Mixed Integer Linear Program — Control Systems/Optimization

Mixed-Integer Linear Program

Control Systems / Optimization

Solves min_x f'·x subject to A·x ≤ b, Aeq·x = beq and lb ≤ x ≤ ub, with the variables listed in Integer Variables restricted to whole numbers, on every step – MATLAB's intlinprog. Integer variables are how on/off decisions, counts and discrete choices enter an optimization: unit commitment, scheduling, a hybrid controller's mode. The constraint matrices and bounds are settings; the linear term f and the inequality right-hand side b arrive on ports. The method is branch and bound: each node solves the linear relaxation with an interior point, branches on the most fractional integer variable, and prunes a node that cannot beat the best whole-number point found.

Ports

• f – the linear term, an [n,1] column with one entry per variable.
• b – the right-hand side of A·x ≤ b, a [p,1] column with one entry per row of A. When A is empty this port is unused and must be connected to any [1,1] signal.
• x – the minimizer, [n,1], its integer entries exact whole numbers.
• fval – the objective at x, f'·x, [1,1].
• exitflag – [1,1], with intlinprog's values: 1 optimal within the gap tolerances, 2 stopped early (node limit) with a whole-number point, 0 stopped early without one, −2 no feasible point exists, −3 the linear relaxation is unbounded. 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.
• Integer Variables – the indices of the integer variables, 1-based as in intlinprog's intcon: [1 3] makes x1 and x3 whole numbers. At least one; each needs finite lower and upper bounds, at most 1e6 apart.
• Inequality Matrix A – [p,n], or [] for none. Its right-hand side is the b port.
• Equality Matrix Aeq, Equality Vector beq – [q,n] and [q,1], or [] for none; the rows of Aeq must be linearly independent.
• Lower Bounds lb, Upper Bounds ub – [n,1] each or []; -Inf/Inf entries are allowed on continuous variables only.
• Maximum Nodes – the cap on branch-and-bound nodes per step, 1 to 2000; default 1000. Every node is one linear solve, so this bounds the step's cost.
• Integer Tolerance – how far from a whole number a value may be and still count as one. Default 1e-5, intlinprog's.
• Absolute Gap Tolerance, Relative Gap Tolerance – a node is pruned unless it can beat the best point by more than max(absolute, relative·|best|). Defaults 0 and 1e-4, intlinprog's; set both to 0 for the exact optimum.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

At most 12 variables, 12 of them integer, and 60 rows from A, the finite continuous bounds and two rows per integer variable together.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. The block's simulation runs the very program that is printed, so every target does the same arithmetic in the same order. The whole search runs in every generated body; its node loop is Maximum Nodes long and 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.

Simulink bridge

None (Support::None). intlinprog is an Optimization Toolbox function and that 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 f and b alone, and nothing is carried between steps. It is not linear in f and b, so it carries no state space.
• ⚠ x jumps: a whole-number answer changes discontinuously as f and b move, so a target that sees slightly different inputs – the HDL targets' Q16.16 ports – can land on a different optimum when two tie.
• ⚠ This is not intlinprog's algorithm, whose code is sealed: where several points share the optimal value, the two can return different x with the same fval.
• ⚠ An exit flag other than 1 or 2 leaves x defined but not a solution: it is the last relaxation's point, where MATLAB returns an empty x.