least square of the following experimental data: xy 1.0 1.05 2.0 1.25 3.0 1.55 4.0 1.59

  14. Problem Formulation and mesh, contour

  15. The Importance of One-dimensional Problem - The Iterative Optimization Procedure Consider a objective function Min f(X)= x12+x22, with an initial point X0=(-4,-1) and a direction (1,0), what is the optimum at this direction, i.e. X1=X0 +*(1,0). This is a one -dimensional search for .

  16. 3-2 Steepest Decent Approach – Cauchy’s Method • A direction d=[x1,x2,..,xn]’ is a n-dimensional vector, a line search is an approach such that from a based point x0, find *, for all , x0+ *d is an optimum. • Theorem: the direction d=-[f/x1,f/x2, .., f/xn]’ is the steepest decent direction. Proof: Consider a two dimensional system: f(x1,x2), the movement ds2=dx12+dx22, and f= (f/x1)dx1+ (f/x2)dx2= (f/x1)dx1+ (f/x2)(ds2-dx12)1/2. Then to maximize the change f, we have d(f)/dx1=0. It can be shown that dx1/ds= (f/x1)/((f/x1)2 + (f/x2)2)1/2, dx2/ds= (f/x2)/((f/x1)2+ (f/x2)2)1/2

  17. Example: f(x)=2x13+4x1x23-10x1x2+x23Steepest decent search from the initial point at (2,2); we have d=[6x12+4x23-10x2, 12x1x22-10x1+3x22]’=

  18. Quadratic Function Properties Property #1: Optimal linear search  for a quadratic function:

  19. Quadratic Function Properties- Continued Property #2 fk+1 is orthogonal to sk for a quadratic function

  20. 3-2 Newton’s Method – Best Convergence approaching the optimum

  21. 3-2 Newton’s Method – Best Convergence approaching the optimum

  22. Comparison of Newton’s Method and Cauchy’s Method

  23. Remarks • Newton’s method is much more efficient than Steep Decent approach. Especially, the starting point is getting close to the optimum. • There is a requirement for the positive definite of the hessian in implementing Newton’s method. Otherwise, the search will be diverse. • The evaluation of hessian requires second derivative of the objective function, thus, the number of function evaluation will be drastically increased if a numerical derivative is used. • Steepest decent is very inefficient around the optimum, but very efficient at the very beginning of the search.

  24. Conclusion • Cauchy’s method is very seldom to use because of its inefficiency around the optimum. • Newton’s method is also very seldom to use because of the requirement of second derivative. • A useful approach should be somewhere in between.

  25. 3-3 Method of Powell’s Conjugate Direction • Definition: Let u and v denote two vectors in En. They are said to be orthogonal, if their scalar product equals to zero, i.e. uv=0. Similarly, u and v are said to be mutually conjugative with respect to a matrix A, if u and Av are mutually orthogonal, i.e. uAv=0. • Theorem: If conjugate directions to the hessian are used to any quadratic function of n variables that has a minimum, can be minimized in n steps.

  26. Powell’s Conjugated Direction Method-Continued • Corollary: Generation of conjugated vectors-Extended Parallel Space Property • Given a direction d and two initial point x(1), x(2), we assume:

  27. Powell’s Algorithm • Step 1: Define x0, and set N linearly indep. Directions say: • s(i)=e(i) • Step 2: Perfrom N+1 one directional searches along with s(N), s(1),,s(N) • Step 3: Form the conjugate direction d using the extended parallel space property. • Step 4: Set • Go to Step 2

  28. Powell’s Algorithm (MATLAB) • function opt=powell_n_dim(x0,n,eps,tol,opt_func) • xx=x0;obj_old=feval(opt_func,x0);s=zeros(n,n);obj=obj_old;df=100;absdx=100; • for i=1:n;s(i,i)=1;end • while df>eps/absdx>tol • x_old=xx; • for i=1:n+1 • if(i==n+1) j=1; else j=i; end • ss=s(:,j); • alopt=one_dim_pw(xx,ss',opt_func); • xx=xx+alopt*ss'; • if(i==1) • y1=xx; • end • if(i==n+1) • y2=xx; • end • end • d=y2-y1; • nn=norm(d,'fro'); • for i=1:n • d(i)=d(i)/nn; • end • dd=d'; • alopt=one_dim_pw(xx,dd',opt_func); • xx=xx+alopt*d • dx=xx-x_old; • plot(xx(1),xx(2),'ro') • absdx=norm(dx,'fro'); • obj=feval(opt_func,xx); • df=abs(obj-obj_old); • obj_old=obj; • for i=1:n-1 • s(:,i)=s(:,i+1); • end • s(:,n)=dd; • end • opt=xx

  29. Example: f(x)=x12+25x22

  30. Example 2: Rosenbrock’s function

  31. Example: fittings • x0=[3 3];opt=fminsearch('fittings',x0) • opt = • 2.0884 1.0623

  32. Fittings

  33. Function Fminsearch • >> help fminsearch • FMINSEARCH Multidimensional unconstrained nonlinear minimization (Nelder-Mead). • X = FMINSEARCH(FUN,X0) starts at X0 and finds a local minimizer X of the • function FUN. FUN accepts input X and returns a scalar function value • F evaluated at X. X0 can be a scalar, vector or matrix. • X = FMINSEARCH(FUN,X0,OPTIONS) minimizes with the default optimization • parameters replaced by values in the structure OPTIONS, created • with the OPTIMSET function. See OPTIMSET for details. FMINSEARCH uses • these options: Display, TolX, TolFun, MaxFunEvals, and MaxIter. • X = FMINSEARCH(FUN,X0,OPTIONS,P1,P2,...) provides for additional • arguments which are passed to the objective function, F=feval(FUN,X,P1,P2,...). • Pass an empty matrix for OPTIONS to use the default values. • (Use OPTIONS = [] as a place holder if no options are set.) • [X,FVAL]= FMINSEARCH(...) returns the value of the objective function, • described in FUN, at X. • [X,FVAL,EXITFLAG] = FMINSEARCH(...) returns a string EXITFLAG that • describes the exit condition of FMINSEARCH. • If EXITFLAG is: • 1 then FMINSEARCH converged with a solution X. • 0 then the maximum number of iterations was reached. • [X,FVAL,EXITFLAG,OUTPUT] = FMINSEARCH(...) returns a structure • OUTPUT with the number of iterations taken in OUTPUT.iterations. • Examples • FUN can be specified using @: • X = fminsearch(@sin,3) • finds a minimum of the SIN function near 3. • In this case, SIN is a function that returns a scalar function value • SIN evaluated at X. • FUN can also be an inline object: • X = fminsearch(inline('norm(x)'),[1;2;3]) • returns a minimum near [0;0;0]. • FMINSEARCH uses the Nelder-Mead simplex (direct search) method. • See also OPTIMSET, FMINBND, @, INLINE.

  34. P2 P1 Q … D L Pipeline design

  35. 3-4 Conjugated Gradient Method Fletcher-Reeves’s Method (FRP) • Given s1=-f1, s2=-f2+s1 • It can be shown that if • where, s2 is started at the optimum of the line search along the direction s1. And fa+bTxk+1/2xkTHxk, then s2 is conjugated to s1 with respect to H • Most general iteration:

  36. Example: (The Fletcher-Reeves Method for Rosenbrock’s function )

  37. 3-5 Quasi-Newton’s Method (Variable Matrix Method) • The variable matric approach is using an approximate Hessian matrix without taking second derivative of the objective function, the Newton’s direction can be approximated by: xk+1=xk-kAkfk • As Ak=I Cauchy’s direction • As Ak=H-1 Newton’s direction • In the variable matric approach, Ak can be updated to approximate the inverse of the Hessian by iterative approaches

  38. 3-5 Quasi-Newton’s Method (Variable Matrix Method)- Continued • Given the objective function f(x)a+bx+ 1/2xTHx, then f b+Hx, and fk+1- f kH(xk+1- xk ), or xk+1- xk= H-1gk where gk =fk+1- f k • Let Ak+1=H-1 = Ak+ Ak  xk+1- xk= (Ak+ Ak )gk •  Ak gk = xk- Ak gk • Finally,

  39. Quasi-Newton’s Method – Continued • Choose y= xk, z= Ak gk (The Davidon-Flecher-Powell’s method • Broyden-Flecher-Shanno’s Method • Summary: • In the first run, given initial conditions and convergence criteria, and evaluate the gradient • First direction is on the negative of the gradient, Ak =I • In the iteration phase, evaluate the new gradient and find gk, xk, and solve Ak using the equation in the previous slide • The new direction is on sk+1=Ak+1f k+1 • Check convergence, and continue the iteration phase

  40. Example: (The BFS Method for Rosenbrock’s function )

  41. Trial Point Base Point Heuristic Approaches (1) s-2 Simplex Approach • Main Idea • Rules • 1: Straddled: Selected the “worse” vertex generated in the previous iteration, then choose instead the vertex with the next highest function value • 2. Cycling: In a given vertex unchanged more than M iterations, reduce the size by some factor • 3. Termination: The search is terminated when the simlex gets small enough.

  42. Example: Rosenbrock Function

  43. z xnew xnew xnew xc xc xc X(h) X(h) X(h) xnew z z xc X(h) Simplex Approach: Nelder and Mead xnew

  44. Comparisons of Different Approach for Solving Rosenbrock function

  45. Heuristic Approach (Reduction Sampling with Random Search) • Step 1: Given initial sampling radius and initial center of the generate a random N points within the region. • Step 2: Get the best of the N sampling as the new center of the circle. • Step 3: Reduce the radius of the search circle, if the radius is not small enough, go back to step 2, otherwise stop.

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