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Chapter 6 Method of Successive Quadratic Programming Professor Shi-Shang Jang National Tsing-Hua University Chemical Engineering Department

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6-1 Quadratic Programming Problems Problem:

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Classical Constrained Regression Problem Where β is an m×1 vector of regression coefficients to be estimated, e is an n×1 vector of error variables, Y is an n×1 observations on the dependent variable and X is an n×m matrix of observations on the independent variables. It is clear that the classical regression problem is a quadratic programming problem.

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Example- A numerical problem

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Example-A scientist ’ s model A scientist has observed a certain quantity Q as a function of t. She has a good reason to believe that there is a physical law relating t and Q that takes the form: Q(t)=asint+bcost+c She wants to have the “ best ” possible idea of the value of the coefficients a,b and c using the results of her n experiments:

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Example-A scientist ’ s model (t 1,q 1 ; t 2,q 2 ; …, t n,q n ). Taking into account that her experiments are not perfect and also perhaps that the model is not rigorous enough, she does not expect to find a perfect fit. So defines an error term: E i =q i -q(t i ) Her idea is to minimize:

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Example-A scientist ’ s model What if a=3;b=2;c=5 with 15 experimental data?

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Classical Least Square Fitting Problem: Given a set of data (x 1,y 1 ; x 2,y 2 ; …, x n,y n ). Find a set of parameters ={a, b, c, … } such that: e i 2 =y i -(ax i +bx i 2 +..) And i e i 2 is minimized The solution of the unconstrained case: =(X T X) -1 X T Y=quasi_inverse(X)Y

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Example: Curve Fitting >> x=linspace(0,5); >> for i=1:100 y(i)=3*x(i)^2+2*x(i)+5; y(i)=y(i)+randn(1,1); xx(i,1)=x(i)*x(i);xx(i,2)=x(i);xx(i,3)=1; end >> theta=inv(xx'*xx)*xx'*y' theta = 2.9983 2.1215 4.5839 >> for i=1:100 yy(i)=theta(1)*x(i)^2+theta(2)*x(i)+theta(3); end >> plot(x,yy)

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Example: Curve Fitting-Continued

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Program QUADPROG QUADPROG Quadratic programming. X=QUADPROG(H,f,A,b) solves the quadratic programming problem: min 0.5*x'*H*x + f'*x subject to: A*x <= b x X=QUADPROG(H,f,A,b,Aeq,beq) solves the problem above while additionally satisfying the equality constraints Aeq*x = beq. X=QUADPROG(H,f,A,b,Aeq,beq,LB,UB) defines a set of lower and upper bounds on the design variables, X, so that the solution is in the range LB <= X <= UB. Use empty matrices for LB and UB if no bounds exist. Set LB(i) = -Inf if X(i) is unbounded below; set UB(i) = Inf if X(i) is unbounded above.

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Curve Fitting Using Quadprog x=linspace(0,5); for i=1:100 y(i)=3*x(i)^2+2*x(i)+5; y(i)=y(i)+randn(1); xx(i,1)=x(i)*x(i);xx(i,2)=x(i);xx(i,3)=1; end a=eye(100); a1=zeros(1,100); H=[a1; a1; a1; a]; f=zeros(103,1); HH=[f f f H]; Aeq=[xx eye(100)];beq=y'; A=[];b=[]; X=QUADPROG(HH,f,A,b,Aeq,beq) ; XX=X'; XX(1) XX(2) XX(3)

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The Solution Method-Complementary Pivot Problem K-T Condition for a QP:

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The Solution Method-Complementary Pivot Problem Find vectors w and z such that

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The Solution Method-Complementary Pivot Problem The solution of the complementary pivot problem is equivalent to find a K-T point of the original QP problem and hence the solution is sufficiently found since the QP is a convex objected (quadratic function) and qualified constrained (linear constraints).

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6-2 The Successive QP Approach Consider a quadratic function: The objective is to find d=(x-x 0 ) such that the following problem can be minimized

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Example

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Algorithm Step 1: Formulate the QP problem Step 2: Solve the QP problem, and set x(t+1)=x(t)+ *d Step 3: Check Convergency

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Remarks Original SQP is subjected to the following two problems: (1) The second derivative of the objective function is generally difficult to obtain. (2) There is no guarantee that the second derivative of any function at any point is positive definite. In case the hessian is not positive definite, then the QP is failed.

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6-3 Quadratic Approximation of the Lagrangian Function

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Algorithm-The Variable Metric Method Given initial estimates x 0,u 0,v 0, and a symmetric positive definitive matrix H 0 Step 1: Solve the problem: Step 2: Select the step size α along d(t), and set x (t+1) =x (t) + α d (t)

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Algorithm-The Variable Metric Method (VMCON) Step 3: Check convergence Step 4: Update H(t) using the gradient difference BSF update formulation:

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Line Search on the SQP direction Given d from QP, instead of using the original objective function, it is more useful to implement the following penalty function:

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Numerical Example

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MATLAB Program - VMCON % DEFINE GLOBAL VARIABLES global mu global sigma global vv global uu global dd global X0 % % INITIALIZE THE PROBLEM H=eye(2);X0=[2;1];uu=0;vv=0; dx=100;mu=0;sigma=0;iter=0; % % THE WHILE LOOP % while abs(dx)>0.000001 iter=iter+1; % % FIND THE GRADIENT df % [df,dlgn]=sqp_Lagrangian(X0); % % DEFINE Aq,beq such that Aq*d=beq % A, b such that Ad<b

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MATLAB Program – VMCON-Continued A=[-1 -1];b=(X0(1)+X0(2)-1); Aeq=[X0(2) X0(1)]; beq=(X0(1)*X0(2)-2); % % SOLVE THE SUB-QUADRATIC PROBLEM FOR d % [ss,FVAL,EXITFLAG,OUTPUT,LAMBDA]=QUADPROG(H,df,A,b,Aeq,beq); dd=ss'; % % GET THE LARAGNGE=MULTIPLIERS % uu=LAMBDA.ineqlin; vv=LAMBDA.eqlin; % % PREPARE THE LINE SEARCH % mu=max(abs(vv),0.5*(mu+abs(vv))); sigma=max(abs(uu),0.5*(sigma+abs(uu))); a0=0.1; % % LINE SEARCH ALONG dd WITH THE PENALTY FUNCTION AS THE OBJ % alopt = FMINSEARCH('sqp_penalty',a0);

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MATLAB Program – VMCON-Continued % % GET THE NEW POINT % X=X0+alopt*dd'; % % PREPARE TO UPDATE THE APPROXIMATE HESSIAN % z=X-X0; [df,dlgn_1]=sqp_Lagrangian(X0); [df,dlgn_2]=sqp_Lagrangian(X); y=dlgn_2-dlgn_1; zz=z'*H*z; zzp=0.2*zz; zy=z'*y; if(zy>zzp) theta=1; else theta=0.8*z'*H*z/(zz-zy); end w=theta*y+(1-theta)*H*z; % % UPDATE THE APPROXIMATE HESSIAN % H=H-(H*z*z'*H)/zz+w*w'/(z'*w);

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MATLAB Program – VMCON-Continued % % CALCULATE THE OBJ AND EQUALITY CONSTRAINT AT THE NEW POINT % hh=X(1)*X(2)-2; ddx=X-X0; obj=sqp_obj(X); % % UPDATE THE NEW POINT % X0=X; % % CHECK THE CONVERGENCY % dx=sqrt(ddx(1)^2+ddx(2)^2); end % % OUTPUT THE RESULT % iter X obj hh

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MATLAB Program Using Fmincon > help fmincon FMINCON Finds a constrained minimum of a function of several variables. FMINCON solves problems of the form: min F(X) subject to: A*X <= B, Aeq*X = Beq (linear constraints) X C(X) <= 0, Ceq(X) = 0 (nonlinear constraints) LB <= X <= UB

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MATLAB Program Using Fmincon- Continued X=FMINCON(FUN,X0,A,B,Aeq,Beq,LB,UB,NON LCON) subjects the minimization to the constraints defined in NONLCON. The function NONLCON accepts X and returns the vectors C and Ceq, representing the nonlinear inequalities and equalities respectively. FMINCON minimizes FUN such that C(X)<=0 and Ceq(X)=0.

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MATLAB Program Using Fmincon- Example X0=[2;1]; A=[-1 -1]; B=1; Aeq=[];Beq=[];LB=[];UB=[]; X=FMINCON('sqp',X0,A,B,Aeq,Beq,LB,UB,'sqp_nlcon') function [c,ceq]=sqp_nlcon(x) c=[]; ceq=x(1)*x(2)-2;

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