Adding Heavy-Ball Momentum to Distributed Constrained Optimization for Energy Resource Management Subject to Time-Delays and Quantization

Mohammadreza Doostmohammadian; Sergio Pequito

eess.SYOct 17, 2026

Adding Heavy-Ball Momentum to Distributed Constrained Optimization for Energy Resource Management Subject to Time-Delays and Quantization

Authors: Mohammadreza Doostmohammadian, Sergio Pequito

Abstract

This paper studies a hybrid discrete-time signed gradient-consensus method with heavy-ball momentum, paired communication delays, and fixed logarithmic quantization. The method preserves the aggregate supply-demand equality at every iteration. For a static connected undirected graph and bounded delays, a Lyapunov-Krasovskii analysis establishes practical stability by isolating the discontinuous quantization error as an additive disturbance and closing a computable invariant-sublevel argument. The resulting ergodic residual bound is quadratic in the quantization resolution, and exact convergence to the smooth penalized equality-constrained optimizer is recovered in the unquantized case. For delay-free unquantized quadratic costs, an exact modal analysis identifies the heavy-ball stability region and optimal parameters, with the predicted rate validated on a fixed sparse network. A reproducible delayed experiment reports all constants and a certificate whose guaranteed steady-state level lies below the reported initialization. A separate mixed generator-battery experiment exercises the signed update directly. Finite smooth penalties approximate, but do not generally enforce, the local box constraints exactly.

Figures & tables

Figure 1: Uniform quantization (left) and logarithmic quantization (right).

Type	$\alpha_{i}$ ($/h)	$\beta_{i}$ ($/MWh)	$\gamma_{i}$ ($/MW 2 h)
A	561	2.0	0.04
B	310	3.0	0.03
C	78	4.0	0.035
D	561	4.0	0.03
E	78	2.5	0.04

Table 1: Power cost parameters for different energy node types [ 61 , 62 ] .

Panel of Figure 2 — Figure 2: Effect of momentum $\mu$ on convergence ( $n=40$ , delay-free). Top-left: cost residual for $\mu=0,0.25,0.5,0.75$ . Others: power allocations $z_{i}(k)$ showing constant weighted average (anytime feasibility), and momentum variables $w_{i}(k)$ converging to zero.

Figure 3: Observed heavy-ball and zero-momentum errors on a fixed sparse quadratic network. Dashed curves are exact zero-initial-momentum worst-mode recurrence bounds computed from the actual positive spectrum of

Q^{1/2}LQ^{1/2}

; the displayed

r_{\rm HB}

is an asymptotic root-modulus factor, and the heavy-ball bound retains the polynomial factor caused by repeated roots at the two extreme modes.

Figure 4: Empirical mixed signed validation with 12 generators (

b_{i}=+1

) and 8 batteries (

b_{i}=-1

) on a fixed sparse graph. Left: normalized cost gap and signed-gradient disagreement. Right: numerical error in the conserved coupling equality

b^{\top}z(k)=D

. The gradual increase in the equality-error trace remains at floating-point scale and reflects accumulated roundoff in an algebraically cancelling update, not loss of the conservation invariant.

Figure 5: Cost residual under different log-scale quantization levels $\rho$ .

Figure 6: Comparison of uniform [ 63 ] and log-scale quantization at $\rho=0.0156$ .

Algorithm	Run Time	Algorithm	Run Time
Linear [ 25 ]	0.742 s	Accelerated [ 64 ]	0.797 s
Single-bit [ 65 ]	0.791 s	Fixed-time [ 39 ]	1.147 s
Saturated [ 32 ]	0.926 s	This work	0.747 s

Table 2: Wall-clock time (5000 iterations) for different DERMS algorithms.

Figure 7: Cost residual comparison with existing DERMS methods.

Implemented problem and algorithm parameters
$n$	25	$\lambda_{2}=\lambda_{n}$	1	$d_{\max}$	0.96
$v$	0.03	$u$	0.55	$\bar{\tau}$	2
$\rho$	$10^{-4}$	$\eta_{\tau}$	0.0095	$\mu$	0.09
Proof auxiliaries (analysis only)
${\omega_{w}}$	4.2	${\omega_{H}}$	0.12	$(\epsilon,\gamma,\xi)$	$(39,2,0.125)$
Derived certificate quantities

Table 3: Complete numerical certificate for Theorem 1 .

Figure 8: Retuned theorem-certified delayed experiment. Left: composite-cost gaps for the certified

\mu=0.09

dynamics and a separately verified

\mu=0

baseline. Center: running average of

\Phi+\|w\|^{2}+H

and its finite-

N

bound

[\mathcal{V}(0)/N+\varepsilon_{\rho}]/\min_{i}\widetilde{\beta}_{i}

. Right: equality-feasibility errors; their gradual increase remains at floating-point scale and reflects accumulated roundoff in algebraically cancelling updates, not loss of the conservation invariant. Parameters and coefficients are in Table 3 .

Figure 9: Sufficient

(\eta_{\tau},\mu)

region for the fixed auxiliary parameters in Table 3 ; the red point is the reported certificate.

Panel of Figure 10 — Figure 10: Delayed stress test ( $n=25$ , $\eta_{\tau}=0.15$ ). Top-left: cost residuals for $\bar{\tau}\in\{2,5,10,16\}$ . Top-right: allocation states for $\bar{\tau}=10$ , with constant weighted average. Bottom: momentum variables for $\bar{\tau}=10$ . The nonsmooth penalty and $\mu=0.8$ place these runs outside Theorem 1 .

Figure 11: Cost residual for nonquadratic smooth strongly convex costs with $\mu=0$ (dashed) and $\mu=0.5$ (solid). In this empirical run, momentum improves the transient and both curves approach the same residual floor.

Distributed Optimization

Distributed optimization coordinates computation across multiple agents or machines while balancing convergence, communication, privacy, and reliability.

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Low-Bit Distributed Constrained Optimization with Application to Optimal Scheduling and Allocation

Mohammadreza Doostmohammadian, Sergio Pequito

Distributed Optimization Constrained Optimization Sign-Based Optimization

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This paper proposes a distributed resource allocation algorithm in which each agent communicates only the sign of its local cost gradient, a single bit per directed link per iteration. The algorithm is based on Laplacian-gradient dynamics and preserves the equality (demand) constraint at every iterate, guaranteeing anytime feasibility irrespective of termination time. First, we provide a complete Lyapunov-based convergence analysis for the fixed-step-size regime: the iterates are shown to enter, in at most O(1/\varepsilon^2) iterations, a neighborhood of the optimizer whose radius scales as O(τ) in the decision variable and O(τ^2) in the cost residual, where

τ

is the step-size. Crucially, the step-size bound is a time-invariant network constant (depending only on the graph Laplacian spectrum and the cost curvature bounds), so no adaptive tuning is required. Second, we characterize the invariant set under B-connected, weight-balanced digraphs, strictly weaker than the all-time connectivity and doubly-stochastic weights assumed in most prior work, and show that any odd, sign-preserving nonlinearity yields the same invariant-set and descent structure, with the single-bit protocol being the extreme communication-minimizing instance of a broader nonlinear class. Third, we address local box constraints via smooth penalty relaxation and quantify explicitly how the penalty weight interacts with the step-size bound and the optimality gap. Simulations on economic dispatch (1000 generators) and CPU scheduling (500 servers) demonstrate that the single-bit protocol achieves 5-60 times reduction in total communication cost relative to multi-bit baselines, at the price of a tunable and explicitly bounded optimality gap. A diminishing step-size schedule is shown empirically to eliminate this gap; its formal convergence analysis is identified as an open problem.

Abstract

Distributed Optimization

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