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Distributed Cross-Layer Optimization for Covert Multi-Hop, Multi-Modal Networks: Exponentially Fast Convergence and Robust Tracking

Sirin Chakraborty, Andrea Panebianco, Yuchen Tian, Kevin S Chan, Fikadu Dagefu, Yin Sun, Ness B. Shroff

arXiv:2608.28469v1cs.ITeess.SP

TL;DR

Covert multi-hop, multi-modal network control must jointly optimize several cross-layer decisions despite non-concave DEP constraints. The paper constructs a concave log-DEP lower bound and applies PP-ADMM, proving global Q-linear convergence and reporting robust tracking under channel fading and Willie mobility.

  • Problem

    Existing formulations do not jointly optimize congestion control, multipath routing, fractional scheduling, and power control under aggregate per-Willie, per-modality DEP constraints.

  • Method

    The paper constructs the tightest concave lower bound on log-DEP to obtain a conservative convex problem and solves it with distributed PP-ADMM.

  • Results

    PP-ADMM achieves global Q-linear convergence to the optimal solution set under standard regularity conditions, with numerical confirmation of robust tracking under channel fading and Willie mobility.

  • Takeaways & Limitations

    The framework unifies hard DEP constraints with covertness-utility maximization in distributed cross-layer covert network control.

Abstract

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This paper develops the first distributed cross-layer algorithm for joint congestion control, routing, scheduling, and power control in covert multi-hop, multi-modal wireless networks, where adversarial wardens (Willies) monitor radio modalities via energy detection. The Detection Error Probability (DEP), the probability that a Willie fails to reliably detect ongoing transmissions, is generally non-concave in the transmit powers, making DEP-based covert network optimization challenging. We resolve this by constructing the tightest concave lower bound on the log-DEP, yielding a conservative convex problem that guarantees satisfaction of the original DEP constraints and unifies hard covertness constraints and covertness-utility maximization in a single problem. We develop a Parallel Proximal Alternating Direction Method of Multipliers (PP-ADMM) algorithm for the resulting cross-layer problem and prove global Q-linear convergence, i.e., exponentially fast convergence, to the set of optimal solutions under standard regularity conditions. Numerical results confirm linear convergence and demonstrate robust tracking performance under channel fading and Willie mobility.

I. INTRODUCTION

The paper formulates covert multi-hop, multi-modal control as a distributed cross-layer optimization problem and addresses the non-concavity of DEP-based constraints. It introduces a conservative convexification and PP-ADMM, with convergence and tracking results under changing conditions.

  • DEP-based covert optimization is challenging because DEP is generally non-concave in transmit powers.
  • The tightest concave lower bound on log-DEP yields a conservative convex problem that guarantees the original DEP constraints.
  • The resulting framework jointly optimizes congestion control, routing, scheduling, and power control across heterogeneous radio modalities.
  • PP-ADMM updates primal blocks in parallel with proximal regularization and then updates dual variables by gradient ascent.The parallel updates cover congestion control, routing, power–rate control, and scheduling.
  • Global Q-linear convergence to the optimal solution set is proved under standard regularity conditions.
  • Numerical results confirm exponentially fast convergence and robust tracking under channel fading and Willie mobility.

III. CONCAVE COVERTNESS BOUND AND CROSS-LAYER COVERT NETWORK CONTROL PROBLEMS

The section replaces non-concave DEP constraints with a tight concave lower bound on log-DEP, producing conservative cross-layer optimization formulations that retain covertness guarantees.

  • The construction supports hard-covertness and covertness-utility formulations within cross-layer network control.
  • DEP is generally non-concave in transmit powers, complicating covert network optimization.
  • The log-DEP is nearly concave but becomes convex for sufficiently large aggregate SNR.
  • The bound gL(s) retains the concave region of hL(s) and replaces its convex tail with a tangent line.
  • gL(s) is concave and satisfies gL(s) ≤ hL(s) = ln[DEP*(s, L)].
  • Because aggregate SNR is affine in transmit powers, the resulting Willie-specific bound is concave in those powers.

B. Problem 1: NUM with Hard Covertness Constraints

Problem 1 maximizes network utility subject to flow, rate, capacity, scheduling, power, and DEP constraints, while distributed reformulation supports local scheduling updates and consensus.

  • The first formulation maximizes total network utility while imposing hard covertness constraints.
  • Constraints enforce flow conservation, aggregate-rate consistency, link capacity, node-exclusive scheduling, Willie SNR definitions, and DEP thresholds.
  • The concave log-DEP lower bound guarantees that the original DEP constraints are satisfied conservatively.
  • A second formulation adds strongly concave covertness utilities while retaining the constraints of Problem 1.
  • Local scheduling copies let nodes update scheduling variables locally and exchange information only with neighbors.
  • Consensus of endpoint scheduling copies recovers shared link schedules and preserves local capacity and node-exclusive constraints.

B. Augmented Lagrangian Formulation

The augmented-Lagrangian design organizes the distributed problem into primal blocks, penalizes linear coupling constraints, and enables parallel proximal updates stabilized by regularization.

  • B. Augmented Lagrangian Formulation: Dual variables are assigned to the linear equality constraints and remain unconstrained.
  • B. Augmented Lagrangian Formulation: The formulation stacks primal variables for congestion, routing, power, rates, scheduling copies, and Willie-related auxiliaries into block vectors.
  • B. Augmented Lagrangian Formulation: The augmented Lagrangian uses penalty parameter ρ to penalize the dualized equality constraints.
  • B. Augmented Lagrangian Formulation: Nonlinear capacity and covertness functions enter through hard constraints rather than the penalized equality terms.
  • C. Parallel Proximal ADMM Algorithm: At each iteration, congestion, routing, power-rate, and scheduling blocks are updated in parallel using other blocks’ previous iterates.
  • C. Parallel Proximal ADMM Algorithm: Proximal regularization stabilizes parallel updates and ensures strong concavity of each primal subproblem.
  • C. Parallel Proximal ADMM Algorithm: The algorithm then performs dual ascent after solving the primal block subproblems.
  • C. Parallel Proximal ADMM Algorithm: The PP-ADMM update rules specify separate source-node congestion and routing subproblems within the distributed iteration.

3) Power Control, Aggregate Rate, and Scheduling:

Power, rate, and scheduling variables are updated jointly or locally within PP-ADMM, while convergence follows globally under explicit feasibility, parameter, and KKT regularity assumptions.

  • 3) Power Control, Aggregate Rate, and Scheduling:: Each node jointly updates outgoing-link transmit powers, aggregate rates, and local scheduling variables because capacity couples them.
  • 3) Power Control, Aggregate Rate, and Scheduling:: All nodes solve these local updates in parallel using iteration-k information from neighboring nodes.
  • 3) Power Control, Aggregate Rate, and Scheduling:: Willie auxiliary variables are updated through parallel scalar subproblems, with covertness utility omitted for the hard-constraint problem.
  • 3) Power Control, Aggregate Rate, and Scheduling:: During iterations, link endpoints use a common feasible schedule; at convergence, their local copies agree.
  • 3) Power Control, Aggregate Rate, and Scheduling:: The dual variables are updated by gradient ascent with dual step size τ.
  • Convergence Analysis: The KKT residual vanishes exactly at optimal primal-dual pairs, linking residual reduction to convergence toward the solution set.
  • Convergence Analysis: Global linear convergence is established under Slater feasibility, admissible step-size and proximal-weight conditions, and a local KKT error bound.
  • Convergence Analysis: Theorem 1 states that every initial point generates iterates converging globally and linearly to the optimal primal-dual solution set.

VI. NUMERICAL ANALYSIS

Numerical experiments show PP-ADMM converges exponentially fast to the optimum under static channels and closely tracks time-varying optima under Willie mobility and fading.

  • Experimental setup: The experiments use a five-node, eight-link, two-modality network over VHF and UHF, with two flows and two Willies.All results solve Problem (18).
  • Exponentially Fast Convergence under Static Channels: The total throughput of two flows under static channels converges to the optimum.Existing covert-routing works are not directly comparable because they address different cross-layer problems.
  • Experimental setup: 1 ms algorithm iterations, 5 m/s Willie mobility, and Jakes-model temporal fading define the dynamic tracking setting.Static experiments fix distances and fading gains.
  • Exponentially Fast Convergence under Static Channels: The Lyapunov ratio under static channels confirms the exponentially fast convergence predicted by Theorem 1.Figure 2 plots the ratio V_k/V_0.
  • Robust Tracking under Willie Mobility and Channel Fading: Under Willie mobility and channel fading, total throughput and Willie 1’s VHF DEP closely track their evolving optimal values.Figures 4–5 evaluate tracking for these two metrics.
  • Conclusion: The paper concludes that distributed multi-hop, multi-modal covert control achieves linear convergence and robust tracking under Willie mobility.Future work considers multi-antenna transmissions, Willie cooperation, and higher mobility speeds.

APPENDIX A PROOF OF LEMMA 1

Appendix A establishes regularity and asymptotic properties of the log-DEP, including continuity on the nonnegative domain and eventual convexity at large SNR.

  • Regularity: For s > 0, h_L(s) = ln D(s) is twice continuously differentiable because D(s) remains positive and the incomplete gamma functions are smooth.The proof establishes positivity of D(s) and smoothness on (0,∞).
  • Behavior near zero: At zero aggregate SNR, silence and transmission have identical observation distributions, giving D(0) = 1.Therefore h_L(s) extends continuously to [0,∞).
  • Behavior near zero: The proof derives small-s expansions and differentiates them to characterize the limiting behavior of h_L and its derivatives near zero.These steps use expansions of incomplete gamma functions and continuity arguments.
  • Large-s behavior: For large s, the proof separately expands the lower and upper incomplete gamma terms before combining them and differentiating the resulting log-DEP expression.The expansion controls the asymptotic behavior of D(s) and h_L(s).
  • Large-s behavior: The second derivative satisfies h′′_L(s) > 0 for sufficiently large s, so h_L(s) is eventually convex.This completes part (c) of Lemma 1.

APPENDIX B EQUIVALENCE AND BLOCKWISE LINEAR COUPLING

Appendix B proves that endpoint-local scheduling constraints are equivalent to the original formulation and organizes the linear couplings into primal-block contributions.

  • Equivalence: Replacing constraints (16d)–(16e) with (16k)–(16m) preserves feasible network variables and objective values for Problems (16) and (18).The scheduling variables are recoverable from endpoint-local copies.
  • Equivalence: The equivalence proof constructs local scheduling copies from an original feasible point and recovers the original variables from endpoint-local constraints.Both directions retain the remaining variables and constraints.
  • Equivalence: Because local scheduling copies and q_ml do not enter either objective, equivalent feasible representations have identical objective values and corresponding optima.This establishes transfer of optimal solutions between formulations.
  • Blockwise coupling: The formulation partitions variables into B primal update blocks, assigning power and rate variables to transmitting nodes and local scheduling copies to incident-link blocks.Empty node–modality blocks are omitted.
  • Blockwise coupling: Only the linear equalities for flow, rate, SNR, and consensus are dualized and represented by complete residual vectors.Their scalar and vector residuals support the augmented-Lagrangian construction.
  • Blockwise coupling: Blockwise constraint contributions may be nonzero individually while summing to zero at an optimal solution.The complete residual vectors measure equality violations, whereas blockwise differences enter the Lyapunov function.
  • Blockwise coupling: With fixed topology and coefficients, each blockwise constraint contribution is a fixed linear function of its corresponding primal variables.These linear maps are used in the convergence analysis.

APPENDIX C PROOF OF THEOREM 1

Appendix C establishes the convergence proof framework by linking KKT residuals, Lyapunov decrease, and bounded PP-ADMM iterates to linear convergence toward a saddle point.

  • Proof setup: The proof defines Euclidean projections, stacked dual variables, primal update blocks, and blockwise linear equality contributions for the KKT analysis.These objects provide the notation for the residual and Lyapunov arguments.
  • KKT characterization: The feasible sets are defined separately for x, r, (p, q, y^(n)), and s, enabling blockwise stationarity conditions.The joint set H covers power, rate, and local scheduling variables.
  • KKT characterization: The KKT residual map contains four stationarity rows and four equality-constraint rows for flow, rate, SNR, and consensus conditions.Projection identities characterize stationarity over each primal feasible set.
  • KKT characterization: Under convexity and Slater’s condition, a zero KKT residual is equivalent to primal–dual optimality.The endpoint-local formulation supplies the needed convex representation.
  • Proof setup: Each primal block combines its objective contribution with a convex feasible-set indicator, while proximal and augmented terms yield positive-definite block metrics.The construction uses α > 0, ρ > 0, and τ > 0.
  • Linear convergence: The proof combines a uniform KKT error bound, Lyapunov decrease, residual control by iterate changes, and compactness to establish linear convergence.These are organized as four successive proof steps.
  • Linear convergence: Proximal terms make every primal subproblem coercive and strongly convex, while affine dual updates ensure a well-defined deterministic iterate sequence.This holds from any finite initial point under fixed problem data.

1) Step 1: A uniform KKT error bound holds over compact sets:

The proof establishes a uniform KKT error bound on a compact set containing optimal saddle points, then combines bounded iterates, residual control, and Lyapunov descent to prove linear convergence.

  • Step 1: A uniform KKT error bound holds over compact sets:: On any compact set intersecting the optimal saddle-point set, the KKT residual uniformly bounds distance to that set.The residual is zero exactly at optimal primal–dual pairs.
  • Step 1: A uniform KKT error bound holds over compact sets:: Each PP-ADMM iteration decreases the weighted Lyapunov error, and this decrease controls the change between consecutive iterates.The proof uses sufficient descent and a one-step residual bound to connect iterate changes with optimality error.
  • Step 1: A uniform KKT error bound holds over compact sets:: The PP-ADMM iterates and complete primal–dual sequence remain uniformly bounded within a compact set containing a saddle point.Positive definiteness of G bounds the sequence, including the dual iterates.
  • Step 1: A uniform KKT error bound holds over compact sets:: The iterates form a Cauchy sequence whose limit has zero KKT residual and therefore belongs to the optimal saddle-point set.Continuity of the residual maps the convergent sequence to an optimal primal–dual point.
  • Step 1: A uniform KKT error bound holds over compact sets:: The weighted Lyapunov sequence converges Q-linearly, while the complete primal–dual sequence converges R-linearly to a saddle point.The primal component is an optimal solution with the same optimal objective value.

APPENDIX D PROOF OF LEMMA 2

The appendix proves the uniform KKT error bound by combining local error bounds near saddle points with a positive residual bound away from them on a compact set.

  • APPENDIX D PROOF OF LEMMA 2: The proof covers saddle points in the compact set with finitely many neighborhoods where local KKT error bounds apply.Compactness of the saddle-point intersection permits a finite subcover.
  • APPENDIX D PROOF OF LEMMA 2: The region outside those neighborhoods is compact and contains no saddle point, so the continuous KKT residual has a strictly positive minimum there.A zero residual would imply membership in the saddle-point set, contradicting the region’s construction.
  • APPENDIX D PROOF OF LEMMA 2: A reference saddle point in the compact set provides a uniform distance bound for points outside the neighborhoods.Combining this bound with the positive residual minimum controls distance to the saddle-point set.
  • APPENDIX D PROOF OF LEMMA 2: The block optimality conditions are compared with KKT conditions, converted into G-norm inequalities, and summed across blocks.Monotonicity of each convex block subdifferential enables the comparison.
  • APPENDIX D PROOF OF LEMMA 2: Young’s inequality bounds the primal–dual cross term, while positive-definite quadratic forms yield sufficient Lyapunov descent.The descent matrix collects coefficients for primal and dual iterate changes.

6) Step 6: Positive Definiteness and the Descent Constant

The proof establishes positive definiteness of the descent matrices and bounds the KKT residual by consecutive iterate changes, completing the linear-convergence argument.

  • 6) Step 6: Positive Definiteness and the Descent Constant: The descent matrix collects quadratic coefficients for primal and dual iterate changes in the sufficient-descent inequality.Its block structure separates the contributions of the primal blocks and the dual variables.
  • 6) Step 6: Positive Definiteness and the Descent Constant: Under the stated parameter conditions, both G and H are positive definite, producing a valid descent constant for the Lyapunov analysis.The proof explicitly uses the positive definiteness of these matrices to establish the descent inequality.
  • 6) Step 6: Positive Definiteness and the Descent Constant: Each block update’s optimality condition is rewritten using the current dual iterate so remaining terms depend only on consecutive iterate changes.This representation prepares the residual bounds used in the convergence proof.
  • 6) Step 6: Positive Definiteness and the Descent Constant: The stacked remaining block terms are bounded by a constant multiple of the G-norm of the iterate change.Positive definiteness of G links the Euclidean norm to the G-norm.
  • 6) Step 6: Positive Definiteness and the Descent Constant: The dual update bounds equality residuals, and projection nonexpansiveness bounds stationarity residuals, yielding a complete KKT residual bound.The resulting inequality shows that residuals vanish as PP-ADMM iterates stop changing.
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