Finite-budget projected-gradient loop with state-adjoint residual certificates

Assume Countable Choice. Let $E\subset {\mathbb{R}}^d$, $d\ge 2$, be a bounded polyhedral open control domain with a finite matching affine simplicial mesh as used in The L2 projection onto a cellwise constant box clips cell averages. Let $U_h\subset U=L^2(E)$ be the finite-dimensional space of cellwise-constant controls on this mesh, and let $Q_h=U_h\cap Q_{ad}$ be nonempty with bounds constant on each control cell. The target is the exact continuous reduced objective $\widehat{J}$ of Linear coercive elliptic PDE control problem and reduced objective, restricted to $Q_h$; write $Ju(q)=c+Sq$ and $g(q)=\mathrm{\nabla }\widehat{J}(q)=B^{\ast }p(q)+\alpha q$. Choose finite certified bounds ${\beta }_B\ge \mathrm{\parallel }B\mathrm{\parallel }$ and $L_h\ge \mathrm{\parallel }S{\mathrm{\mid }}_{U_h}{\mathrm{\parallel }}^2+\alpha >0$. Given a finite budget $N\ge 1$, tolerance $\tau \ge 0$, and initial $q_0\in Q_h$, repeat for $k=0,\dots ,N-1$:

Fix a finite-dimensional conforming state space $V_h\subset V$ and the discrete primal and transposed adjoint equations satisfying the first-Strang hypotheses in State and adjoint Strang bounds give a reduced-gradient error certificate at each queried control. An additional input is a residual-evaluation routine $\mathcal{C}$. For every queried $q_k$, it must return in finite time either a valid tuple $({\tilde{u}}_k,{\tilde{p}}_k,{\epsilon }_{u,k},{\epsilon }_{p,k})$ or failure. A valid tuple has ${\tilde{u}}_k,{\tilde{p}}_k\in V_h$ solving the specified discrete primal and transposed adjoint equations, with the adjoint right-hand side formed using ${\tilde{u}}_k$, and nonnegative finite bounds that dominate the complete primal and adjoint first-Strang right-hand sides. Those bounds retain all form, geometry, quadrature, and load defects, including the adjoint load contribution $\mathrm{\parallel }J{\mathrm{\parallel }}^2{\epsilon }_{u,k}$ from the state error. The finite-return condition is part of the algorithm's admissible input contract; if the backend cannot certify the equations and bounds, it returns failure.

A second input is a projection-evaluation routine $\mathcal{P}$. Write $q_{k,K}:=q_k{\mathrm{\mid }}_K$. For every queried $(q_k,{\tilde{p}}_k,L_h)$ it must return in finite time either failure or finite real cell values $z_{k,K}$ and a finite real $r_k\ge 0$ satisfying $z_{k,K}={\mathrm{clip}}_{[q_a{\mathrm{\mid }}_K,q_b{\mathrm{\mid }}_K]}(q_{k,K}-\frac{\alpha q_{k,K}+\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\int }_K{B}^{\ast }{\tilde{p}}_k}{L_h}),r_k^2={\sum }_{K\in {\mathcal{T}}_h}\mathrm{\mid }K\mathrm{\mid }\mathrm{\mid }{q}_{k,K}-z_{k,K}{\mathrm{\mid }}^2\mathrm{.}$ Thus the returned cellwise function is exactly $P_{Q_h}(q_k-{\tilde{g}}_k\mathrm{/}{L}_h)$ and $r_k$ is its exact step norm by The L2 projection onto a cellwise constant box clips cell averages. Computing the required cell averages, clipped values and norm is part of this finite-return backend contract; it is not inferred from boundedness of $B$, and no finite representation of the entire function $B^{\ast }{\tilde{p}}_k$ is assumed. Unavailable exact values must be reported as failure, not left pending.

1. Call $\mathcal{C}(q_k)$. If it returns failure or no valid bounds, return $(q_k,\mathtt{\text{inconclusive}},+\mathrm{\infty })$. Otherwise use its conforming finite-element state and adjoint approximations ${\tilde{u}}_k,{\tilde{p}}_k$ and complete Strang bounds ${\epsilon }_{u,k},{\epsilon }_{p,k}$. Set ${\delta }_k={\beta }_B{\epsilon }_{p,k}$ and ${\tilde{g}}_k=B^{\ast }{\tilde{p}}_k+\alpha q_k$.
2. Call $\mathcal{P}(q_k,{\tilde{p}}_k,L_h)$. If it returns failure or no valid cell values and norm, return $(q_k,\mathtt{\text{inconclusive}},+\mathrm{\infty })$. Otherwise set ${\tilde{q}}_{k+1}{\mathrm{\mid }}_K=z_{k,K}$ and ${\tilde{\rho }}_k=r_k$ from its valid return.
3. If ${\tilde{\rho }}_k+{\delta }_k\mathrm{/}{L}_h\le \tau$, return $(q_k,\mathtt{\text{certified}},{\tilde{\rho }}_k+{\delta }_k\mathrm{/}{L}_h)$.
4. Otherwise set $q_{k+1}={\tilde{q}}_{k+1}$ and continue.

If all $N$ iterations finish without that test, return the last evaluated iterate $(q_{N-1},\mathtt{\text{budget-exhausted}},{\tilde{\rho }}_{N-1}+{\delta }_{N-1}\mathrm{/}{L}_h)$. Both routines' finite-return contracts ensure that an unavailable state/adjoint bound or projection/residual value is reported as failure rather than left pending. The reported number certifies a projected-gradient residual for the exact continuous PDE objective restricted to $Q_h$; it is not a continuous-control discretization error bound.