Frequency-Varying Optimization (FVO)

• Frequency-Varying Optimization (FVO) is a collection of methods where the frequency variable is dynamic, acting as a moving constraint or evolving spectral field rather than a static parameter.
• It leverages differential and transport structures to track moving optima, ensuring rapid convergence under time-varying and delivery-time constraints in various engineering applications.
• FVO has practical impact across disciplines—from coordinating power system responses and enhancing time-frequency analysis to enabling spatially adaptive neural representations and optimizing ultrafast waveform shaping.

Frequency-Varying Optimization (FVO) denotes a class of optimization problems in which the relevant frequency variable is not treated as fixed, global, or purely parametric, but instead enters the optimization as a changing constraint, a moving spectral law, a spatially varying field, or a frequency-coupled operator. In the most explicit use of the term, FVO is a control framework for coordinating frequency response services in power systems (Xu et al., 23 Sep 2025). Closely related formulations appear in optimization-based time-frequency analysis for signals with fast-varying instantaneous frequency (Kowalski et al., 2015), in implicit neural representations that explicitly model a spatially varying local frequency field (Versace, 23 Nov 2025), in waveform shaping through a measured frequency-frequency reflection matrix in ultrafast time-varying media (Hooper et al., 18 Aug 2025), and in online and distributed optimization methods that track time-varying optima with uncertainty-aware perturbation or chirpy probing (Aarnoudse et al., 27 Aug 2025, Li et al., 26 Sep 2025). This suggests that FVO is best understood as a family of frequency-aware optimization paradigms rather than a single universally fixed formalism.

1. Scope and terminological usage

The literature currently associates FVO with several technically distinct, though structurally related, problem classes. In all of them, the optimization is driven by a nonstationary frequency law or by an object whose behavior varies across frequency, time, or space.

Domain	Representative formulation	Frequency-varying object
Power-system control (Xu et al., 23 Sep 2025)	ARU coordination under a delivery curve	Equality constraint $h(\Delta\omega_0(t))\,c_{agg}$
Time-frequency analysis (Kowalski et al., 2015)	Tycoon convex functional	Chirp term $G(t,\omega)\partial_\omega F(t,\omega)$
Implicit neural representations (Versace, 23 Nov 2025)	NSTR	Local spectrum field $S(x)$
Time-varying media (Hooper et al., 18 Aug 2025)	Reflection-matrix waveform shaping	Frequency-coupling operator $\boldsymbol{R}$

In the power-systems formulation, FVO is positioned as a specific subclass of time-varying optimization (TVO): the time variation is driven by grid frequency dynamics and, more specifically, by the delivery requirement curve of a frequency response service. The defining feature is a frequency-varying equality constraint rather than a generic exogenous time index. By contrast, the time-frequency analysis literature frames the problem as recovery of an ideal time-frequency representation by minimizing a functional that encodes reconstruction, concentration, and differential consistency for fast-varying instantaneous frequency. In INR research, the comparable idea is that useful frequencies are not constant across space and should therefore be represented and optimized as a spatial field. In waveform shaping for ultrafast time-varying media, the optimization variable is a spectral waveform acted on by a measured frequency-coupling matrix.

A common misconception is that frequency variation here always means a larger static Fourier basis. The surveyed works do not support that interpretation. Some of them instead optimize a moving equality target, a local spectral field, or an eigenchannel of a frequency-coupling operator. This suggests that “frequency-varying” refers less to basis size than to the fact that spectral structure itself is dynamic or location dependent.

2. Power-system FVO as a control problem

In power systems, FVO is formulated for an Aggregated Response Unit (ARU) whose required aggregate response varies with grid frequency (Xu et al., 23 Sep 2025). If asset $i$ delivers response relative to its baseline as

$x_i(t)=P_i(t)-P_i(0),$

and its feasible set is

$$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$

then the optimization problem is

$$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$

Here $\Delta\omega_0(t)$ is treated as an exogenous input, $c_{agg}$ is the aggregate contracted capacity, and $G(t,\omega)\partial_\omega F(t,\omega)$0 is the service delivery curve. For Dynamic Moderation (DM), the paper gives the service shape explicitly: deadband $G(t,\omega)\partial_\omega F(t,\omega)$1 Hz, no response; sublinear delivery between 0.015 and 0.1 Hz; linear delivery from 0.1 to 0.2 Hz; and full delivery at $G(t,\omega)\partial_\omega F(t,\omega)$2 Hz.

Under the assumptions that each $G(t,\omega)\partial_\omega F(t,\omega)$3 is twice continuously differentiable and strongly convex in $G(t,\omega)\partial_\omega F(t,\omega)$4, continuously differentiable in $G(t,\omega)\partial_\omega F(t,\omega)$5, and that a Slater condition holds for all $G(t,\omega)\partial_\omega F(t,\omega)$6, the optimizer is unique at each time. The key difficulty is that instantaneous satisfaction of the frequency-varying equality constraint is generally infeasible. The problem is therefore reformulated as Tracking of the Optimal Trajectory (TOT).

For negligible asset dynamics, the model is

$G(t,\omega)\partial_\omega F(t,\omega)$7

with tracking goal

$G(t,\omega)\partial_\omega F(t,\omega)$8

For BESSs with non-negligible dynamics, the model becomes

$G(t,\omega)\partial_\omega F(t,\omega)$9

with tracking goal

$S(x)$0

The proposed TOT controllers are projected fixed-time feedback laws augmented with feedforward terms that compensate for the fact that the optimizer moves with $S(x)$1. The convergence statements are fixed-time guarantees: for TOT-1,

$S(x)$2

and for TOT-2,

$S(x)$3

This fixed-time structure is central because the services impose maximum delivery times rather than asymptotic objectives. The paper maps these directly to NESO’s services: $S(x)$4 s for DR and $S(x)$5 s for both DM and DC.

The numerical results are framed around feasibility, delivery-time compliance, and scalability. In an IEEE 14-bus system with an ARU of 6 BESSs and a 50 MW contracted quantity, the DC case under a 1 pu load increase converges at about $S(x)$6 s with tracking time $S(x)$7 s, satisfying $S(x)$8 s and $S(x)$9 s. For DM under fluctuating load, convergence is about $\boldsymbol{R}$0 s. For DR with BESS dynamics included, convergence is about $\boldsymbol{R}$1 s, well within the 10 s deadline. In an IEEE 39-bus scalability study, Algorithm 2 has the smallest frequency-response mismatch, nearly zero, while a benchmark primal-dual projected gradient algorithm fails to track satisfactorily. The framework is also explicitly stated to be readily distributed through local estimates of the dual variable and a distributed estimator for the feedforward term.

3. Convex time-frequency optimization for fast-varying instantaneous frequency

A substantially earlier formulation with the same structural character appears in optimization-based time-frequency analysis (Kowalski et al., 2015). The paper does not use the acronym FVO, but it presents an optimization problem whose purpose is to recover a time-frequency representation that tracks fast-varying instantaneous frequency. The signal model is

$\boldsymbol{R}$2

with $\boldsymbol{R}$3 and $\boldsymbol{R}$4. The adaptive harmonic model imposes smoothness, boundedness, and frequency-separation conditions, including

$\boldsymbol{R}$5

The ideal time-frequency representation is

$\boldsymbol{R}$6

and the ideal time-varying power spectrum is

$\boldsymbol{R}$7

For fast-varying instantaneous frequency, the key differential consistency relation is

$\boldsymbol{R}$8

The central convex functional is then

$\boldsymbol{R}$9

This formulation combines four requirements that the paper identifies as properties of an ideal representation: signal reconstruction, sharp concentration in the time-frequency plane, differential consistency, and sparsity. The $i$0 term enforces concentration, while the extra transport-like term $i$1 is what allows the method to track fast frequency bending more effectively than ordinary synchrosqueezing. The resulting algorithm is called Tycoon, for “Time-frequency bY COnvex OptimizatioN.” It alternates between minimizing in $i$2 and in the chirp variable, using FISTA for the convex but nonsmooth subproblem in $i$3. For fixed $i$4, the accelerated convergence rate stated for FISTA is

$i$5

as compared with $i$6 for basic forward-backward splitting.

The paper also emphasizes identifiability. Under the adaptive harmonic model, the amplitude-phase decomposition is unique up to small $i$7 or $i$8 errors. In numerical experiments, Tycoon produces a much sharper ridge and better tracks the instantaneous frequency for a single-component fast-varying signal motivated by atrial fibrillation. On a clean two-component example, the OT-based metric reports mean $i$9 std values of $x_i(t)=P_i(t)-P_i(0),$0 for Tycoon, $x_i(t)=P_i(t)-P_i(0),$1 for EMD-HS, $x_i(t)=P_i(t)-P_i(0),$2 for STFT, $x_i(t)=P_i(t)-P_i(0),$3 for synchrosqueezed STFT, and $x_i(t)=P_i(t)-P_i(0),$4 for synchrosqueezed CWT. In the noisy case, Tycoon reports $x_i(t)=P_i(t)-P_i(0),$5, compared with $x_i(t)=P_i(t)-P_i(0),$6 for EEMD-HS, $x_i(t)=P_i(t)-P_i(0),$7 for STFT, $x_i(t)=P_i(t)-P_i(0),$8 for synchrosqueezed STFT, and $x_i(t)=P_i(t)-P_i(0),$9 for synchrosqueezed CWT. The paper concludes that the method is effective for fast-varying IF signals, but computationally expensive, with further work needed on efficiency, parameter tuning, and noise theory.

4. Spatially varying spectrum in implicit neural representations

In implicit neural representations, a closely related development is Neural Spectral Transport Representation (NSTR), which explicitly models a spatially varying local frequency field (Versace, 23 Nov 2025). The starting claim is that standard INR families—Fourier-feature MLPs, SIREN, and multiresolution hash grids—implicitly assume a global and stationary spectral basis. According to the paper, this is a poor match for signals whose frequency characteristics vary significantly across space, producing local high-frequency textures, smooth regions, and frequency drift phenomena. The stated consequences are unnecessary over-parameterization in smooth regions, underfitting or aliasing in high-frequency areas, slower optimization due to spectral mismatch, and poor scalability for heterogeneous signals.

NSTR introduces a learnable local spectrum field $$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$0 and a frequency transport network $$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$1 constrained by

$$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$2

The signal is reconstructed by spatially modulating a compact set of global sinusoidal bases: $$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$3 The local spectrum is parameterized as

$$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$4

where $$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$5 comes from interpolation over a coarse learnable grid and $$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$6 is a lightweight MLP. The two regularizers highlighted in the paper are PDE consistency, which enforces spectral transport structure, and smoothness regularization, which discourages noisy spectrum fields.

This decomposition separates global oscillatory structure, carried by the fixed sinusoidal bases, from local spectral variation, carried by $$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$7. The paper explicitly frames this as strong local adaptivity with a compact set of global sinusoidal bases. In the language of FVO, the optimization is not over a stationary frequency palette but over a frequency field with structured spatial dynamics.

The experimental evidence is presented as an accuracy-parameter and convergence advantage. On CelebA-HQ image regression, NSTR achieves 35.7 PSNR with 0.3M parameters, compared with Fourier MLP at 30.1 PSNR with 1.2M parameters, SIREN at 31.4 PSNR with 1.2M parameters, and Instant-NGP at 33.5 PSNR with 0.5M parameters. In audio reconstruction, it improves SNR by +3.5 dB over SIREN. On ShapeNet chairs and cars, Chamfer distance drops by 28–42% relative to SIREN-based DeepSDF. In compact NeRFs, parameter count is reduced by 2×–4× and training is sped up by about 1.5×. Training details reported in the paper are Adam with lr $$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$8, 20k–50k iterations depending on the dataset, $$\Omega_i=\{x_i(t)\mid P_i^{\min}\le P_i(0)+x_i(t)\le P_i^{\max}\},$$9 global frequencies, a 2-layer MLP of width 64 for $$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$0, a 3-layer MLP for $$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$1, and PDE weight $$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$2 unless otherwise stated. Ablations show that removing the PDE loss causes noisy $$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$3, worse convergence, and a 1–2 dB PSNR drop in image fitting.

Interpretability is an explicit part of the framework. The paper proposes visualizing

$$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$4

and reports high-magnitude Jacobians near structural boundaries, divergent spectral flows in textured regions, and stable low-residual behavior in smooth regions. This makes the model interpretable as a spectral transport system rather than a black-box implicit decoder.

5. Frequency-domain waveform optimization in ultrafast time-varying media

A distinct but mathematically related use of frequency-varying optimization appears in waveform shaping for ultrafast time-varying media (Hooper et al., 18 Aug 2025). The central object is a measured frequency-frequency reflection matrix. For a time-varying termination, the input-output relation is

$$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$5

which becomes, after Fourier transformation,

$$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$6

Under periodic modulation, this reduces to

$$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$7

The matrix $$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$8 encodes how an incident spectral component at one frequency is scattered into another. In a static medium it would be diagonal; in the time-varying case it is generally dense because modulation couples frequencies.

The matrix is measured experimentally by injecting continuous-wave signals over $$\begin{aligned} \min_{x(t)\in\mathbb R^n}\quad &\sum_{i=1}^n f_i(x_i(t),\Delta\omega_0(t))\ \text{s.t.}\quad &\sum_{i=1}^n x_i(t)=h(\Delta\omega_0(t))\,c_{agg},\ &x_i(t)\in\Omega_i,\quad i=1,\dots,n. \end{aligned}$$9 MHz to $\Delta\omega_0(t)$0 GHz in $\Delta\omega_0(t)$1 MHz steps, recording the reflected waveform in the time domain, Fourier transforming the reflected signal, and using each reflected spectrum as one column of $\Delta\omega_0(t)$2. The platform is a single-mode transmission line terminated by a rapidly modulated ring resonator containing varactor diodes, with a control-voltage modulation period of $\Delta\omega_0(t)$3. The experiments use sinusoidal modulation at $\Delta\omega_0(t)$4 MHz and randomized periodic modulation built from 100 frequency components from 1–100 MHz.

Once $\Delta\omega_0(t)$5 is known, the optimization problems become linear-algebraic. The eigen-decomposition

$\Delta\omega_0(t)$6

defines eigenpulses $\Delta\omega_0(t)$7 through

$\Delta\omega_0(t)$8

These are incident waveforms whose reflected spectra are unchanged in shape up to scaling. Reflected power is

$\Delta\omega_0(t)$9

so the maximally and minimally reflected waveforms are the eigenvectors of $c_{agg}$0 associated with the largest and smallest eigenvalues, or equivalently the extreme right singular vectors of the SVD of $c_{agg}$1.

The paper then defines a contrast-operator approach for concentrating reflected energy into chosen spectral regions. With masked matrices

$c_{agg}$2

the contrast operator is

$c_{agg}$3

with pseudoinverse if needed. Its dominant eigenvector gives the waveform that maximizes spectral contrast between the target and suppressed bands.

The reported physical consequences are notable. In the random-modulation experiment, the smallest eigenvalue of $c_{agg}$4 is about

$c_{agg}$5

with the matrix normalized so the largest eigenvalue is 1, implying over $c_{agg}$6 absorption of incident power. Using the contrast operator, reflected power can be concentrated into selected frequency bands by more than four orders of magnitude relative to the suppressed bands. The paper emphasizes that the demonstrated waveforms and responses are experimentally measured rather than purely numerical.

6. Online tracking, uncertainty, and distributed frequency-aware search

A further branch of the literature concerns optimization algorithms for moving optima when the process is unknown, time-varying, or distributed. In uncertainty-based perturb-and-observe, the problem is an unknown, discrete-time, time-varying scalar cost $c_{agg}$7 over a finite equidistant input set $c_{agg}$8, with noisy measurements $c_{agg}$9 and a unique optimizer $G(t,\omega)\partial_\omega F(t,\omega)$00 at each instant (Aarnoudse et al., 27 Aug 2025). Standard P&O uses

$G(t,\omega)\partial_\omega F(t,\omega)$01

The proposed uncertainty-based P&O builds an online probabilistic model of the cost at each candidate input, discounts older measurements by a forgetting factor $G(t,\omega)\partial_\omega F(t,\omega)$02, and perturbs only when this is expected to lead to improved performance over a certain time horizon. Its one-step-ahead estimate has Gaussian mean and variance

$G(t,\omega)\partial_\omega F(t,\omega)$03

The convergence result for standard P&O states that the iterates remain in the neighborhood

$G(t,\omega)\partial_\omega F(t,\omega)$04

For uP&O, the paper proves that the same neighborhood is recovered in the limit $G(t,\omega)\partial_\omega F(t,\omega)$05. In a photovoltaic-array simulation over 300 time steps, uP&O uses 91 perturbations, compared with 165 for standard P&O, while achieving 2.4% higher realized power than standard P&O and 8% higher than using the best constant input $G(t,\omega)\partial_\omega F(t,\omega)$06.

In distributed time-varying optimization via unbiased extremum seeking, the setting is a network of $G(t,\omega)\partial_\omega F(t,\omega)$07 agents on a weight-balanced, strongly connected directed graph, with local costs $G(t,\omega)\partial_\omega F(t,\omega)$08 and global objective

$G(t,\omega)\partial_\omega F(t,\omega)$09

The optimizer

$G(t,\omega)\partial_\omega F(t,\omega)$10

moves with an unknown exogenous signal $G(t,\omega)\partial_\omega F(t,\omega)$11 (Li et al., 26 Sep 2025). The method is gradient-free: each agent uses only its own real-time objective measurements and neighbor-shared data. The probing signal can have time-varying frequency in the chirpy form

$G(t,\omega)\partial_\omega F(t,\omega)$12

Three choices of $G(t,\omega)\partial_\omega F(t,\omega)$13 produce different convergence types: asymptotic,

$G(t,\omega)\partial_\omega F(t,\omega)$14

exponential,

$G(t,\omega)\partial_\omega F(t,\omega)$15

and prescribed-time,

$G(t,\omega)\partial_\omega F(t,\omega)$16

The theoretical analysis uses Lie bracket averaging, Lyapunov functions, and LMI conditions. The constant-frequency algorithm yields asymptotic tracking of the form

$G(t,\omega)\partial_\omega F(t,\omega)$17

with rate

$G(t,\omega)\partial_\omega F(t,\omega)$18

The chirpy version extends this to asymptotic, exponential, or prescribed-time convergence depending on the growth function. A notable point made by the paper is that in the distributed setting the local minimizers generally do not coincide with the global minimizer, so local gradients at the global optimizer are typically nonzero; this is one of the main obstacles relative to non-distributed extremum seeking.

Taken together, these online and distributed methods broaden the meaning of FVO beyond explicit spectral variables. They show that once the optimum itself varies because of aging information, exogenous signals, or probing frequencies, optimization becomes inseparable from estimation and tracking.

7. Cross-cutting principles and recurring misconceptions

Across the surveyed works, three structural motifs recur. First, the optimization target is typically a moving object: an optimal asset trajectory in power systems, a time-frequency ridge, a local spectrum field, or an optimal waveform over spectral channels. Second, differential or transport structure is used to regularize that motion: $G(t,\omega)\partial_\omega F(t,\omega)$19 in Tycoon, $G(t,\omega)\partial_\omega F(t,\omega)$20 in NSTR, and feedforward terms compensating for $G(t,\omega)\partial_\omega F(t,\omega)$21 and $G(t,\omega)\partial_\omega F(t,\omega)$22 in TOT. Third, the computational problem is usually not a one-shot solve but an online tracking problem subject to noise, dynamics, or delivery-time constraints.

Several misconceptions are not supported by the available literature. FVO is not simply static optimization expressed in the frequency domain. In the ARU setting, the optimization itself changes with the service delivery curve; in Tycoon, the representation must satisfy a differential law adapted to chirp; in NSTR, the effective spectrum changes across space; and in ultrafast media, the objective is defined through a frequency-coupling operator. FVO is also not inherently gradient-based: uP&O relies on uncertainty-aware perturbation, and distributed unbiased extremum seeking uses only function measurements. Nor does frequency awareness necessarily require expansion of the global basis everywhere: NSTR instead uses a compact set of global sinusoidal bases together with spatial modulation.

This suggests that the unifying content of FVO lies in how optimization is organized around nonstationary spectral structure. In some domains that structure is physically measured, as in a reflection matrix; in others it is prescribed by a service curve; in others it is inferred as a latent field or chirp law. The main technical consequence is that optimality must be coupled to transport, tracking, sparsity, or distributed coordination rather than treated as a static equilibrium.