Nonnegative Least Squares — Control Systems/Optimization

Nonnegative Least Squares

Control Systems / Optimization

Solves min_x ||C·x − d|| subject to x ≥ 0 on every step – MATLAB's lsqnonneg(C, d). The matrix C is a setting, the right-hand side d arrives on the port, and the answer leaves with the residual norm ||C·x − d||² and an exit flag. The method is the Lawson-Hanson active set: it grows the set of positive entries one at a time, taking the entry whose multiplier w = C'(d − C·x) is largest, and backs off whenever an entry would go negative.

Ports

• d – the right-hand side, an [m,1] column with one entry per ROW of C.
• x – the minimizer, [n,1], one entry per COLUMN of C; every entry is zero or positive.
• resnorm – ||C·x − d||², [1,1], as lsqnonneg's second output: on an exit flag of 0 it is the residual of the last accepted iterate, not of the x that comes out.
• exitflag – [1,1]: 1 the answer is optimal, 0 the iteration limit was reached. Read it before using x.

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.
• Coefficient Matrix C – the matrix, [m,n] with m ≥ n and at most 48 rows and 12 columns. It must have full column rank, so that the answer is unique; a C whose columns are numerically dependent is refused with that reason when the model is built.
• X Tolerance – lsqnonneg's TolX: an entry of x below it is taken as zero and leaves the positive set, and a multiplier below it stops the search. 0 selects MATLAB's own default, 10·eps·||C||₁·max(m,n).
• Maximum Iterations – the limit on INNER iterations, which is what lsqnonneg itself limits; 0 selects its 3n. When the limit is reached the exit flag is 0 and x is the intermediate solution of the positive set, exactly as lsqnonneg returns it – which may hold entries that are zero or negative.
• 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 the same program the block runs, so every target does the same arithmetic in the same order as the simulation. C, its Gram matrix C'C and the resolved tolerance are baked in as constants at export time; re-export after changing them.

The whole solve runs in every generated body, and its cost is bounded rather than merely finite: at most (n+1)·(K+1)+1 passes, each one a Cholesky of an n×n matrix, where K is the resolved iteration limit. The loop is that length in the emitted code, and it exits early through a flag.

The three HDL targets carry the arithmetic in real and quantize only at the ports: simulation-only, not offered as synthesizable. A Cholesky has a division by a pivot and a square root, and the active set is a data-dependent branch, none of which belongs in a Q16.16 datapath.

Simulink bridge

None (Support::None). lsqnonneg is a MATLAB function and the Optimization Toolbox ships no Simulink library at all, so there is no path a diagram could name. The bridge reports this block rather than dropping it silently, and it has no parity testbench; code export verification covers all ten languages.

Notes

• Algebraic: the answer depends on this step's d alone, and nothing is carried between steps. It is not linear in d, so it carries no state space.
• ⚠ The subproblem is solved through the normal equations (a Cholesky of the masked C'C), where MATLAB's backslash uses a QR factorization. That squares the condition number of C, so a nearly dependent C loses digits this way that MATLAB would keep. Measured over 1450 random problems against R2026a: worst |Δx|/(1+|x|) 2.7e-13.
• ⚠ An exit flag of 0 does not mean "nearly converged": x is then lsqnonneg's intermediate z and may hold negative entries, and resnorm belongs to the previous iterate. That is what MATLAB returns, and it is reproduced rather than tidied.
• The measured agreement with R2026a, the problem sets behind it and the emitted MATLAB body's own check are on the source banner.