Feasibility-Preserving Acceleration

• The paper establishes that acceleration schemes, such as extrapolation-based and component-identification methods, rigorously maintain feasibility constraints at every iterate, achieving 2x–4x speedups in problems like sparse affine feasibility.
• The work demonstrates that sequential optimization methods like AA(d)-FSLP leverage projected linearizations and trust-region corrections to ensure all intermediate solutions remain feasible while reducing constraint evaluations.
• The concept extends to real-time control, quantum dynamics, and accelerator physics, where feasibility-preserving strategies uphold safety and physical constraints, enabling high-fidelity, experimentally viable performance.

A feasibility-preserving acceleration result refers to a rigorous guarantee that an acceleration scheme—intended to improve convergence, computational efficiency, or dynamical timescales—does not compromise feasibility constraints inherent in the original problem formulation. Such results are central to guaranteeing that all iterates, trajectories, or outputs generated by the accelerated scheme remain within the admissible (feasible) set defined by physical, mathematical, or algorithmic constraints, thus enabling both faster performance and correctness in constrained optimization, control, or dynamical system contexts.

1. Feasibility-Preserving Acceleration in Projection and Fixed-Point Methods

In projection-based or fixed-point iterative algorithms for feasibility problems, classical acceleration techniques often risk violating feasibility during intermediate steps, even if they converge to feasible solutions asymptotically. Feasibility-preserving acceleration results rigorously define acceleration schemes that maintain feasibility at every iteration, not just in the limit.

In "Global convergence and acceleration of projection methods for feasibility problems involving union convex sets" (Alcantara et al., 2022), two principal acceleration frameworks are introduced:

• Extrapolation-based acceleration: Inserts a “momentum” step before projection, with the extrapolation parameter $t_k$ chosen so as to decrease a Lyapunov function and preserve feasibility.
• Component-identification acceleration: Monitors which convex subset is active at each iterate, and switches to a specialized subproblem when convergence stagnates, using only operations guaranteed to maintain feasibility.

The main convergence theorem states that if the set-valued operator $T$ governing the feasible set is upper semicontinuous and has a coercive Lyapunov function, then every cluster point of the accelerated sequence is a fixed point of $T$, and, under calmness, the entire sequence converges. Both Accelerated Alternating Projections (AMAP) and Accelerated Averaged Projections (AMAveP) exhibit these properties, yielding 2x–4x speedups (and more) in feasibility problems such as sparse affine feasibility and the linear complementarity problem, with all iterates provably feasible at every step.

2. Feasibility-Preserving Acceleration in Sequential Optimization

For nonlinear programs, Feasible Sequential Linear Programming (FSLP) provides a globally convergent trust-region algorithm ensuring feasibility of every intermediate solution via projected linearizations and correction steps. In "Anderson Accelerated Feasible Sequential Linear Programming" (Kiessling et al., 2022), Anderson Acceleration (AA($d$)) is shown to accelerate the zero-order (projection-based) inner updates of FSLP while rigorously retaining feasibility. This is achieved by:

• Constructing the AA($d$) affine combination only over feasible iterates.
• Projecting the accelerated step back into the trust region, which by design encloses the feasible set guaranteed by the linearization validity.

Theoretical results show local linear convergence and demonstrate that, under the strong-regularity assumption, AA($d$)-FSLP remains feasible throughout all inner and outer iterations. Empirical results confirm 2x–4x reductions in constraint evaluations and solve time versus classical FSLP, with zero feasibility violations.

3. Feasibility-Preserving Acceleration in Control via Control Barrier Functions

In real-time optimal control, Quadratic Programs (QPs) formulated with Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs) are widely used to enforce safety and dynamics constraints. Tight input bounds or rapidly changing safety requirements can render such QPs infeasible, jeopardizing both safety and control performance.

"Feasibility Guaranteed Traffic Merging Control Using Control Barrier Functions" (Xu et al., 2022) describes an acceleration-by-tracking approach where, at each timestep, the optimal unconstrained solution is tracked via QPs incorporating CLF and CBF constraints. To guarantee feasibility of every QP (and thus every control action), additional feasibility CBFs are derived that preempt “closure” of the admissible set by propagating the effect of input bounds through the CBFs:

• The feasibility CBFs impose that the next-step CBF upper bound always lies above the lower input bound, thereby ensuring overlap between admissible control values from safety and actuator constraints.
• The QP is solved over the intersection of all safety and feasibility CBF halfspaces, always yielding a nonempty feasible set under mild assumptions.

This ensures that each real-time QP is always feasible, eliminating infeasibility-induced failures or safety violations regardless of operating conditions. Simulation studies show strict satisfaction of all constraints and smooth, conservative control action near constraint boundaries.

4. Feasibility-Preserving Acceleration in Dynamical Quantum Systems

In the context of quantum state transfer and quantum Zeno dynamics, a process is feasible if system evolution remains confined to a “protected” subspace (the Zeno subspace). Traditional shortcuts to adiabaticity often compromise this property by introducing nonphysical auxiliary terms.

"Flexible and experimentally feasible shortcut to quantum Zeno dynamic passage" (Li et al., 2016) introduces an acceleration scheme where the control Hamiltonian $H_\mathrm{acc}(t) = \sum_j u_j(t) H_{c_j}$ is constructed only from experimentally realizable interactions, and the time-dependent controls $u_j(t)$ are designed to monotonically decrease a Lyapunov function $V(t) = \operatorname{Tr}[H_I^2 \rho]$. The feasibility-preserving result is that these controls ensure $V(t)\to0$, i.e., the state is always steered into (and remains in) the Zeno subspace throughout the accelerated process.

In contrast to invariant-based schemes, this approach relaxes the commutator constraints and allows acceleration using only physically available interactions, guaranteeing both speedup and preservation of the protected dynamics essential to quantum information protocols such as high-fidelity entanglement generation.

5. Feasibility-Preserving Target and Channel Design in Accelerators

In particle accelerators, feasibility-preserving acceleration refers to parameter choices and structural design that ensure material, beam, and electromagnetic constraints are never violated, permitting high-gradient acceleration without compromising beam quality, structural integrity, or target survivability.

For example, in "TeV/m Nano-Accelerator: Investigation on Feasibility of CNT-Channeling Acceleration at Fermilab" (Shin et al., 2015), simulation-based and analytic assessments confirm that by tuning carbon nanotube (CNT) channel dimensions (tens to hundreds of nanometers), transverse phase-space acceptance is maximized, thermal and mechanical loads remain below damage thresholds for MW-class beams, and GV–TeV/m gradients are achievable with existing beam qualities. Channel radii can be selected to guarantee phase-stability and survivability of both the channel and beam in projected experimental runs, thus feasibility is preserved under aggressive acceleration parameters.

Similarly, in "Laser ion acceleration from tailored solid targets with micron-scale channels" (Lezhnin et al., 2022), theoretical modeling and multidimensional PIC simulations define optimal target channel lengths, densities, and radii that maximize ion energies (via two-stage combined TNSA+RPA), but always within material, laser, and plasma-parameter feasibility constraints. This approach yields upward of 1 GeV with 15% conversion efficiency, demonstrating robust feasibility-preserving acceleration under experimental conditions matching current petawatt-class facilities.

6. Numerical and Experimental Evidence

Across the reviewed domains, feasibility-preserving acceleration results are supported by extensive numerical validation:

• Iteration counts, wall-clock times, and constraint satisfaction metrics for accelerated projection and sequential LP methods indicate order-of-magnitude reductions in computational cost, with no feasibility loss (Alcantara et al., 2022, Kiessling et al., 2022).
• In constrained control, online simulations of traffic merging scenarios demonstrate strict satisfaction of all safety and input constraints when using feasibility CBFs, whereas standard unconstrained OCBF schemes fail frequently under tight merging conditions (Xu et al., 2022).
• Multidimensional electromagnetic PIC simulations in advanced accelerator studies confirm that predicted ion/proton energy gains are realizable while keeping material and electromagnetic state variables within feasible ranges (Shin et al., 2015, Lezhnin et al., 2022).
• Quantum dynamics simulations exhibit accelerated population transfer with fidelity above 95% and with all wavefunction components remaining confined to the Zeno subspace using flexible, realizable Hamiltonians (Li et al., 2016).

7. Significance and Generalization

Feasibility-preserving acceleration results establish a pathway for rigorously accelerating systems (algorithmic, control, or physical) while certifying that essential feasibility (physical, mathematical, or safety) constraints are globally and locally guaranteed at every step or instant. These results remove barriers to adopting more aggressive or efficient acceleration strategies in high-dimensional, safety-critical, or experimentally constrained systems. The methodologies span fixed-point theory, Lyapunov analysis, optimal control, sequential programming, and advanced accelerator physics, demonstrating broad applicability and a mature theoretical and practical foundation across mathematical programming, control engineering, and physical sciences.