In the following, we describe how to model a problem of the above form with FORCES Pro. First make sure that the FORCES Pro client is on the MATLAB/Python path. Click here for more details.
Python users first have to import the FORCES Pro module.
Multistage struct
First, an empty struct/class has to be initialized, which contains all fields needed and initialises matrices and vectors to empty matrices. The command
stages = MultistageProblem(N);
stages = MultistageProblem(N)
creates such an empty structure/class of length \(N\). Once this structure/class has been created, the corresponding matrices, vectors and dimensions can be set for each element of stages.
Dimensions
In order to define the dimensions of the stage variables \(z_i\), the number of lower and upper bounds, the number of polytopic inequality constraints and the number of quadratic constraints use the following fields:
stages(i).dims.n = ...; % length of stage variable zi
stages(i).dims.r = ...; % number of equality constraints
stages(i).dims.l = ...; % number of lower bounds
stages(i).dims.u = ...; % number of upper bounds
stages(i).dims.p = ...; % number of polytopic constraints
stages(i).dims.q = ...; % number of quadratic constraints
stages.dims[ i ]['n'] = ... # length of stage variable zi
stages.dims[ i ]['r'] = ... # number of equality constraints
stages.dims[ i ]['l'] = ... # number of lower bounds
stages.dims[ i ]['u'] = ... # number of upper bounds
stages.dims[ i ]['p'] = ... # number of polytopic constraints
stages.dims[ i ]['q'] = ... # number of quadratic constraints
Cost function
The cost function is, for each stage, defined by the matrix \(H_i\) and the vector \(f_i\). These can be set by
stages(i).cost.H = ...; % Hessian
stages(i).cost.f = ...; % linear term
stages.cost[i]['H'] = ... # Hessian
stages.cost[i]['f'] = ... # linear term
Note: whenever one of these terms is zero, you have to set them to zero (otherwise the default of an empty matrix is assumed, which is different from a zero matrix).
Equality Constraints
The equality constraints for each stage, which are given by the matrices \(C_i\), \(D_i\) and the vector \(c_i\), have to be provided in the following form:
stages(i).eq.C = ...;
stages(i).eq.c = ...;
stages(i).eq.D = ...;
stages.eq[ i ]['C'] = ...
stages.eq[ i ]['c'] = ...
stages.eq[ i ]['D'] = ...
Lower and upper bounds
Lower and upper bounds have to be set in sparse format, i.e. an index vector lbIdx/ubIdx that defines the elements of the stage variable \(z_i\) has to be provided, along with the corresponding upper/lower bound lb/ub:
stages(i).ineq.b.lbidx = ...; % index vector for lower bounds
stages(i).ineq.b.lb = ...; % lower bounds
stages(i).ineq.b.ubidx = ...; % index vector for upper bounds
stages(i).ineq.b.ub = ...; % upper bounds
stages.ineq[ i ]['b']['lbidx'] = ... # index vector for lower bounds
stages.ineq[ i ]['b']['lb'] = ... # lower bounds
stages.ineq[ i ]['b']['ubidx'] = ... # index vector for upper bounds
stages.ineq[ i ]['b']['ub'] = ... # upper bounds
Both lb and lbIdx must have length stages(i).dims.l / stages.dims[ i ]['l'], and both ub and ubIdx must have length stages(i).dims.u / stages.dims[ i ]['u'].
Polytopic constraints
In order to define the inequality \(A_i z_i \leq b_i \), use
stages(i).ineq.p.A = ...; % Jacobian of linear inequality
stages(i).ineq.p.b = ...; % RHS of linear inequality
stages.ineq[ i ]['A'] = ... # Jacobian of linear inequality
stages.ineq[ i ]['b'] = ... # RHS of linear inequality
The matrix A must have stages(i).dims.p / stages.dims[ i ]['p'] rows and stages(i).dims.n / stages.dims[ i ]['n'] columns. The vector b must have stages(i).dims.p / stages.dims[ i ]['p'] rows.
Quadratic constraints
Similar to lower and upper bounds, quadratic constraints are given in sparse form by means of an index vector, which determines on which variables the corresponding quadratic constraint acts.
stages(i).ineq.q.idx = { idx1, idx2, …}; % index vectors
stages(i).ineq.q.Q = { Q1, Q2, …}; % Hessians
stages(i).ineq.q.l = { L1, L2, …}; % linear terms
stages(i).ineq.q.r = [ r1; r2; … ]; % RHSs
stages.ineq[ i ]['q']['idx'] = ... # index vectors
stages.ineq[ i ]['q']['Q'] = ... # Hessians
stages.ineq[ i ]['q']['l'] = ... # linear terms
stages.ineq[ i ]['q']['r'] = ... # RHSs
If the index vector idx1 has length \(m_1\), then the matrix Q must be square and of size \(m_1 \times m_1\), the column vector l1 must be of size \(m_1\) and r1 is a scalar. Of course this dimension rules apply to all further quadratic constraints that might be present. Note that \(L1\), \(L2\) etc. are column vectors in MATLAB!
Since multiple quadratic constraints can be present per stage, in MATLAB we make use of the cell notation for the Hessian, linear terms, and index vectors. In Python we make use of Python object arrays for the Hessians, linear terms, and index vectors.
Example
To express the two quadratic constraints
$$ \begin{align}
z_{3,3}^2 + 2z_{3,5}^2 - z_{3,5} &\leq 3 \\
5z_{3,1}^2 &\leq 1
\end{align}
$$
on the third stage variable, use the code
stages(3).ineq.q.idx = { [3 5], [1] } % index vectors
stages(3).ineq.q.Q = { [1 0; 0 2], [5] }; % Hessians
stages(3).ineq.q.l = { [0; -1], [0] }; % linear terms
stages(3).ineq.q.r = [ 3; 1 ]; % RHSs
stages.ineq[3-1]['q']['idx'] = np.zeros((2,), dtype=object) % index vectors
stages.ineq[3-1]['q']['idx'][0] = np.array([3,5])
stages.ineq[3-1]['q']['idx'][1] = np.array([1])
stages.ineq[3-1]['q']['Q'] = np.zeros((2,), dtype=object) % Hessians
stages.ineq[3-1]['q']['Q'][0] = np.array([1.0 0],[0 2.0])
stages.ineq[3-1]['q']['Q'][1] = np.array([5])
stages.ineq[3-1]['q']['l'] = np.zeros((2,), dtype=object) % linear terms
stages.ineq[3-1]['q']['l'][0] = np.array([0], [-1])
stages.ineq[3-1]['q']['l'][1] = np.array([0])
stages.ineq[3-1]['q']['r'] = np.array([3],[1]) % RHSs
Binary Constraints
To declare binary variables, you can use the bidx field of the stages struct or object. For example, the following code declares variables 3 and 7 of stage 1 to be binary:
stages.bidx[1] = np.array([3, 7])
That's it! You can now generate a solver that will take into account the binary constraints on these variables. If binary variables are declared, FORCES Pro will add a branch-and-bound procedure to the standard convex solver it generates.