Constrained linear least-squares problems, with unknowns in multiplier matrix

I have three known vectors v_1, v_2 and v_3 and an unknown vector C

v_1 = [47;23;4;3;9;8;3];
v_2 = [32;17;5;21;8;3;11];
v_3 = [22;9;6;34;2;2;0];
V = sym('v',[7 1]);

the four vectors are related by the following equations

syms a b c d
eqn1 = v_2 == a*v_1+b*V;
eqn2 = v_3 == c*v_1+d*V;

So in total I have 11 unknowns (a,b,c,d and v1-v7) and 14 equations (7 from eqn1 and 7 from eqn2). The sum of a+b (and c+d) must be equal to 1, and the sum of v1-v7 must give 100.

This is not an exact relationship, but I would like to solve the problem as closely as possible. For this reason, based on my experiences with a similar problem (but with C as a known vector), I thought of solving the problem using lsqlin:

C = [v_1,V]

Aeq = ones(1,2);
beq = 1;
lb = zeros(1,2);
ub = ones(1,2);

x = lsqlin(C,v_2,[],[],Aeq,beq,lb,ub)

However, I have now run into two problems: (1) lsqlin requires C to be data type double (2) I want to consider both eqn1 and eqn2 simultaneously to solve my problem.

Does anyone have any advice on how to deal with this problem?

Answer 1

[Torsten]

You must use a nonlinear optimizer (in this case: fmincon) because products of the parameters to be estimated are involved (b*V, d*V). lsqlin is not suited for this case.

Take the below code as a start.

v_1 = [47;23;4;3;9;8;3];
v_2 = [32;17;5;21;8;3;11];
v_3 = [22;9;6;34;2;2;0];
fun = @(p)[v_2-p(1)*v_1-p(2)*p(5:11);v_3-p(3)*v_1-p(4)*p(5:11)];
f = @(p) sum(fun(p).^2);
Aeq = [0 0 0 0 1 1 1 1 1 1 1;1 1 0 0 0 0 0 0 0 0 0; 0 0 1 1 0 0 0 0 0 0 0];
beq = [100;1;1];
lb = [0 0 0 0 -Inf -Inf -Inf -Inf -Inf -Inf -Inf].';
ub = [1 1 1 1 Inf Inf Inf Inf Inf Inf Inf ].';
p0 = [0.5 0.5 0.5 0.5 100/7*ones(1,7)].';
p = fmincon(f,p0,[],[],Aeq,beq,lb,ub)

[Gino Sartori]

Thanks a lot!