Unbalanced Optimization: Theory and Applications

• Unbalanced optimization is a modeling stance that replaces standard balance conditions with formulations capturing inherent asymmetries in power systems, networks, and optimal transport.
• Key applications include unbalanced power flow in three-phase grids, consensus algorithms over directed networks, and optimal transport with relaxed marginal constraints for improved feasibility.
• Methodologies such as successive linear approximations, SDP relaxations, and push-sum corrections provide actionable strategies for addressing imbalances in real-world optimization problems.

Searching arXiv for papers on unbalanced optimization and closely related formulations. Unbalanced optimization denotes a family of optimization formulations in which a standard balance condition is absent, relaxed, or intentionally replaced by an asymmetric model. In power systems, unbalanced optimal power flow is a class of optimization problems subject to the steady state physics of three-phase power grids with nonnegligible phase unbalance (Geth et al., 2021). In distributed optimization over digraphs, “unbalanced” means that $1^T L \neq 0^T$, or more generally that only row-stochastic or column-stochastic mixing is available, so standard consensus protocols do not preserve averages (Zhang et al., 2022). In optimal transport, unbalanced formulations relax hard marginal constraints and permit mass to be created or destroyed at a penalty (Arase et al., 2023). In combinatorial optimization and classification, the term also appears in asymmetric penalty encodings of inequality constraints and in optimization methods tailored to imbalanced data (Montanez-Barrera et al., 2022, Medlin et al., 2024). A plausible organizing view is that these literatures share a common move: they replace a convenient symmetry assumption by a model that matches the governing physics, communication topology, data distribution, or constraint geometry more faithfully.

1. Structural meanings of unbalance

In three-phase distribution networks, unbalance is physical. The network is steady-state, three-phase, radial or weakly meshed; conductor transposition is neglected, so lines may be phase-unbalanced. Voltages $V_i \in \mathbb{C}^3$, branch currents $I_{ij} \in \mathbb{C}^3$, shunt currents, and complex-power flows are all phase-indexed, and the optimization must respect phase-specific Kirchhoff–Ohm relations rather than a balanced single-phase surrogate (Geth et al., 2021).

In multi-agent optimization, unbalance is topological. For a fixed strongly-connected directed graph with Laplacian $L$, “unbalanced” means $1^T L \neq 0^T$; incoming and outgoing total weights at each node do not match, and consensus mechanisms built for doubly-stochastic mixing cease to be exact. This is the source of push-sum corrections, topology balancing estimators, and row/column stochastic split schemes (Zhang et al., 2022, Xie et al., 2016).

In optimal transport, unbalance is measure-theoretic. Classical OT imposes exact marginal constraints; unbalanced OT instead penalizes deviations of transported marginals from prescribed source and target measures by divergences such as KL, or, in dynamical formulations, augments the continuity equation with a reaction term $w\mu$ that encodes mass creation or dissipation (Arase et al., 2023, Wu et al., 4 Apr 2025).

In QUBO encodings, unbalance is penalization asymmetry. The penalty is larger when an inequality constraint is not achieved than when it is, so one avoids slack variables and still places the optimal feasible solution near the ground-state cost Hamiltonian (Montanez-Barrera et al., 2022). In imbalanced-data learning, unbalance refers to class proportions and to subset-selection rules that reject majority datapoints unable to improve model loss (Medlin et al., 2024).

2. Three-phase unbalanced power-system optimization

The core formulations for three-wire unbalanced OPF are the branch-flow model and the bus-injection model. For a branch $i \to j$, Ohm’s law is

$V_i - V_j = Z_{ij} I_{ij},$

with shunt currents and power-flow matrices defined phase-wise; nodal balance is enforced by

$$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$

Because most solvers do not natively accept complex-valued variables, these equations are rewritten in real form. Rectangular coordinates split voltages and currents into real and imaginary parts, polar coordinates yield trigonometric nonconvexities, and lifted SDP formulations use $W_i = V_i V_i^H$ with $V_i \in \mathbb{C}^3$0 and $V_i \in \mathbb{C}^3$1 (Geth et al., 2021).

A distinctive contribution of the same line of work is the explicit treatment of engineering bounds across formulations. Voltage-magnitude bounds become quadratic constraints in rectangular coordinates and diagonal bounds in lifted SDP variables; current-magnitude, apparent-power, and voltage-angle-difference bounds are likewise translated into semantically consistent forms. The paper emphasizes that consensus on bound semantics is a pressing need for benchmarking. Its formulations were validated against OpenDSS on the IEEE 13-node, 34-node and 123-node unbalanced test feeders using PowerModelsDistribution.jl, with worst maximum phase-voltage error $V_i \in \mathbb{C}^3$2 pu for the baseline AC-NLP, exact rank-1 SDP relaxations on all radial feeders tested, and runtimes of $V_i \in \mathbb{C}^3$3 s for Ipopt and $V_i \in \mathbb{C}^3$4 s for Mosek on IEEE-123 (Geth et al., 2021).

Scalability and feasibility motivate a second thread. In distribution-level OPF, the nonlinear programming model can have convergence issues and may not scale well for unbalanced systems. The proposed PSLP method uses successive linear approximation and solves multiple LP subproblems, while ISOCP uses SOCP relaxations with directional constraints so that feasible power-flow solutions are recovered over multiple iterations. On the IEEE 123-bus feeder, the reported runtime is approximately $V_i \in \mathbb{C}^3$5 s for NLP, $V_i \in \mathbb{C}^3$6 s for PSLP, and $V_i \in \mathbb{C}^3$7 s for ISOCP; on the R3-12.47-2 feeder, NLP requires $V_i \in \mathbb{C}^3$8–$V_i \in \mathbb{C}^3$9 s, PSLP $I_{ij} \in \mathbb{C}^3$0–$I_{ij} \in \mathbb{C}^3$1 s, and ISOCP $I_{ij} \in \mathbb{C}^3$2–$I_{ij} \in \mathbb{C}^3$3 s, with all methods agreeing on substation power within $I_{ij} \in \mathbb{C}^3$4 MW (Jha et al., 2021).

Operational control problems inherit the same unbalanced structure. In the two-layer Volt/VAR framework, the upper layer solves a receding-horizon MIQP using a generalized linearized branch-flow model that removes the bilinear term $I_{ij} \in \mathbb{C}^3$5, while the lower layer deploys a decentralized integral-like reactive-power update

$I_{ij} \in \mathbb{C}^3$6

with sufficient stability condition $I_{ij} \in \mathbb{C}^3$7. On the unbalanced IEEE 123-Node Test Feeder, the upper-layer solve time is $I_{ij} \in \mathbb{C}^3$8 s for $I_{ij} \in \mathbb{C}^3$9 h and $L$0 s for $L$1 h, versus $L$2 s and $L$3 s for the non-convex formulation, and the two-layer scheme holds all phases within $L$4 p.u. at all times (Guo et al., 2019).

Rolling-horizon microgrid energy management extends the same philosophy. The framework in a modified IEEE 34-bus three-phase microgrid preserves full current-based unbalanced physics, derives a convex QCP and a linearized LP from a nonlinear nonconvex model, and replans over windows with $L$5 h and discount factor $L$6. Over $L$7 Monte Carlo runs, the convex QCP average cost is approximately $L$8 kr/day, the 4-sided LP approximately $L$9 kr/day, and a two-stage stochastic baseline approximately $1^T L \neq 0^T$0 kr/day, implying rolling-horizon savings of approximately $1^T L \neq 0^T$1; voltage stays in $1^T L \neq 0^T$2 p.u. in $1^T L \neq 0^T$3 of hours (Cortés et al., 19 Mar 2025).

3. Distributed optimization over unbalanced directed networks

The distributed-optimization literature treats unbalance as a property of the communication graph rather than the objective. A representative continuous-time formulation considers

$1^T L \neq 0^T$4

where each $1^T L \neq 0^T$5 is differentiable with globally Lipschitz gradient, possibly nonconvex, and the global minimizer $1^T L \neq 0^T$6 is unique. The graph is fixed, strongly connected, directed, and unbalanced, so the left eigenvector $1^T L \neq 0^T$7 satisfying $1^T L \neq 0^T$8 is generally unknown to the nodes. The proposed fully distributed algorithm uses a topology-balancing estimator $1^T L \neq 0^T$9, divides $w\mu$0 by $w\mu$1, and introduces adaptive gains $w\mu$2 and $w\mu$3. Its analysis passes through a perturbed-system form, input-to-state stability with vanishing perturbation, and asymptotic convergence of every $w\mu$4 to $w\mu$5, without requiring knowledge of the smallest strong convexity constant or of $w\mu$6 (Zhang et al., 2022).

For strongly convex smooth objectives, linear-rate methods have been constructed directly on row-stochastic or column-stochastic networks. The ADBB method combines Barzilai–Borwein step-sizes,

$w\mu$7

with multi-consensus inner loops and gradient tracking. Under Lipschitz-gradient and strong-convexity assumptions, each $w\mu$8 lies in $w\mu$9, and the iterates converge linearly. In mushroom classification with $i \to j$0 agents on an unbalanced graph, ADBB achieves the highest test accuracy after $i \to j$1 and $i \to j$2 iterations, approximately $i \to j$3 average versus $i \to j$4 for the benchmarks ABm, AB, ADD-OPT, FROST, and DEXTRA (Hu et al., 2023).

Online convex optimization over unbalanced networks adds temporal variation and global constraints. DUST addresses

$i \to j$5

over time-varying directed graphs with column-stochastic mixing. Each agent maintains a primal variable $i \to j$6, a constraint-tracking variable $i \to j$7, a local dual estimate $i \to j$8, and a push-sum weight $i \to j$9, with $V_i - V_j = Z_{ij} I_{ij},$0. With $V_i - V_j = Z_{ij} I_{ij},$1 and $V_i - V_j = Z_{ij} I_{ij},$2, DUST attains dynamic regret

$V_i - V_j = Z_{ij} I_{ij},$3

and cumulative constraint violation $V_i - V_j = Z_{ij} I_{ij},$4, improved to $V_i - V_j = Z_{ij} I_{ij},$5 under Slater’s condition (Wang et al., 2023).

Time-varying unbalanced topologies with local constraints have also been handled by paired row-stochastic and column-stochastic matrix sequences. Liu et al. combine a row-stochastic state-mixing step, a Polyak-style projection-correction for local inequality constraints, and a column-stochastic gradient-tracking variable $V_i - V_j = Z_{ij} I_{ij},$6. Under uniformly jointly strongly connected graph sequences, compact local sets, and a diminishing step-size $V_i - V_j = Z_{ij} I_{ij},$7, the method achieves consensus, feasibility, and convergence to an optimal solution, with averaged rate $V_i - V_j = Z_{ij} I_{ij},$8 for the optimality gap and consensus violation (Liu et al., 2022).

When explicit subgradients are unavailable, the push-sum idea persists. Xiong et al. construct row-stochastic matrices from a column-stochastic $V_i - V_j = Z_{ij} I_{ij},$9 by an auxiliary scalar $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$0, then combine consensus with a local randomized zeroth-order oracle. With $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$1, $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$2, and $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$3, the dynamic regret is $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$4, and the paper reports that the zeroth-order version has essentially the same rate as the subgradient-based counterpart (Xiong et al., 2022).

A different resolution is epigraph reformulation. Xie et al. transform $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$5 with local inequality constraints into an epigraph form where all nodes share the same linear objective $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$6, thereby eliminating the Perron-weight issue of unbalanced graphs. The resulting distributed random-fixed projection algorithm uses only row-stochastic weights and converges almost surely to a common optimal solution over uniformly jointly strongly connected time-varying digraphs (Xie et al., 2016).

For finite-sum problems, Push-LSVRG-UP combines push-sum, loopless SVRG, and uncoordinated triggered probabilities $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$7. Under a constant step-size

$$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$8

the method converges linearly and requires only $$\sum_{\ell:i\to j}\operatorname{diag}(S_{ij}) + \sum_{\ell:k\to i}\operatorname{diag}(S_{ki}) + \sum_{u\in \mathrm{Units}(i)} [P_{u,i}+jQ_{u,i}] + \operatorname{diag}(V_i Y_i^H V_i^H)=0.$$9 storage per agent. This places unbalanced digraph optimization in the same algorithmic class as modern variance-reduced distributed learning (Hu et al., 2023).

4. Unbalanced optimal transport and related control formulations

In optimal transport, unbalance is formalized by relaxing the marginal constraints. For source and target fertilities $W_i = V_i V_i^H$0, $W_i = V_i V_i^H$1, cost matrix $W_i = V_i V_i^H$2, entropic regularization $W_i = V_i V_i^H$3, and KL penalties $W_i = V_i V_i^H$4, unbalanced OT solves

$W_i = V_i V_i^H$5

Balanced OT enforces $W_i = V_i V_i^H$6 and $W_i = V_i V_i^H$7, partial OT imposes hard inequalities plus a fixed transported mass, and unbalanced OT penalizes marginal deviations instead of enforcing them exactly. The corresponding generalized Sinkhorn–Knopp scaling algorithm converges linearly to the unique optimum (Arase et al., 2023).

This relaxation is useful in applications where null alignment or unequal information content is intrinsic. In monolingual word alignment, the same formulation is used to accommodate words with no corresponding counterparts. The paper reports that balanced, partial, and unbalanced OT are natural and powerful approaches even without tailor-made techniques, and that the generic OT-based methods are competitive against state-of-the-art systems specially designed for word alignment, particularly on datasets with high null alignment frequencies (Arase et al., 2023).

Gazdieva et al. develop a continuous unbalanced entropic OT solver that avoids minimax training. Their plan class has the form

$W_i = V_i V_i^H$8

and the key result is a tight upper bound that turns the UEOT problem into a single-level objective

$W_i = V_i V_i^H$9

With unnormalized Gaussian mixtures for $V_i \in \mathbb{C}^3$00 and $V_i \in \mathbb{C}^3$01, the method has no inner loops, no adversarial critic, and no nested maximizations. The paper proves universal approximation of UEOT solutions and gives generalization bounds of order $V_i \in \mathbb{C}^3$02. Reported runtimes are a few minutes on a 4-core CPU for 2-D toy mixtures and under ten minutes on CPU for $V_i \in \mathbb{C}^3$03, $V_i \in \mathbb{C}^3$04, and $V_i \in \mathbb{C}^3$05 steps (Gazdieva et al., 2023).

Dynamical unbalanced OT pushes the same idea into PDE-constrained control. The primal variables are a nonnegative measure $V_i \in \mathbb{C}^3$06, a transport velocity $V_i \in \mathbb{C}^3$07, and a mass creation/dissipation rate $V_i \in \mathbb{C}^3$08, linked by

$V_i \in \mathbb{C}^3$09

The cost is $V_i \in \mathbb{C}^3$10 subject to $V_i \in \mathbb{C}^3$11. After relaxation through Young measures and Fenchel–Rockafellar duality, the problem becomes an HJB-type supremum over smooth subsolutions,

$V_i \in \mathbb{C}^3$12

and the paper proposes a first-order augmented-Lagrangian algorithm in which each iteration solves one linear PDE and performs one pointwise projection onto a convex set (Wu et al., 4 Apr 2025).

Reinforcement learning admits an analogous UOT formulation over occupancy measures. For a policy $V_i \in \mathbb{C}^3$13, the discounted occupancy measure is $V_i \in \mathbb{C}^3$14, and the optimization augments reward maximization by marginal penalties on state visitations and action executions. The objective is written as a Bregman-divergence problem

$V_i \in \mathbb{C}^3$15

and solved by Dykstra’s algorithm via Bregman projections. In large state or action spaces, the paper derives an actor–critic saddle-point form in dual potentials $V_i \in \mathbb{C}^3$16, $V_i \in \mathbb{C}^3$17, and $V_i \in \mathbb{C}^3$18, thus importing unbalanced transport directly into constrained policy optimization (Givchi et al., 2021).

5. Learning-based and stochastic optimization under imbalance

Recent work embeds unbalance directly into learned surrogates rather than approximating it away. For real-time BESS dispatch in unbalanced three-phase grids, a heterogeneous graph $V_i \in \mathbb{C}^3$19 is built with node types for buses, storage units, external-grid connections, loads, and lines, and with features that include phase voltages, unbalanced loads, BESS states, and line mutual couplings. A physics-informed loss adds soft penalties for SoC, voltage, and external-grid constraints. On the CIGRE 18-bus distribution system, the reported bus voltage MSEs are $V_i \in \mathbb{C}^3$20 for GCN, $V_i \in \mathbb{C}^3$21 for GAT, $V_i \in \mathbb{C}^3$22 for GPS, and $V_i \in \mathbb{C}^3$23 for SAGE; the constraint-violation MSE for SoC and C-rate is reduced from $V_i \in \mathbb{C}^3$24–$V_i \in \mathbb{C}^3$25 without the physics loss to nearly zero with it (Ma et al., 10 Dec 2025).

A different learning-based acceleration appears in stochastic Volt-VAR optimization. In the two-stage stochastic VVO model for a modified IEEE 123-bus three-phase unbalanced network with 4 OLTCs, 6 tie switches, and 14 PVs, the second-stage recourse

$V_i \in \mathbb{C}^3$26

is approximated by a neural network embedded into a MILP through big-$V_i \in \mathbb{C}^3$27 ReLU constraints. Across 100 test instances with $V_i \in \mathbb{C}^3$28, the average solve times are $V_i \in \mathbb{C}^3$29 s for the full extensive MILP, $V_i \in \mathbb{C}^3$30 s for Progressive Hedging, and $V_i \in \mathbb{C}^3$31 s for the neural MILP. The average optimality gap of the neural method versus exact is $V_i \in \mathbb{C}^3$32, always $V_i \in \mathbb{C}^3$33, and the surrogate MILP time is essentially constant regardless of scenario count (Shao et al., 27 Oct 2025).

Optimization for imbalanced data uses a different mechanism. In the bilevel formulation for classification, the upper level selects a majority subset $V_i \in \mathbb{C}^3$34, while the lower level trains network weights

$V_i \in \mathbb{C}^3$35

The MUBO heuristic accepts a candidate majority batch $V_i \in \mathbb{C}^3$36 if the resulting majority-only loss $V_i \in \mathbb{C}^3$37 is no larger than the previous threshold $V_i \in \mathbb{C}^3$38. The paper reports average F1 improvements on several UCI datasets: Gisette $V_i \in \mathbb{C}^3$39 versus best prior $V_i \in \mathbb{C}^3$40, Abalone $V_i \in \mathbb{C}^3$41 versus $V_i \in \mathbb{C}^3$42, Connect4 $V_i \in \mathbb{C}^3$43 versus $V_i \in \mathbb{C}^3$44, Spambase approximately $V_i \in \mathbb{C}^3$45, and Ionosphere $V_i \in \mathbb{C}^3$46 versus a GAN baseline $V_i \in \mathbb{C}^3$47, the last serving as an explicit small-data limitation (Medlin et al., 2024).

A plausible implication is that learning-based unbalanced optimization now splits into two regimes. In one regime, the learned model is constrained to respect physical feasibility through penalty terms or MILP embeddings; in the other, the optimization itself is a mechanism for selecting or reweighting data under class asymmetry. The data block supports both patterns, but they address different meanings of “unbalanced.”

6. Constraint encoding, misconceptions, and open directions

A recurring misconception is that “unbalanced” names one unified mathematical object. The surveyed papers instead use the term for at least four non-equivalent departures from balance: phase asymmetry in power networks, non-doubly-stochastic directed communication, relaxed measure conservation, and asymmetric inequality penalization. Another misconception is that relaxing a balanced model automatically preserves feasibility. In unbalanced D-OPF, the literature explicitly notes that existing relaxation techniques do not guarantee a feasible power flow solution, which is why iterative LP and SOCP schemes are designed to recover optimal and feasible solutions over multiple iterations (Jha et al., 2021).

Constraint handling illustrates the same diversity. In QUBO models, inequality constraints are commonly encoded with slack variables, but this increases the number of qubits and operations. The unbalanced penalization method replaces the slack-variable encoding of

$V_i \in \mathbb{C}^3$48

by

$V_i \in \mathbb{C}^3$49

with larger penalization when the inequality constraint is not achieved than when it is. On D-Wave hybrid solvers for bin packing, the paper reports solutions for up to 29 items with unbalanced penalization, whereas the slack-variables approach handles up to 11 items (Montanez-Barrera et al., 2022).

Benchmarking and semantics remain central. In unbalanced OPF, the lack of definitions of current and voltage bounds across different classes of representations is identified as a key problem, and benchmarks are described as a cornerstone for the development and validation of scalable and reproducible optimization models and tools (Geth et al., 2021). In distributed optimization, open directions explicitly include time-varying or switching digraphs, discrete-time counterparts, event-triggered communication, and more general agent dynamics (Zhang et al., 2022). In neural stochastic VVO, the remaining approximation gap is attributed to finite training data and MLP expressivity, with larger networks, more data, or end-to-end fine-tuning proposed as extensions (Shao et al., 27 Oct 2025). In quantum unbalanced penalization, open questions include rigorous penalty-weight selection and theoretical performance guarantees (Montanez-Barrera et al., 2022).

Across these literatures, the dominant methodological pattern is not merely “handling imbalance,” but making imbalance first-class in the optimization model. In power systems that means retaining three-phase couplings and explicit bound semantics; in networked optimization it means correcting nonconservative mixing rather than assuming doubly-stochastic communication; in transport it means penalizing marginal mismatch or adding source/sink dynamics; and in combinatorial or data-imbalance settings it means asymmetric penalties or bilevel subset selection. This suggests that unbalanced optimization is best understood as a modeling stance: balance is treated as a special case, not the default.