I am new to the field of optimization and I need help in the following optimization problem. I have tried to solve it using normal coding to make sure that I got he correct results. However, the results I got are different and I am not sure my way of analysis is correct or not. This is a short description of the problem:
The objective function shown in the picture is used to find the optimal temperature of the insulating system that minimizes the total cost over a given horizon.
[This image provides the mathematical description of the objective function and the constraints] (https://i.stack.imgur.com/yidrO.png)
The data of the problems are as follow:
1-
Problem data:
A=1.07×10^8
h=1
T_ref=87.5
N=20
p1=0.001;
p2=0.0037;
This is the curve I want to obtain
2- Optimization variable:
u_t
3- Model type:
The model is a nonlinear cost function with non-linear constraints and it is solved using non-linear solver SNOPT.
4-The meaning of the symbols in the objective and constrained functions
The optimization is performed over a prediction horizon of N years.
T_ref is The reference temperature.
Represent the degree of polymerization in the kth year.
X_DP Represents the temperature of the insulating system in the kth year.
h is the time step (1 year) of the discrete-time model.
R is the ratio of the load loss at the rated load to the no-load loss.
E is the activation energy.
A is the pre-exponential constant.
beta is a linear coefficient representing the cost due to the decrement of the temperature.
I have developed the source code in MATLAB, this code is used to check if my analysis is correct or not.
I have tried to initialize the Ut value in its increasing or decreasing states so that I can have the curves similar to the original one. [This is the curve I obtained] (https://i.stack.imgur.com/KVv2q.png)
I have tried to simulate the problem using conventional coding without optimization and I got the figure shown above.
close all; clear all;
h=1;
N=20;
a=250;
R=8.314;
A=1.07*10^8;
E=111000;
Tref=87.5;
p1=0.0019;
p2=0.0037;
p3=0.0037;
Utt=[80,80.7894736842105,81.5789473684211,82.3684210526316,83.1578947368421,... % The value of Utt given here represent the temperature increament over a predictive horizon.
83.9473684210526,84.7368421052632,85.5263157894737,86.3157894736842,...
87.1052631578947,87.8947368421053,88.6842105263158,89.4736842105263,...
90.2631578947369,91.0526315789474,91.8421052631579,92.6315789473684,...
93.4210526315790,94.2105263157895,95];
Utt1 = [95,94.2105263157895,93.4210526315790,92.6315789473684,91.8421052631579,... % The value of Utt1 given here represent the temperature decreament over a predictive horizon.
91.0526315789474,90.2631578947369,89.4736842105263,88.6842105263158,...
87.8947368421053,87.1052631578947,86.3157894736842,85.5263157894737,...
84.7368421052632,83.9473684210526,83.1578947368421,82.3684210526316,...
81.5789473684211,80.7894736842105,80];
Ut1=zeros(1,N);
Ut2=zeros(1,N);
Xdp =zeros(N,N);
Xdp(1,1)=1000;
Xdp1 =zeros(N,N);
Xdp1(1,1)=1000;
for L=1:N-1
for k=1:N-1
%vt(k+L)=Ut(k-L+1);
Xdq(k+1,L) =(1/Xdp(k,L))+A*exp((-1*E)/(R*(Utt(k)+273)))*24*365*h;
Xdp(k+1,L)=1/(Xdq(k+1,L));
Xdp(k,L+1)=1/(Xdq(k+1,L));
Xdq1(k+1,L) =(1/Xdp1(k,L))+A*exp((-1*E)/(R*(Utt1(k)+273)))*24*365*h;
Xdp1(k+1,L)=1/(Xdq1(k+1,L));
Xdp1(k,L+1)=1/(Xdq1(k+1,L));
end
end
% MATLAB code
for j =1:N-1
Ut1(j)= -p1*(Utt(j)-Tref);
Ut2(j)= -p2*(Utt1(j)-Tref);
end
sum00=sum(Ut1);
sum01=sum(Ut2);
X1=1./Xdp(:,1);
Xf=1./Xdp(:,20);
Total= table(X1,Xf);
Tdiff =a*(Total.Xf-Total.X1);
X22=1./Xdp1(:,1);
X2f=1./Xdp1(:,20);
Total22= table(X22,X2f);
Tdiff22 =a*(Total22.X2f-Total22.X22);
obj=(sum00+(Tdiff));
ob1 = min(obj);
obj2=sum01+Tdiff22;
ob2 = min(obj2);
plot(Utt,obj,'-o');
hold on
plot(Utt1,obj)
this equations i wan to transfer in my Mathematica environment and calculate the mean of date the values of this equations
I have solved using SCIP a convex mathematical model with binary variables, linear Objective function and a set of linear constraints amended with a single non-linear constraint making the model as a non-linear binary problem.
In the output file provided by SCIP, there is a term named as: First LP value and a value has been assigned to. I cannot figure out what is exactly the meaning of First LP value for my specific nonlinear problem ?? I appreciate any explanation in detail.
For solving nonlinear problems, SCIP solves linear programming relaxations (LPs) that describe an outer approximation of the feasible region. The first LP value is the value of the optimal solution to the initial LP that was solved at the root node, after presolving, but before any separation.
I am attempting to solve a system of non-linear equations of the form below, using numpy:
a(y-2.7)(1-exp(-a*z)) = (x-2.7)(1-exp(-z))
b(w-2.7)(1-exp(-b*z)) = (x-2.7)(1-exp(-z))
c(w-2.7)(1-exp(-b*z)) = (y-2.7)(1-exp(-a*z)
d([y+w]/2-2.7)(1-exp(-d*z)) = (x-2.7)(1-exp(-z))
Obviously there are as many equations as unknowns in the system. The values a,b,c,d are constants for the system above. This is the simplest system, there will be more equations in other cases.
The solutions to these equations have a similar order of magnitude and so I am aware that the Levenberg-Marquardt algorithm can be used to solve the system given a set of initial values for the unknown values. I am sure that scipy.optimize can be used with default values all 1s for the unknowns w,x,y,z.
SciPy has a set of nonlinear solvers that would probably work for your application. They are part of scipy.optimize and are specifically designed for nonlinear systems. The documentation can be found here. For an in-depth discussion of how to use the solvers, see the previous S.O. discussion on the topic here.
Does anybody know which simplex-like algorithm CPLEX uses to solve quadratic programs. what is the so called Quadratic Simplex it is using?
Thank you in advance,
Mehdi
I'm not sure what CPLEX uses but the Simplex Method has been modified by Philip Wolfe to solve quadratic programming. In a nut shell, this is what it does:
Given a quadratic programming problem: QPP. p'x + 1/2x'Cx with constrains Ax = b
C has to be symmetric positive definite (positive semi-definite might work as well)
generate linear constraints using the Karush-Kuhn-Tucker conditions
modify the Simplex method in a way such that complementary slackness holds when choosing the pivot columns.
proceed with the other usual Simplex method steps
For more detailed information, please take a look at this paper:
http://pages.cs.wisc.edu/~brecht/cs838docs/wolfe-qp.pdf
Hope this helps.