Constrained Nonlinear Minimization — Control Systems/Optimization

Constrained Nonlinear Minimization

Control Systems / Optimization

Finds the x that minimizes a polynomial subject to polynomial equality and inequality constraints and to bounds, every coefficient arriving on a port – MATLAB's fmincon with its active-set algorithm, solved again on every sample:

minimize over x   f(x)   subject to   ceq_k(x) = 0,   c_k(x) ≤ 0,   lb ≤ x ≤ ub

where f, every ceq_k and every c_k is Σ_t a_t · x₁^e(t,1) · … · x_n^e(t,n), each with its own coefficients a over ONE shared set of terms.

The exponents e are configuration – the problem's shape, which fixes the loop bounds every export unrolls. The coefficients are the signal: one column per function, stacked into a single port. So the objective and the constraints can change on every sample while their form does not.

The search is fmincon's active-set method: a sequential quadratic programming iteration with a BFGS Hessian approximation, an active-set QP with its own feasibility phase, a merit line search and a first-order optimality test. Gradients are exact, because a polynomial's gradient is another polynomial.

The factory setting is fmincon's documented example: Rosenbrock's function 100(x₂ − x₁²)² + (1 − x₁)² inside the unit disk x₁² + x₂² ≤ 1, from [0; 0], over the terms [0 2; 2 1; 4 0; 0 0; 1 0; 2 0] (x₂², x₁²x₂, x₁⁴, 1, x₁, x₁²). Feed c the column [100; −200; 100; 1; −2; 1; 1; 0; 0; −1; 0; 1] and the answer is x = (0.7864, 0.6177).

Ports

• c – the coefficients, a column [(1 + p + q)·T,1] for p equalities and q inequalities: rows 1..T are the objective's, one per ROW of Term Exponents in the same order, then T rows for each equality in turn, then T rows for each inequality. A row vector is refused.
• x – the minimizer, [n,1] (n = the columns of Term Exponents).
• fval – f(x), a scalar [1,1].
• exitflag – a scalar [1,1], fmincon's own: 1 the first-order optimality measure is below Optimality Tolerance and the constraints hold within Constraint Tolerance; 4 the search direction became smaller than twice Step Tolerance; 5 the predicted change in f became smaller than Function Tolerance (both with the constraints met); 0 a limit below stopped the search; −2 no feasible point was found.

Parameters

• Options – empty (default), or the path of a Solver Options block (Home/Solver Options), MATLAB’s options argument: for the run, every option it sets replaces this block’s parameter of the same name; one it leaves at default, or one this block does not have, changes nothing.
• Term Exponents – a [T,n] matrix of whole numbers from 0 to 4: row t, column i is the power of x_i in term t. T is 1 to 8, n is 1 to 4, and no term may total more than degree 4. The caps are the export's, not the search's – see Code export.
• Number of Equalities – p, the equality constraints ceq(x) = 0: from 0 to 3, and fewer than n.
• Number of Inequalities – q, the inequality constraints c(x) ≤ 0: from 0 to 6.
• Start Point – x0, n entries. A start outside a bound is moved onto it first, as fmincon does. The search is LOCAL, so on a problem with several local minima this chooses which one is found.
• Lower Bounds lb, Upper Bounds ub – [n,1] each, or [] for none; -Inf/Inf entries leave a variable unbounded on that side. No lower bound may exceed its upper bound.
• Step Tolerance – StepTolerance, positive; default 1e-6. The search stops with flag 4 once the largest entry of the QP step is below twice this.
• Function Tolerance – FunctionTolerance, positive; default 1e-6. The search stops with flag 5 once the step's predicted change in f is below this.
• Optimality Tolerance – OptimalityTolerance, positive; default 1e-6: the first-order measure – the largest entry of the Lagrangian's gradient, or of a multiplier times its constraint – that stops the search with flag 1.
• Constraint Tolerance – ConstraintTolerance, positive; default 1e-6: how far a constraint may be violated and still count as met.
• Maximum Iterations – MaxIterations, 1 to 2000, default 400, MATLAB's own.
• Maximum Function Evaluations – MaxFunctionEvaluations, 0 to 2000. 0 (the default) means MATLAB's own for this algorithm, 100·n.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. Each prints the same description of the iteration that the simulation itself runs, so every target does the same arithmetic in the same order as the simulation.

The exponent pattern is unrolled once: each term's monomial and its n partial derivatives, shared by every function. The bounds are written into the arithmetic, so they are not tunable on the generated code. The iteration is emitted with fixed trip counts and flags that stop the arithmetic once the answer is found, because none of the ten has an unbounded loop a hardware target could also carry. The caps above keep the Java method under the language's 64 KB limit.

⚠ The three HDL targets run the search in simulation-only real arithmetic, quantizing only at the port boundaries: a QP divides and compares across many orders of magnitude, which a Q16.16 datapath does not carry.

Simulink bridge

None (Support::None). The Optimization Toolbox ships no Simulink library at all and fmincon is a MATLAB function, so there is no block to map onto; the bridge reports this block rather than dropping it silently, and it has no parity testbench. Code export verification still covers it across all ten languages.

Notes

• Stateless: the search restarts from Start Point on every sample, so the answer depends only on the coefficients present at that step.
• ⚠ Active-set, not interior-point. fmincon's default algorithm is interior-point; this block reproduces optimoptions('fmincon','Algorithm','active-set'), which on a problem with several local minima may find a different one.
• ⚠ LOCAL. The method finds a point where no small feasible step lowers f; on a non-convex problem that depends on Start Point.
• ⚠ If f is unbounded below on the feasible set – an odd-degree term with nothing to hold it – there is no answer, and the point reported is wherever the evaluation limit stopped the search (flag 0).
• eps has no spelling in nine of the ten targets, so MATLAB's eps and the constants built from it are the exact binary values here.