How to set positive lambdas in quadprog?

Hi,

I am working on a simple solver that would maximise a function that is attached.

%%quadprog 
% Initial coeffiecients
% a_bi = [3; 4; 3];
% b_bi = [9; 7; 10];
% a_si = [3; 4; 3];
% b_si = [2; 2; 1];
% The entire utility functions look like this
% u_1 = -3*x_b1^2 + 9*x_b1 - p_ 1*x_b1 + p_ 1*x_s1 - 3*x_s1 - 2*x_s1
% u_2 = -4*x_b2^2 + 7*x_b2 - p_ 2*x_b2 + p_ 2*x_s2 - 4*x_s2 - 2*x_s2
% u_3 = -3*x_b3^2 + 10*x_b3 - p_ 3*x_b3 + p_ 3*x_s3 - 3*x_s3 - 1*x_s3
% Here we change all signs to - so that we get the maximum instead of minimum
% Looks like
% -u_1 = 3*x_b1^2 - 9*x_b1 + p_ 1*x_b1 - p_ 1*x_s1 + 3*x_s1 + 2*x_s1
%    x_b1  x_b2  x_b3  x_s1  x_s2  x_s3  f12  f13  f23
H = [   3,    0,    0,    0,    0,    0,   0,   0,   0;  % -a_b1
      0,    4,    0,    0,    0,    0,   0,   0,   0;  % -a_b2
      0,    0,    3,    0,    0,    0,   0,   0,   0;  % -a_b3
      0,    0,    0,    3,    0,    0,   0,   0,   0;  % -a_s1
      0,    0,    0,    0,    4,    0,   0,   0,   0;  % -a_s2
      0,    0,    0,    0,    0,    3,   0,   0,   0;  % -a_s3
      0,    0,    0,    0,    0,    0,   0,   0,   0;
      0,    0,    0,    0,    0,    0,   0,   0,   0;
      0,    0,    0,    0,    0,    0,   0,   0,   0
  ];
%   -b_b1 -b_b2 -b_b3 -b_s1 -b_s2 -b_s3  f12  f13  f23
f = [  -9;   -7;  -10;    2;    2;    1;   0;   0;   0];
%    x_b1  x_b2  x_b3  x_s1  x_s2  x_s3  f12  f13  f23
Aeq = [-1,    0,    0,    1,    0,    0,  -1,  -1,   0;  % x_s1 - x_b1 = f12 + f13
      0,   -1,    0,    0,    1,    1,   1,   0,  -1;  % x_s2 - x_b2 = -f12 + f23
      0,    0,   -1,    0,    0,    1,   0,   1,   1;  % x_s3 - x_b3 = -f13 - f23
     -1,   -1,   -1,    1,    1,    1,   0,   0,   0   % -x_b1 - x_b2 - x_b3 + x_s1 + x_s2 + x_s3 = 0
  ];
beq = [0; 0; 0; 0];
options = optimoptions ('quadprog', 'Algorithm', 'interior-point-convex', 'Display', 'off')
[x, fval, exitflag, output, lambda] = quadprog (H, f, A, b, Aeq, beq, [], [], [], options);
x;

In the process of optimisation I get correct values of x. However, the lambdas are negative. They are supposed to be positive because I am solving a natural problem which cannot have negative volumes or values.

I have two questions: 1. Could these negative lambdas be a consequence of minimising negative as opposed to maximising positive? If yes, how to fix it. 2. What is a way to set a constraint on lambdas to ensure that they are nonnegative?

Thank you.