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  1. Introduction To Non–linear Optimization PART I

  2. Optimization Tree Figure 1: Optimization tree.

  3. What is Optimization? • Optimization is an iterative process by which a desired solution • (max/min) of the problem can be found while satisfying all its • constraint or bounded conditions. Figure 2: Optimum solution is found while satisfying its constraint (derivative must be zero at optimum). • Optimization problem could be linear or non-linear. • Non –linear optimization is accomplished by numerical ‘Search Methods’. • Search methods are used iteratively before a solution is achieved. • The search procedure is termed as algorithm.

  4. What is Optimization?(Cont.) • Linear problem – solved by Simplex or Graphical methods. • The solution of the linear problem lies on boundaries of the feasible region. Figure 3: Solution of linear problem Figure 4: Three dimensional solution of non-linear problem • Non-linear problem solution lies within and on the boundaries of the feasible region.

  5. Maximize X1 + 1.5 X2 Subject to:X1 + X2 ≤ 1500.25 X1 + 0.5 X2 ≤ 50X1 ≥ 50X2 ≥ 25X1 ≥0, X2 ≥0 Fundamentals of Non-Linear Optimization • Single Objective function f(x) • Maximization • Minimization • Design Variables, xi , i=0,1,2,3….. • Constraints • Inequality • Equality Figure 5: Example of design variables and constraints used in non-linear optimization. • Optimal points • Local minima/maxima points: A point or Solution x* is at local point if there is no other x in its Neighborhood less than x* • Global minima/maxima points: A point or Solution x** is at global point if there is no other x in entire search space less than x**

  6. Fundamentals of Non-Linear Optimization (Cont.) Figure 7: Local point is equal to global point if the function is convex. Figure 6: Global versus local optimization.

  7. Fundamentals of Non-Linear Optimization (Cont.) • Function f is convex if f(Xa) is less than value of the corresponding • point joining f(X1) and f(X2). • Convexity condition – Hessian 2nd order derivative) matrix of • function f must be positive semi definite ( eigen values +ve or zero). Figure 8: Convex and nonconvex set Figure 9: Convex function

  8. Mathematical Background • Slop or gradient of the objective function f – represent the • direction in which the function will decrease/increase most rapidly • Taylor series expansion • Jacobian – matrix of gradient of f with respect to several variables

  9. Mathematical Background (Cont.) • First order Condition (FOC) • Hessian – Second derivative of f of several variables • Second order condition (SOC) • Eigen values of H(X*) are all positive • Determinants of all lower order of H(X*) are +ve

  10. Optimization Algorithm • Deterministic - specific rules to move from one iteration to next , • gradient, Hessian • Stochastic – probalistic rules are used for subsequent iteration • Optimal Design – Engineering Design based on optimization algorithm • Lagrangian method – sum of objective function and linear combination of the constraints.

  11. Optimization Methods • Deterministic • Direct Search – Use Objective function values to locate minimum • Gradient Based – first or second order of objective function. • Minimization objective function f(x) is used with –ve sign – • f(x) for maximization problem. • Single Variable • Newton – Raphson is Gradient based technique (FOC) • Golden Search – step size reducing iterative method • Multivariable Techniques ( Make use of Single variable Techniques • specially Golden Section) • Unconstrained Optimization • a.) Powell Method – Quadratic (degree 2) objective function polynomial is • non-gradient based. • b.) Gradient Based – Steepest Descent (FOC) or Least Square minimum • (LMS) • c.) Hessian Based -Conjugate Gradient (FOC) and BFGS (SOC)

  12. Optimization Methods …Constrained • Constrained Optimization • a.) Indirect approach – by transforming into unconstrained • problem. • b.) Exterior Penalty Function (EPF) and Augmented Lagrange • Multiplier • c.) Direct Method Sequential Linear Programming (SLP), SQP and • Steepest Generalized Reduced Gradient Method (GRG) • Figure 10: Descent Gradient or LMS

  13. Optimization Methods (Cont.) • Global Optimization – Stochastic techniques • Simulated Annealing (SA) method – minimum energy principle of cooling metal crystalline structure • Genetic Algorithm (GA) – Survival of the fittest • principle based upon evolutionary theory

  14. Optimization Methods (Example) Multivariable Gradient based optimization J is the cost function to be minimized in two dimension The contours of the J paraboloid shrinks as it is decrease function retval = Example6_1(x) % example 6.1 retval = 3 + (x(1) - 1.5*x(2))^2 + (x(2) - 2)^2; >> SteepestDescent('Example6_1', [0.5 0.5], 20, 0.0001, 0, 1, 20) Where [0.5 0.5] -initial guess value 20 -No. of iteration 0.001 -Golden search tol. 0 -initial step size 1 -step interval 20 -scanning step >> ans 2.7585 1.8960 Figure 11: Multivariable Gradient based optimization Figure 12: Steepest Descent

  15. MATLAB Optimization Toolbox PART II

  16. Presentation Outline • Introduction • Function Optimization • Optimization Toolbox • Routines / Algorithms available • Minimization Problems • Unconstrained • Constrained • Example • The Algorithm Description • Multiobjective Optimization • Optimal PID Control Example

  17. Function Optimization • Optimization concerns the minimization or maximization of functions • Standard Optimization Problem: Subject to: Equality Constraints Inequality Constraints Side Constraints Where: is the objective function, which measure and evaluate the performance of a system. In a standard problem, we are minimizing the function. For maximization, it is equivalent to minimization of the –ve of the objective function. is a column vector of design variables, which can affect the performance of the system.

  18. Function Optimization (Cont.) • Constraints – Limitation to the design space. Can be linear or nonlinear, explicit or implicit functions Equality Constraints Inequality Constraints Most algorithm require less than!!! Side Constraints

  19. Optimization Toolbox • Is a collection of functions that extend the capability of MATLAB. • The toolbox includes routines for: • Unconstrained optimization • Constrained nonlinear optimization, including goal attainment • problems, minimax problems, and semi-infinite minimization • problems • Quadratic and linear programming • Nonlinear least squares and curve fitting • Nonlinear systems of equations solving • Constrained linear least squares • Specialized algorithms for large scale problems

  20. Minimization Algorithm

  21. Minimization Algorithm (Cont.)

  22. Equation Solving Algorithms

  23. Least-Squares Algorithms

  24. Implementing Opt. Toolbox • Most of these optimization routines require the definition of an M- file containing the function, f, to be minimized. • Maximization is achieved by supplying the routines with –f. • Optimization options passed to the routines change optimization parameters. • Default optimization parameters can be changed through an options structure.

  25. Unconstrained Minimization • Consider the problem of finding a set of values [x1 x2]T that solves • Steps: • Create an M-file that returns the function value (Objective • Function). Call it objfun.m • Then, invoke the unconstrained minimization routine. Use fminunc

  26. Step 1 – Obj. Function function f = objfun(x) f=exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1); Objective function

  27. Step 2 – Invoke Routine Starting with a guess x0 = [-1,1]; options = optimset(‘LargeScale’,’off’); [xmin,feval,exitflag,output]= fminunc(‘objfun’,x0,options); Optimization parameters settings Output arguments Input arguments

  28. Results xmin = 0.5000 -1.0000 feval = 1.3028e-010 exitflag = 1 output = iterations: 7 funcCount: 40 stepsize: 1 firstorderopt: 8.1998e-004 algorithm: 'medium-scale: Quasi-Newton line search' Minimum point of design variables Objective function value Exitflag tells if the algorithm is converged. If exitflag > 0, then local minimum is found Some other information

  29. More on fminunc– Input [xmin,feval,exitflag,output,grad,hessian]= fminunc(fun,x0,options,P1,P2,…) fun : Return a function of objective function. x0 : Starts with an initial guess. The guess must be a vector of size of number of design variables. Option : To set some of the optimization parameters. (More after few slides) P1,P2,… : To pass additional parameters.

  30. More on fminunc– Output [xmin,feval,exitflag,output,grad,hessian]= fminunc(fun,x0,options,P1,P2,…) • xmin : Vector of the minimum point (optimal point). The size is the number of design variables. • feval : The objective function value of at the optimal point. • exitflag : A value shows whether the optimization routine is terminated successfully. (converged if >0) • Output : This structure gives more details about the optimization • grad : The gradient value at the optimal point. • hessian : The hessian value of at the optimal point

  31. Options Setting – optimset Options = optimset(‘param1’,value1, ‘param2’,value2,…) • The routines in Optimization Toolbox has a set of default optimization parameters. • However, the toolbox allows you to alter some of those parameters, for example: the tolerance, the step size, the gradient or hessian values, the max. number of iterations etc. • There are also a list of features available, for example: displaying the values at each iterations, compare the user supply gradient or hessian, etc. • You can also choose the algorithm you wish to use.

  32. Options Setting (Cont.) Options = optimset(‘param1’,value1, ‘param2’,value2,…) • Type help optimset in command window, a list of options setting available will be displayed. • How to read? For example: LargeScale - Use large-scale algorithm if possible [ {on} | off ] The default is with { } Value (value1) Parameter (param1)

  33. Options Setting (Cont.) Options = optimset(‘param1’,value1, ‘param2’,value2,…) LargeScale - Use large-scale algorithm if possible [ {on} | off ] Since the default is on, if we would like to turn off, we just type: Options = optimset(‘LargeScale’, ‘off’) and pass to the input of fminunc.

  34. Useful Option Settings Highly recommended to use!!! • Display - Level of display [ off | iter | notify | final ] • MaxIter - Maximum number of iterations allowed [ positive integer ] • TolCon - Termination tolerance on the constraint violation [ positive scalar ] • TolFun - Termination tolerance on the function value [ positive scalar ] • TolX - Termination tolerance on X [ positive scalar ]

  35. fminuncandfminsearch • fminunc uses algorithm with gradient and hessian information. • Two modes: • Large-Scale: interior-reflective Newton • Medium-Scale: quasi-Newton (BFGS) • Not preferred in solving highly discontinuous functions. • This function may only give local solutions.. • fminsearch is generally less efficient than fminunc for problems of order greater than two. However, when the problem is highly discontinuous, fminsearch may be more robust. • This is a direct search method that does not use numerical or analytic gradients as in fminunc. • This function may only give local solutions.

  36. Constrained Minimization Vector of Lagrange Multiplier at optimal point [xmin,feval,exitflag,output,lambda,grad,hessian] = fmincon(fun,x0,A,B,Aeq,Beq,LB,UB,NONLCON,options,P1,P2,…)

  37. Example function f = myfun(x) f=-x(1)*x(2)*x(3); Subject to:

  38. Example (Cont.) For Create a function call nonlcon which returns 2 constraint vectors [C,Ceq] function [C,Ceq]=nonlcon(x) C=2*x(1)^2+x(2); Ceq=[]; Remember to return a null Matrix if the constraint does not apply

  39. Example (Cont.) Initial guess (3 design variables) x0=[10;10;10]; A=[-1 -2 -2;1 2 2]; B=[0 72]'; LB = [0 0 0]'; UB = [30 30 30]'; [x,feval]=fmincon(@myfun,x0,A,B,[],[],LB,UB,@nonlcon) CAREFUL!!! fmincon(fun,x0,A,B,Aeq,Beq,LB,UB,NONLCON,options,P1,P2,…)

  40. Warning: Large-scale (trust region) method does not currently solve this type of problem, switching to medium-scale (line search). > In D:\Programs\MATLAB6p1\toolbox\optim\fmincon.m at line 213 In D:\usr\CHINTANG\OptToolbox\min_con.m at line 6 Optimization terminated successfully: Magnitude of directional derivative in search direction less than 2*options.TolFun and maximum constraint violation is less than options.TolCon Active Constraints: 2 9 x = 0.00050378663220 0.00000000000000 30.00000000000000 feval = -4.657237250542452e-035 Const. 1 Const. 2 Const. 3 Const. 5 Const. 4 Const. 6 Const. 7 Const. 8 Const. 9 Sequence: A,B,Aeq,Beq,LB,UB,C,Ceq Example (Cont.)

  41. Multiobjective Optimization • Previous examples involved problems with a single objective function. • Now let us look at solving problem with multiobjective function by lsqnonlin. • Example is taken for data curve fitting • In curve fitting problem the the error is reduced at each time step producing multiobjective function.

  42. lsqnonlin in Matlab – Curve fitting • clc; %recfit.m • clear; • global data; • data= [ 0.6000 0.999 • 0.6500 0.998 • 0.7000 0.997 • 0.7500 0.995 • 0.8000 0.982 • 0.8500 0.975 • 0.9000 0.932 • 0.9500 0.862 • 1.0000 0.714 • 1.0500 0.520 • 1.1000 0.287 • 1.1500 0.134 • 1.2000 0.0623 • 1.2500 0.0245 • 1.3000 0.0100 • 1.3500 0.0040 • 1.4000 0.0015 • 1.4500 0.0007 • 1.5000 0.0003 ]; % experimental data,`1st coloum x, 2nd coloum R • x=data(:,1); • Rexp=data(:,2); • plot(x,Rexp,'ro'); % plot the experimental data • hold on • b0=[1.0 1.0]; % start values for the parameters • b=lsqnonlin('recfun',b0) % run the lsqnonlin with start value b0, returned parameter values stored in b • Rcal=1./(1+exp(1.0986/b(1)*(x-b(2)))); % calculate the fitted value with parameter b plot(x,Rcal,'b'); % plot the fitted value on the same graph Find b1 and b2 >>recfit >>b =0.0603 1.0513 %recfun.m function y=recfun(b) global data; x=data(:,1); Rexp=data(:,2); Rcal=1./(1+exp(1.0986/b(1)*(x-b(2)))); % the calculated value from the model %y=sum((Rcal-Rexp).^2); y=Rcal-Rexp; % the sum of the square of the difference %between calculated value and experimental value • Link to this Page • Short tutorial Model Fitting last edited on 26 October 2003 at 7:22 pm by westlake.che.gatech.edu

  43. Simulink Example Jeff_fly basket.mdl Shooting a flying box Eq. of ball motion in z horz. direction Eq. of ball motion in h vert. direction Aerodynamic drag force Angle of ball

  44. Simulink example – shooting ball %% Start_flyBasketBall.m InitialGuess= pi/2.5 ; X = fminsearch('Distflysim', InitialGuess)*180/pi; fprintf('\nShoot at %f deg \n', X); function P = Distflysim(theta_0) F0=25.0; %N cart_mass=2; %kg x_dot_max=50; %m/sec ro_air=1.224; %kg/m^3 h0=0.5; %m z0=0; Cd=1; r_ball=0.05; %m A_ball=pi*r_ball^2; ball_mass=0.1; %kg g=-9.8; %m/sec^2 theta_0; %rad V0=50; %m/secF0=15.0; %N AeroFac=Cd*A_ball*ro_air/2; theta_0 assignin('base','F0',F0); assignin('base','cart_mass',cart_mass); assignin('base','x_dot_max',x_dot_max); assignin('base','AeroFac',AeroFac); assignin('base','ball_mass',ball_mass); assignin('base','g',g); assignin('base','V0',V0); assignin('base','theta_0',theta_0); % Newrtp=rsimgetrtp('jeff_basket'); % save ShotParams.mat Newrtp; % !jeff_basket -p ShotParams.mat % load jeff_basket; [t,x,y]=sim('jeff_flybasket',[0 10]); np=max(size(y)); xf=y(np,1); zf=y(np,2); %hf=y(np,3); P=(xf-zf)^2;%+(hf-25)^2; % BasketflyBallnit1.m F0=25.0; %N cart_mass=1; %kg x_dot_max=50; %m/sec ro_air=1.224; %kg/m^3 Cd=1; r_ball=0.05; %m A_ball=pi*r_ball^2; ball_mass=0.05; %kg g=-9.8; %m/sec^2 theta_0=pi/2.5; %rad V0=50; %m/sec AeroFac=Cd*A_ball*ro_air/2;

  45. Optimization toolbox for use with MATLAB, User Guide, The MathWorks Inc. 2006 2. Applied Optimization with MATLAB Programming, P. Venkataraman, Wiley InterScience, 2002 3. Optimization for Engieering Design, Kalyanmoy Deb, Prentice Hall, 1996.4. http://mathdemos.gcsu.edu/mathdemos/maxmin/max_min.html5. http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory /MaxMin.html6. http://users.powernet.co.uk/kienzle/octave/optim.html7. http://www.cse.uiuc.edu/eot/modules/optimization/SteepestDescent/ REFERENCES