1. Ali Molavi Iran University Of Science and Technology April 28
Sostool Toolbox
Ali Molavi
Iran University Of Science and Technology
April 28

2. Intruduction Sum of Squares Polynomials
provide tractable relaxations for many hard optimization problems
SOS can be solved using the powerful tools of convex optimization

3. Solving sum of squares problem
SOSTOOLS can solve two kinds of sum of squares programs
Feasibility
Optimization
To define and solve a SOSP
Initialize the SOSP
Declare the SOSP variables.
Define the SOSP constraints.
Set objective function (for optimization problems)
Call solver.
Get solutions 1

4. Initializing a Sum of Squares Program
symbolic objects
>> syms x y;
Polynomial variables
>> pvar x y;
polynomial objects
>> p = (x^4)+(5*x*y)+(x*y^2) 2

5. Initializing a Sum of Squares Program
>> Program1 = sosprogram([x;y]);
[x,y] vector containing Independent variable
p=(x^2)+(x*y)+(x*z)+(m*h) [x;y;z;m;h]
decision variables
Scalar decision variables
Polynomial variables
Sum of squares variables
Matrix of polynomial variables
Matrix of sum of squares variables 3

6. Declare the SOSP variables
Scalar decision variables or constant variable
>>Program1 = sosdecvar(Program1,[a;b]);
>>Program1 = sosprogram([x;y],[a;b]);
Example
P=ax+by
>> syms x y a b;
>> Program = sosprogram([x;y],[a,b]);
>> Program1 = sosdecvar(Program1,[a;b]); 4

7. Declare the SOSP variables
Scalar Polynomial Variables
polynomials with unknown coefficients
>> Program1 = sosdecvar(Program1,[a;b;c]);
>> v = a*x^2 + b*x*y + c*y^2;
[Program1,v] = sospolyvar(Program1,[x^2; x*y; y^2]; ’ wscoeff’);
V=coeff_1*x^2+coeff_2*x*y+coeff_3*y^2 5

8. Declare the SOSP variables
Constructing Vectors of Monomials VEC = monomials([x; y],[1 2 3]) VEC = mpmonomials({[x1; x2],[y1; y2],[z1]},{1:2,1,3}) 6

9. Declare the SOSP variables
Sum of Squares Variables
[Program1,p] = sossosvar(Program1,[x; y]);
P=coeff_4*y^2+(coeff_2+coeff_3)*x*y+coeff_1*x^2
[Q,Z,f] = findsos(P); 7

10. Declare the SOSP variables
Matrix Variables
[prog,P] = sospolymatrixvar(prog,monomials(x,0),[2 2]);
[prog,Q]=sospolymatrixvar(prog,monomials([x1],[1]),[2 2], ’symmetric’);
Matrix dimention 8

11. Define the SOSP constraints
Adding Constraints
Equality constraints
Program1 = soseq(Program1,diff(p,x)-x^2);
Inequality Constraints
Program1 = sosineq(Program1,diff(p,x)-x^2);
Inequlity constraints with an interval
Program2 = sosineq(Program2,diff(p,x)-x^2,[-1 2]); 9

12. Define the SOSP constraints
Matrix Inequality Constraints syms theta1 theta2; or >>pvar theta1 theta2 theta = [theta1; theta2]; prog = sosprogram(theta); [prog,M]=sospolymatrixvar(prog,monomials(theta,0:2),[3,3] ,’symmetric’); There are 2 case prog = sosmatrixineq(prog,M,’quadraticMineq’); prog = sosmatrixineq(prog,M,’Mineq’); 10

13. Define the SOSP constraints
Example
Lyapunove Function V(X)=X’PX
For Stability
We can define “eps” then eps<<1
A’P+PA<0
P>0
A’P+PA<-eps*I
P>eps*I 11

14. Define the SOSP constraints
syms x; G = rss(4,2,2); A = G.a; eps = 1e-6; I = eye(4); prog = sosprogram(x); [prog,P]=sospolymatrixvar(prog,monomials(x,0),[4,4],’symm etric’); prog = sosmatrixineq(prog,P-eps*I,’quadraticMineq’); d= A’*P+P*A; prog = sosmatrixineq(prog,-d-eps*I,’quadraticMineq’); 12

15. Setting an Objective Function and calling solver
Setting objective function
Program1 = sossetobj(Program1,a-b);
(a ,b )are decision variable
Calling solver
Program1 = sossolve(Program1,options);
Getting Solutions
SOLp1 = sosgetsol(Program6,p1,5);
P1 is in fact SOSP variable
For example a polynomial variable 13

16. Application of sum of Squares programming
Sum of Square Test
Given a polynomial p(x), determine if p(x)>0 is feasible
syms x1 x2;
vartable = [x1, x2];
prog = sosprogram(vartable)
% define the inequality
p = 2*x1^4 + 2*x1^3*x2 - x1^2*x2^2 + 5*x2^4;
prog = sosineq(prog,p);
prog = sossolve(prog); % call solver
[Q,Z]=findsos(p,’rational’); % P=Z’QZ 14

17. Application of sum of Squares programming
Lyapunov Function Search
Finding a V(X)>0
Derivative of V(X) 15

18. Application of sum of Squares programming
syms x1 x2 x3; prog = sosprogram([x1;x2;x3]) f=[-x1^3-x1*x3^2;-x2-x1^2*x2;x3+3*x1^2*x33*x3/(x3^2+1)] [prog,V] = sospolyvar(prog,[x1^2; x2^2; x3^2],’wscoeff’); V=coeff_4*x1^2 + coeff_5*x2^2 + coeff_6*x3^2 %For example “eps=1” prog = sosineq(prog,V-(x1^2+x2^2+x3^2)); dd=-(diff(V,x1)*f(1)+diff(V,x2)*f(2)+diff(V,x3)*f(3))*(x3^2+1); prog = sosineq(prog,dd); prog = sossolve(prog); SOLV = sosgetsol(prog,V) 16

19. Application of sum of Squares programming
Another Way to Find Lyapunov Function
syms x1 x2;
V = findlyap([-x1^3+x2; -x1-x2],[x1; x2],2)
The function ‘’findlyap’’ has three argument
The first one Vector Field
The second one Ordering of the independent variable
The third one Degree of lyapunov function 17

20. Application of sum of Squares programming
Global and Constrained Optimization
Minimum ( -Gama ) such that
syms x1 x2 gam;
prog = sosprogram([x1,x2]);
prog = sosdecvar(prog,[gam]);
f = x1+x2-3*x1^2+6*x2*x1
prog = sosineq(prog,(f-gam))
prog = sossetobj(prog,-gam);
prog = sossolve(prog);
SOLgamma = sosgetsol(prog,gam) 18

21. Application of sum of Squares programming
Another way to find minimum of an objective function
syms a b c d;
F=(a^4+1)*(b^4+1)*(c^4+1)*(d^4+1)+2*a + 3*b + 4*c + 5*d;
[bnd,vars,xopt] = findbound(F)
bnd=lower bound for polynomial
Vars=A vector of variable of the polynomial
Xopt=A Point Where The Result is achieved 19

22. Application of sum of Square programming
Minimization of polynomial with constraint
Example:
Minimizing (X1^2)+(X2^2)
subject to
syms x1 x2;
degree = 4;
[gam,vars,opt] = findbound(x1+x2,[x1, x2-0.5],...
[x1^2+x2^2-1, x2-x1^2-0.5],degree); 20