Second Order Cone Program — Control Systems/Optimization

Second-Order Cone Program

Control Systems / Optimization

Solves min_x f'·x subject to K second-order cone constraints ‖A_k·x − b_k‖ ≤ d_k'·x − γ_k, together with A·x ≤ b, Aeq·x = beq and lb ≤ x ≤ ub, on every step – MATLAB's coneprog. Cone constraints are what robust linear programs, norm and distance limits, friction cones and many model-predictive-control constraints come to. The cones and the other constraint matrices are settings; the linear term f and the inequality right-hand side b arrive on ports. The method is a primal-dual interior point on the homogeneous self-dual embedding, with Nesterov-Todd scaling and Mehrotra's predictor-corrector, so an infeasible or unbounded problem is recognized and flagged rather than iterated on forever.

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].
• fval – the objective at x, f'·x, [1,1].
• exitflag – [1,1], with coneprog's values: 1 optimal, 0 the iteration limit was reached, −2 no feasible point exists, −3 the problem is unbounded, −7 the step became too small to continue, −10 the problem is numerically unstable. 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.
• 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 leave a variable unbounded on that side.
• Cone Row Counts – one entry per cone, the number of rows of its A_k: [2 3] is two cones, of 2 and 3 rows. [] for no cone (a linear program).
• Cone Matrices – the A_k stacked on top of each other, [Σ rows, n].
• Cone Vectors – the b_k stacked the same way, one column.
• Cone Directions d – one row d_k' per cone, [K,n].
• Cone Offsets gamma – one γ_k per cone, [K,1].
• Optimality Tolerance – the duality gap to stop at, absolute or relative to |f'·x|. Default 1e-6, coneprog's own.
• Constraint Tolerance – the scaled primal and dual residuals to stop at, and the certificate threshold for infeasibility. Default 1e-6.
• Maximum Iterations – default 200, coneprog's; each iteration factors one small matrix.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

At most 12 variables, 8 cones, 11 equalities, and 60 rows from A, the finite bounds and the cones together (a cone of r rows counts r + 1).

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 solve runs in every generated body; its loop is Maximum Iterations 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). coneprog 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.
• ⚠ This is not coneprog's own algorithm, whose code is sealed: it is an independent interior point that finds the same optimum. At the default tolerances the two agree on every exit flag of the 160 problems measured and on f'·x to about the tolerance; x can differ more where the optimum lies on a cone's curved boundary and the objective is flat along it. Tighten both tolerances for a closer x.
• ⚠ An exit flag other than 1 leaves x defined but not optimal: it is the last iterate (scaled back by the embedding's τ), where MATLAB returns an empty x.
• Each finite bound becomes one more constraint row; lb = ub fixes a variable exactly as an equality would.