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solve optimization with constraints

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ali
ali 2021-2-28
评论： ali 2021-3-1
I want to maximize an equation with 12 variables (The comment in code described the problem). I solved this by excel solver, but in matlab I dont know how to write the constraints. Below is my try:
% Maximize B = 3.9 X1 + 4.2 X2 + 4.2 X3 + 4.5 X4 + 4.1 X5 + 3.6 X6 + 3.1 X7
% + 2.7 X8 + 2.5 X9 + 2.6 X10 + 2.9 X11 +3.6 X12
%Subject to:
% S0 = 5;
% S(t-1) + I(t) - X(t) = S(t) (t =1,...,12) --> how to wirte this constraint?
% 1<= X(t) <= 7 (t = 1,...,12 )
% S(t) <= 10 (t = 1,...,12 )
f = [-3.9 -4.2 -4.2 -4.5 -4.1 -3.6 -3.1 -2.7 -2.5 -2.6 -2.9 -3.6];
lb=[1;1;1;1;1;1;1;1;1;1;1;1];
ub=[7;7;7;7;7;7;7;7;7;7;7;7];
I = [4 3 2 2 1 2 3 3 2 2 2 3];

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Alan Weiss
Alan Weiss 2021-2-28
This problem is similar to Create Multiperiod Inventory Model in Problem-Based Framework. I think that you will find the problem-based approach easy to use.
Alan Weiss
MATLAB mathematical toolbox documentation
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Alan Weiss
Alan Weiss 2021-3-1
Well, it is more awkward, but fairly straightforward. You simply have to keep careful track of variable indices.
Let S(1) through S(12) represent the S variables, and X(1) through X(12) the X variables. You have to put all of the variables into one, typically called x (lower case). Say the mapping is x = [X,S], where all variables are row vectors. Then you can write your constraints all in terms of x in matrices A and Aeq to represent the dynamics. For example, the first equation
S(t-1) + I(t) - X(t) = S(t)
becomes, for t = 1,
S0 + I(1) = S(1) + X(1) = x(1) + x(13)
You can represent this as row 1 in matrix Aeq with beq(1) = S0 + I(1) = 5 + 4. The Aeq row is
[1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] % Aeq(1,1) = Aeq(1,13) = 1
Similarly, for t = 2 the equation is
I(2) = S(2) + X(2) - S(1)
In terms of Aeq and beq you get
Aeq(2,2) = Aeq(2,14) = 1
Aeq(2,13) = -1
beq(2) = I(2) = 3
The matrix Aeq has a simple banded structure. You have to be careful about the bounds on S and X; for example, does S have a lower bound?
You have to represent the f coefficients in terms of x, which means adjoin 12 zeros to the end of your current f vector. OK?
Alan Weiss
MATLAB mathematical toolbox documentation

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