Minimax — Control Systems/Optimization

Minimax

Control Systems / Optimization

Finds the x that makes the largest of m polynomials as small as possible, their coefficients arriving on a port – MATLAB's fminimax, solved again on every sample:

minimize over x   max_i F_i(x),   F_i(x) = Σ_t c_i,t · x₁^e(t,1) · … · x_n^e(t,n),   i = 1 … m

The exponents e are configuration – the problem's shape, which fixes the loop bounds every export unrolls. The m objectives share that ONE term dictionary, and their coefficients c are the signal: one column per objective, stacked into a single port. So the surfaces being balanced can change on every sample while their form does not.

The search is fminimax's own: an artificial variable γ joins x, every objective becomes the constraint F_i(x) − γ ≤ 0, and a sequential quadratic programming method minimizes γ – a BFGS Hessian approximation, an active-set QP with its own feasibility phase, and a merit line search. Gradients are exact, because a polynomial's gradient is another polynomial.

The factory setting is fminimax's documented example: five objectives in two variables over the terms [2 0; 0 2; 1 0; 0 1; 0 0] (x₁², x₂², x₁, x₂, 1) – 2x₁² + x₂² − 48x₁ − 40x₂ + 304, −x₁² − 3x₂², x₁ + 3x₂ − 18, −x₁ − x₂ and x₁ + x₂ − 8 – from [0.1; 0.1]. Its answer is x = (4, 4) with a largest objective of 0.

Ports

• c – the coefficients, a column [m·T,1]: rows 1..T are objective 1's, one per ROW of Term Exponents in the same order, rows T+1..2T objective 2's, and so on. A row vector is refused.
• x – the minimizer, [n,1] (n = the columns of Term Exponents).
• F(x) – every objective at x, [m,1].
• max F – the largest of them, a scalar [1,1] (fminimax's maxfval).
• exitflag – a scalar [1,1], fminimax's own: 4 the search direction became smaller than twice Step Tolerance; 5 the predicted change in γ became smaller than Function Tolerance (both with the constraints met); 0 a limit below stopped the search; −2 no point was found where the constraints hold. fminimax has no flag 1: its first-order test is switched off for this problem.

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 Objectives – m, from 1 to 6. With m = 1 the block minimizes a single polynomial.
• Start Point – x0, n entries. The search is LOCAL, so on a problem with several local minimax points this chooses which one is found.
• 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 γ is below this.
• Constraint Tolerance – ConstraintTolerance, positive; default 1e-6: how far F_i(x) − γ may exceed 0 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, 100·(n+1) – the +1 is γ, which fminimax counts as a variable.
• 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 objective. 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 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 fminimax 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.
• ⚠ LOCAL. The method finds a point where no small step lowers the largest objective; on a non-convex problem that can depend on Start Point.
• ⚠ If the largest objective is unbounded below – every objective can be driven down together, which an odd-degree term with a coefficient that can change sign allows – there is no answer, and the point reported is wherever the evaluation limit stopped the search (flag 0) or where the line search gave up. Keep at least one objective bounded below if every sample is to have a minimax point.
• 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.