Matlab program for convex MPC

how to solve an optimization problem in polynomial time by the primal dual interior point method with example.

Answer 1

[Shubham]

Hi Krishna,

To solve a convex Model Predictive Control (MPC) problem using the primal-dual interior-point method in MATLAB, you can set up the optimization problem in a form that can be solved by this method. Here's a simple example using MATLAB's built-in functions:

% Define the quadratic programming problem
Q = [2 0; 0 2];  % Positive definite matrix
c = [-2; -5];    % Linear term

% Inequality constraints
A = [-1 1; 1 2; 2 1];
b = [2; 6; 8];

% Equality constraints
Aeq = [1 1];
beq = [3];

% Initial guess
x0 = [0; 0];

% Options for the solver
options = optimoptions('quadprog', 'Algorithm', 'interior-point-convex', 'Display', 'iter');

% Solve the quadratic programming problem
[x, fval, exitflag, output] = quadprog(Q, c, A, b, Aeq, beq, [], [], x0, options);

The interior-point-convex algorithm does not accept an initial point.
Ignoring X0.

 Iter            Fval  Primal Infeas    Dual Infeas  Complementarity  
    0   -6.720000e+00   3.996803e-15   5.600000e+00     2.800000e+00  
    1   -7.084989e+00   3.108624e-15   5.724830e-03     1.784682e-01  
    2   -7.124517e+00   1.776357e-15   2.862415e-06     1.532405e-02  
    3   -7.125000e+00   2.220446e-15   1.431207e-09     9.210425e-05  
    4   -7.125000e+00   8.881784e-16   7.156037e-13     4.606072e-08  
    5   -7.125000e+00   0.000000e+00   3.578877e-16     2.303035e-11  

Minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

% Display the results
disp('Optimal solution:');

Optimal solution:

disp(x);

    0.7500
    2.2500

disp('Optimal objective function value:');

Optimal objective function value:

disp(fval);

   -7.1250

This example demonstrates how to set up and solve a convex quadratic programming problem using MATLAB's interior-point method. For a real-world Model Predictive Control (MPC) problem, you would need to define Q, c, A, b, Aeq, and beq based on your specific system dynamics, constraints, and cost function.

For more information on quadprog, refer to the following documentation link:

https://mathworks.com/help/optim/ug/quadprog.html

Hope this helps.