Finite-budget barrier or SQP solve for a transcription NLP

Inputs are a transcription record from Finite-horizon shooting and Runge–Kutta collocation transcription, a primal initial iterate $z_0$ and full multiplier iterate ${\mu }_0=({\nu }_0,{\lambda }_0)$ in the fixed equality/inequality constraint ordering, an optional canonical QP start $(x^0,y^0,s^0)$ with nonnegative primal variables $x^0$, free equality multipliers $y^0$, and nonnegative dual slacks $s^0$, an SQP model builder, and finite nonnegative tolerances ${\tau }_p,{\tau }_d,{\tau }_c,{\tau }_{stage},{\tau }_{quad}$. The stage and quadrature tolerances must match those bound into the transcription record. The KKT tolerances ${\tau }_p,{\tau }_d,{\tau }_c$ bound primal feasibility, stationarity, and multiplier-feasibility/complementarity; in the QP route ${\tau }_c$ also bounds the primal-dual gap. Also supply finite integer SQP and QP update caps $K,N\in {\mathbb{N}}_0$, positive finite integer penalty/backtracking caps $R,J_{max}\in \mathbb{N}$ with $R,J_{max}\ge 1$, merit parameters ${\rho }_0>0$, $\gamma >1$, $0<\beta ,\sigma <1$, and the finite-budget exact-Hessian branch selector and identified active set when that branch is requested. The IVP estimator uses its own declared controller tolerances; its status is not treated as an error certificate.

The SQP model builder is a deterministic finite procedure which, at each queried full state $(z_k,{\nu }_k,{\lambda }_k)$, returns either a failure status or the finite NLP data $f_k,h_k,g_k,D{h}_k,D{g}_k$, its objective gradient, a symmetric positive-definite SQP model matrix $W_k$ when the merit branch is selected, and the exact Lagrangian Hessian when the exact-Hessian branch is selected. The model matrix may depend on the supplied current multiplier vectors. For the exact-Hessian branch with identified active set $I$, evaluate that Hessian with equality multipliers ${\nu }_k$, active inequality multipliers $({\lambda }_k{)}_I$, and zero inactive multipliers; its required positivity is the separate reduced-curvature condition of the exact step, not full positive definiteness. It must use $g_i\le 0$ and nonnegative inequality multipliers. In the merit branch, the associated quadratic subproblem is exactly ${\min }_p\mathrm{\nabla }{f}_k^{T}p+\frac{1}{2}{p}^T{W}_{k}p$ subject to $h_k+D{h}_{k}p=0$ and $g_k+D{g}_{k}p\le 0$. For that branch, the builder also supplies the active equality/inequality row data required by L1 merit globalization with finite penalty and backtracking budgets. Solver tolerances are used as follows: ${\tau }_p$ bounds the norm of equality and positive inequality violations, ${\tau }_d$ bounds the stationarity residual, and ${\tau }_c$ bounds multiplier-feasibility and complementarity residuals. The stage and quadrature tolerances are passed to the transcription's corresponding residual and certified-error checks. The IVP estimator uses its own declared controller tolerances and has a separate status; its embedded difference is not a certified error bound. Any called builder, evaluator, canonicalizer, or solver returns a value or failure status in finite time.

First audit the transcription dimensions, sampled domains, and data status fields. If stage construction failed, return the partial record with its stage and NLP statuses and do not call a solver. If the model satisfies A linear-quadratic transcription with affine constraints is a convex QP, map its finite quadratic objective and affine constraints to canonical LP/QP form: replace each free variable $z_i$ by $z_i^{+}-z_i^{-}$ with $z_i^{+},z_i^{-}\ge 0$; for each affine inequality $a_i^{T}z+d_i\le 0$, add a primal slack ${\xi }_i\ge 0$ and equality $a_i^{T}z+{\xi }_i=-d_i$; retain existing equalities; and transform the quadratic objective by substitution. The transcription record supplies finite stored coefficient arrays. Canonicalize using exact rational arithmetic on their dyadic values, then round each resulting coefficient once to the nearest normal representable value under Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add, preserving exact zeros; any range failure makes the QP route inapplicable. Check the resulting equality matrix by exact rational row elimination; if it is rank deficient or the conversion is otherwise inadmissible, mark the QP route inapplicable and proceed to SQP. The optional start $(x^0,y^0,s^0)$ is in these converted coordinates and is checked constructively: require every primal coordinate, including each added ${\xi }_i$, and every dual slack coordinate $s^0$ to be strictly positive, and require $y^0$ to be finite with the correct equality dimension. If it is absent or fails this check, record strict-start-unavailable and proceed to SQP; do not infer infeasibility or decide whether some other strict start exists. If it passes, call Finite-budget primal-dual Newton methods for LP, convex QP, and symmetric cones once with centering and fraction-to-boundary parameters both $1\mathrm{/}2$, ${ϵ}_p={\tau }_p$, ${ϵ}_d=\min ({\tau }_d,{\tau }_c)$, ${ϵ}_g={\tau }_c$, and cap $N$. The minimum in ${ϵ}_d$ ensures the supplier's single tolerance for both its dual and complementarity residuals meets each requested bound; ${ϵ}_g$ separately bounds the certified primal-dual gap. If it returns invalid-input or strict-start-unavailable, record that QP status and proceed to SQP. Return every other status and audited residuals. certified-gap certifies the resulting stored canonical floating-point QP, interpreted as exact dyadic rationals, under Fraction-to-boundary steps preserve positivity and exact primal-dual feasibility certifies a gap; it is not a certificate for the unsampled continuous problem. A residual-candidate remains a candidate. No other QP status is reinterpreted as success.

If the QP route is inapplicable or lacks a supplied strict start, run the SQP branch only when the transcription status is solver-ready constructed; otherwise return its not-smooth-nlp or model-construction status without an SQP call. Initialize $(z,\nu ,\lambda )=(z_0,{\nu }_0,{\lambda }_0)$ and verify the full multiplier dimensions. An invalid state returns model-build-failure with its last available record. For $k=0,1,\dots ,K$, in order, query the model builder at the entire current state $(z_k,{\nu }_k,{\lambda }_k)$, propagating any builder failure without changing that state, and compute $r_p=(h_k,[g_k{]}_{+}),r_d=\mathrm{\nabla }{f}_k+D{h}_k^T{\nu }_k+D{g}_k^T{\lambda }_k,r_{\lambda }=[-{\lambda }_k{]}_{+},r_c=({\lambda }_{k,i}{g}_{k,i}{)}_i,$ where positive parts are componentwise. If $\mathrm{\parallel }{r}_p\mathrm{\parallel }\le {\tau }_p$, $\mathrm{\parallel }{r}_d\mathrm{\parallel }\le {\tau }_d$, $\mathrm{\parallel }{r}_{\lambda }\mathrm{\parallel }\le {\tau }_c$, and $\mathrm{\parallel }{r}_c\mathrm{\parallel }\le {\tau }_c$, return kkt-residual-met with the current decision and residual record; this is an approximate KKT candidate unless all residuals are exactly zero. If $k=K$, return budget-exhausted. Otherwise, if the exact-Hessian branch is selected and an active set $I$ has been correctly identified, call Full-step exact-Hessian SQP after active-set identification at the current pair $(z_k,({\nu }_k,({\lambda }_k{)}_I))$ with inactive multipliers set to zero, using the exact Hessian evaluated at that same pair. On its successful full Newton-pair return $(z^{+},{\mu }^{+})$, use $z^{+}$, place the equality and active-inequality components of ${\mu }^{+}$ into the full vectors $(\nu ,\lambda )$, and set inactive inequality components to zero; any failure stops with that status without changing the current state. In the merit branch, call L1 merit globalization with finite penalty and backtracking budgets with the current model $W_k$, ${\rho }_0,\gamma ,\beta ,\sigma ,R,J_{max}$. On accepted or accepted-soc, use its actual returned accepted point $z^{+}$, including the SOC displacement when present, and full returned QP multipliers $({\nu }^{QP},{\lambda }^{QP})$ to set $(z,\nu ,\lambda )=(z^{+},{\nu }^{QP},{\lambda }^{QP})$ for the next model-builder query. Retain the complete supplier record, including its retained equality basis and selected inequality support. Do not replace the full vectors by reduced support multipliers or infer the true nonlinear active set from QP support. Any other supplier status stops the loop without changing the current state. The exact-Hessian local branch has only the conditional convergence guarantee in Exact-Hessian SQP converges quadratically on the identified active KKT system.

Every return is a result record with the last available decision variables, the full current SQP multiplier vectors when that branch has been reached, its last supplier record when one exists, defect residuals, path/endpoint constraint residuals, KKT residuals, objective value, and separate nlp_status, implicit_stage_status, ivp_status, quadrature_status, mesh_status, and budget_status fields. Fields not reached are marked not-run; failed evaluations carry their failure status and do not supply a candidate value. The finite caps bound the procedure. It does not guarantee a start, solver success, a global solution of a nonconvex problem, continuous-path feasibility from sampled constraints, or a certified approximation error for the continuous ODE problem.