FORCES Pro supports designing solvers and controllers via MATLAB® and Python scripts. Regardless of the language used, a Simulink® block is always created such that you can plug your advanced formulation directly into your simulation models, or download it to a real-time target platform. Click here for further details on how to deploy the generated solver on an embedded target.

This low-level interface gives advanced optimization users the full flexibility when designing custom optimization solvers and MPC controllers based on non-standard formulations. For help in deciding which interface to use click here.

The FORCES Pro low-level interface supports the class of** convex multistage quadratically constrained programs** (QCQPs) of the form

$$ \begin{align} \text{minimize} \ \ & \sum_{i=1}^N \frac{1}{2} z_i^T H_i z_i + f_i^T z_i & & \text{(separable objective)}\\

\text{subject to} \ \ & D_1 z_1 = c_1 && \text{(initial equality)} \\

& C_{i-1} z_{i-1} + D_{i} z_{i} = c_{i} && \text{(inter-stage equality)} \\

& \underline{z}_i \leq z_i && \text{(lower bounds)}\\

& z_i \leq \bar{z}_i && \text{(upper bounds)}\\

& A_i z_i \leq b_i && \text{(polytopic inequalities)}\\

& z_i^T Q_{i,k} z_i + L_{i,k}^T z_i \leq r_{i,k} && \text{(quadratic inequalities)}

\end{align}$$ for \(i=1,…,N\) and \(k=1,…,M\). To obtain a solver for this optimization problem using the FORCES Pro client, you need to define all data in the problem (the matrices \(H_i\), \(A_i\), \(Q_{i,j}\), \(D_i\), \(C_i\) and the vectors \( \underline{z}_i \), \( \bar{z}_i \), \( b_i \), \( L_{i,k} \), \( r_{i,k} \), \( c_{i} \) ) in a MATLAB® struct or Python dictionary, along with the corresponding dimensions. The following steps will take you through this process. See also Simple MPC Example for a minimalistic example.

**Note: FORCES Pro supports all problem data to be parametric, i.e. to be unknown at code generation time. Read Parametric Problems to learn how to use parameters correctly.**

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.

`from forcespro import *`

First, an empty struct/class has to be initialized, which contains all fields needed and initialises matrices and vectors to empty matrices. The command

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`.

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.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 ]['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

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.f = ...; % linear term

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).

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.D = ...;

stages.eq[ i ]['c'] = ...

stages.eq[ i ]['D'] = ...

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.lb = ...; % lower bounds

stages(i).ineq.b.ubidx = ...; % index vector for upper bounds

stages(i).ineq.b.ub = ...; % upper 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']`.

In order to define the inequality \(A_i z_i \leq b_i \), use

stages(i).ineq.p.b = ...; % RHS 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.

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.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']['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

`
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

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.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 `

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:

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.