Unconstrained Minimization — Control Systems/Optimization

Unconstrained Minimization

Control Systems / Optimization

Finds the x that minimizes a multivariate polynomial whose coefficients arrive on a port – MATLAB's fminunc at its default quasi-Newton (BFGS) algorithm, solved again on every sample:

f(x) = Σ_t c_t · x₁^e(t,1) · … · x_n^e(t,n)

The exponents e are configuration – they are the polynomial's shape, and shape fixes the loop bounds every export unrolls. The coefficients c are the signal, one per term, so the surface being minimized can change on every sample while its form does not. That is the same seam Nelder-Mead Search takes, on purpose: the two blocks accept the same problem and can be put side by side on one diagram.

What this block adds is the derivative. A polynomial's gradient is another polynomial, so it is computed exactly rather than by finite differences, and the search uses it: a BFGS inverse-Hessian approximation, a Wolfe line search, and a first-order optimality test. Nelder-Mead uses no derivative at all and is the block to reach for when the surface is noisy or kinked; this one is the block to reach for when it is smooth, and it gets there in far fewer evaluations.

The factory setting is Rosenbrock's function, expanded: 100(x₂−x₁²)² + (1−x₁)² is the six terms [0 2; 2 1; 4 0; 0 0; 1 0; 2 0] against coefficients [100; −200; 100; 1; −2; 1] from [−1.2; 1] – fminunc's own documented example, so the block out of the library is a worked one. It takes 36 iterations and 46 evaluations there, against Nelder-Mead's 85 and 159 on the same surface.

Ports

• c – the term coefficients, a column [T,1] with one entry per ROW of Term Exponents, in the same order. A row vector is refused, because the block reads the column entry by entry.
• x – the minimizer, [n,1], one entry per variable (n = the columns of Term Exponents).
• f(x) – the value there, a scalar [1,1].
• exitflag – a scalar [1,1], fminunc's own: 1 the gradient test below was met; 2 the step got small first; 5 the line search could not improve on the point it had; 0 a limit below stopped the search; −3 the polynomial fell below −1e20, which for a polynomial means it is unbounded below and has no minimum to find.

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 terms, n is 1 to 4 variables, and no single term may total more than degree 4. Those three caps are the export's, not the search's – see Code export.
• Start Point – x0, fminunc's second argument, a vector of n entries. The search is LOCAL: it descends into the valley it starts in, so on a surface with several minima this parameter chooses which one is reported.
• Optimality Tolerance – optimoptions' OptimalityTolerance, positive; default 1e-6. The search stops with flag 1 once ∥g∥_∞ < tol · (1 + ∥g₁∥_∞) – the largest entry of the gradient, measured RELATIVE to the gradient at the start point, which is fminunc's own test and not an absolute one.
• Step Tolerance – StepTolerance, positive; default 1e-6. Checked only AFTER the gradient test, on the RELATIVE displacement ∥Δx / (1+|x|)∥_∞, and it reports flag 2, not flag 1: a step that got small while the line search was still limited is not a stationary point and this block does not call it one.
• Maximum Iterations – MaxIterations, 1 to 2000, default 400, which is MATLAB's own.
• Maximum Function Evaluations – MaxFunctionEvaluations, same range, default 400. Whichever limit binds first stops the search and the exit flag is 0.
• 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, printed from one description of the iteration, so every target does the same arithmetic in the same order as the simulation.

The exponent pattern is unrolled: a power becomes repeated multiplication, because no target can take a loop bound out of a real array. One evaluation computes f and all n partial derivatives in the same pass – about T·(1+n)·(2+degree) statements – and the body holds three of them, which is why T, n and the degree are capped where they are. The main loop and both halves of the line search are emitted with fixed trip counts and a flag that stops the arithmetic once the answer is found, because none of the ten has an unbounded loop a hardware target could also carry.

⚠ The three HDL targets run the search in simulation-only real arithmetic, quantizing only at the port boundaries: a quasi-Newton step 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 fminunc 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, exactly as its three neighbours do.
• ⚠ LOCAL, and a gradient method does not certify a minimum either. It reports a point where the gradient is small, which on a non-convex surface can be a different valley for a different Start Point – and, in principle, a saddle. That is the method, not a defect.
• ⚠ A polynomial that is unbounded below has no answer to give, and this is the common case with freely varying coefficients: any odd-degree term can dominate. The block reports exit flag −3 rather than a large number dressed up as a minimizer. Keep the highest-degree term EVEN with a coefficient that cannot change sign if every sample is to have a minimum.
• ⚠ Where MATLAB throws, this block answers. If rounding leaves the search direction pointing uphill, fminunc stops with an error ("Search direction is not a descent direction"). A block cannot do that – every port carries a number on every sample – so it stops there and reports flag 5 with the best point it reached.
• eps has no spelling in nine of the ten targets, so MATLAB's eps and sqrt(eps) in the Hessian-update test are the exact binary constants 2⁻⁵² and 2⁻²⁶ here.