Feasibility-Guaranteed L-IPM for OPF Acceleration

• The paper introduces a hybrid L-IPM method that uses an LSTM to predict a near-optimal central-path trajectory and refines it with classical IPM, achieving up to 94% runtime reduction and 85.5% fewer iterations.
• It preserves feasibility by combining grid-informed loss functions with a final rigorous IPM refinement that ensures all OPF constraints are met.
• The method is validated across various system sizes, demonstrating high prediction accuracy (R² up to 0.994) and even accelerated failure detection in stressed scenarios.

Searching arXiv for the exact term and closely related feasibility-preserving interior-point literature. Searching "feasibility-guaranteed learning interior point method". Feasibility-Guaranteed Learning Interior Point Method (L-IPM) is a hybrid interior-point framework in which a learned model accelerates the trajectory of a classical interior-point method (IPM) while feasibility is protected by explicit optimization structure rather than delegated to an unconstrained predictor. In the form introduced for AC and DC optimal power flow (OPF), L-IPM runs only the first few stable IPM iterations, uses a Long Short-Term Memory (LSTM) network to project the remaining central-path trajectory, and then applies a final IPM refinement step under the true OPF constraints; the reported effect is a reduction of solution time by up to $94\%$ and of required iterations by up to $85.5\%$, while maintaining solution accuracy and feasibility (Amani et al., 26 Aug 2025).

1. Origin, problem class, and mathematical setting

The term “Feasibility-Guaranteed Learning Interior Point Method” is used explicitly for ACOPF and DCOPF. The ACOPF objective is written as

$$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$

subject to the complex power-balance equations

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$

together with generator active-power limits,

$$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$

voltage-magnitude limits,

$$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$

and transmission thermal limits,

$$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$

The DCOPF treatment uses the standard linearized OPF variables emphasized in the paper—generator active powers $P_{G,i}$, bus voltage angles $\theta_i$, and line active flows $P_{ij}$ (Amani et al., 26 Aug 2025).

The baseline solver is a classical primal-dual IPM. Inequality constraints are converted into equalities with slack variables: $85.5\%$0 for generation limits, $85.5\%$1 for voltage limits, and $85.5\%$2 for line limits. The barrier-augmented Lagrangian is

$85.5\%$3

Newton’s method is then applied to the KKT equations, so each IPM iteration requires the solution of a large linear system defined by the KKT matrix. The method is motivated by the observation that the early IPM iterations usually have the lowest condition numbers, whereas later iterations become increasingly ill-conditioned and computationally expensive (Amani et al., 26 Aug 2025).

2. Algorithmic structure of L-IPM

L-IPM treats the IPM trajectory itself as a time series. The training input is the first few IPM iterates,

$85.5\%$4

where $85.5\%$5 is the system state at IPM iteration $85.5\%$6 and $85.5\%$7 is the final converged state. The paper sets $85.5\%$8, so the deployed look-back is three IPM iterations. The state representation is described as including generator outputs, bus voltages, Lagrange multipliers, slack variables, and practical system attributes such as load demand, generation constraints, and historical OPF results (Amani et al., 26 Aug 2025).

The recurrent model is an LSTM. The paper gives the standard cell-state and hidden-state recursions,

$85.5\%$9

followed by a fully connected output layer mapping temporal features to OPF variables. It reports that a 2-layer GI-LSTM is effective for small and medium systems and for many larger systems, whereas a 3-layer GI-LSTM is recommended for large-scale OPF with more complex trajectories. The look-back study tested windows from 1 to 5 and selected 3 because 1 lacks sufficient temporal context, 3 gave satisfactory performance, and on the 2869-bus case 3 gave the best performance, while 4 or 5 often gave little improvement (Amani et al., 26 Aug 2025).

The learned map is a direct projection from the early trajectory to a near-terminal state,

$$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$0

or, in sequence notation,

$$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$1

Operationally, the online procedure is: run 3 initial IPM iterations; feed those iterates to the GI-LSTM; obtain a predicted near-solution point; and then resume IPM for final constrained refinement. The method is therefore not a replacement of IPM by a predictor, but a learned mid-solver acceleration of the central-path trajectory (Amani et al., 26 Aug 2025).

3. Meaning of the feasibility guarantee

The feasibility claim in L-IPM is hybrid. The grid-informed LSTM is trained to stay close to feasible OPF structure, but the exact guarantee comes from the final optimization stage. For ACOPF, the training loss is

$$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$2

and for DCOPF,

$$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$3

These losses explicitly penalize deviations in dispatch, voltages or angles, and especially line-flow violations. The paper states that $$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$4 is usually set highest in order to strongly discourage infeasible outputs (Amani et al., 26 Aug 2025).

That said, the paper does not give a theorem that the raw GI-LSTM output is always feasible by itself. There is no explicit constrained decoder, exact projection layer, or proof that the network output satisfies all OPF equalities and inequalities. Instead, the final refinement is formulated as

$$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$5

This means the learned point is a near-feasible or near-optimal proposal, and the returned solution is produced by a final IPM solve under the true OPF constraints. In this sense, “feasibility-guaranteed” refers to the end-to-end method, not to the neural predictor in isolation (Amani et al., 26 Aug 2025).

This distinction is important because the broader interior-point literature uses several different notions of feasibility. In strict-feasible barrier methods, all iterates satisfy interior constraints by construction. In feasible-start LP IPMs, primal and dual affine feasibility can be preserved exactly. In infeasible-start path-following methods, iterates may violate feasibility while residuals are driven to zero. L-IPM belongs to a fourth category: feasibility is promoted statistically by the learned component and recovered exactly by a final certified optimization stage. The paper itself reinforces this interpretation by noting that in extreme stressed 2869-bus scenarios, when the OPF instance is infeasible, the final IPM refinement still fails, although L-IPM reaches that failure faster (Amani et al., 26 Aug 2025).

4. Relation to prior interior-point and learning-based research

The direct predecessor on the learning side is “IPM-LSTM,” which also inserts an LSTM into a primal-dual IPM. There, the network does not predict the solution directly; it approximates the Newton direction by solving the least-squares surrogate

$$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$6

through a recurrent update

$$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$7

The approximate primal-dual point after a fixed number of IPM iterations is then passed to IPOPT as a warm start. The paper is explicit that it does not provide a mechanism guaranteeing feasibility of the learned iterates themselves, and that feasibility and robustness are largely recovered by classical boundary-preserving step-size safeguards and the downstream IPOPT solve. It reports reductions of iterations by up to $$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$8 and solution time by up to $$\min_{\mathbf p}\; \sum_{i \in \mathbb G} \left(a_i p_i^2 + b_i p_i + c_i\right),$$9 compared to the default solver, but it is only partially related to a feasibility-guaranteed L-IPM (Gao et al., 2024).

A different line of work supplies the certification logic that L-IPM lacks internally. In linear programming, inexact predictor-corrector and long-step IPMs show that approximate linear solves can be made theoretically safe if they are paired with an explicit correction vector $$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$0. In the feasible-start setting, one replaces the exact normal-equation solve by

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$1

defines

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$2

and enforces

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$3

Because this identity holds exactly, primal and dual feasibility are preserved even though the Newton system is solved only approximately. In the infeasible-start setting, related correction formulas preserve the residual-line invariant and guarantee convergence to a feasible, approximately optimal solution without increasing outer iteration complexity (Dexter et al., 2022, Chowdhury et al., 2022, Chowdhury et al., 2020).

The nonlinear feasibility literature provides additional design principles. One result shows that reducing primal feasibility at the same asymptotic rate as the barrier parameter $$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$4 keeps the multiplier sequence bounded whenever the limit point has a Lagrange multiplier, even without constraint qualifications. The key rate law is

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$5

with primal and complementarity residuals decreasing in a coupled way. A one-phase IPM for nonconvex optimization operationalizes this by maintaining the exact shifted-feasibility relation

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$6

together with

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$7

thereby reducing primal feasibility and the barrier parameter at the same rate (Haeser et al., 2017, Hinder et al., 2018).

Other feasibility-preserving IPM architectures broaden the design space. An interior proximal gradient method for nonconvex inequality-constrained problems maintains strictly feasible iterates through barrier-domain restriction and an explicit backtracking acceptance test

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$8

so every inner and outer iterate remains strictly interior. A geodesic IPM over symmetric cones preserves cone interiority and exact complementarity by construction through updates of the form

$$s_i = p_i + j q_i = v_i \sum_{j \in \mathbb B} Y_{ij}^* v_j^*,$$9

while ABIP uses a homogeneous self-dual embedding so that the embedded problem is always feasible and has a readily available strictly interior point. In hierarchical least-squares programs, nullspace projections preserve higher-priority active constraints exactly, while line search maintains the sign conditions on slacks and duals (Marchi et al., 2022, Permenter, 2020, Lin et al., 2018, Pfeiffer et al., 2021).

Work	Learned or approximate component	Feasibility mechanism
L-IPM for OPF (Amani et al., 26 Aug 2025)	LSTM central-path projection from first 3 IPM iterations	Grid-informed loss plus final IPM refinement
IPM-LSTM (Gao et al., 2024)	LSTM least-squares Newton-step approximation	Boundary-preserving step sizes and IPOPT warm start
Inexact LP IPMs (Dexter et al., 2022, Chowdhury et al., 2022, Chowdhury et al., 2020)	Approximate linear solves	Exact correction vector $$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$0 and residual certification
Interior proximal gradient IPM (Marchi et al., 2022)	No learning	Strict-feasibility acceptance test $$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$1
Geodesic / HSD IPMs (Permenter, 2020, Lin et al., 2018)	No learning	Interiority by construction or feasibility-independent embedding

A plausible implication is that the strongest blueprint for a genuinely feasibility-guaranteed L-IPM is not an unconstrained predictor, but a predictor wrapped by certification, correction, and fallback logic. The OPF L-IPM demonstrates the hybrid pattern; the LP and nonlinear IPM literature identifies the precise invariants such a wrapper would need.

5. Empirical performance in optimal power flow

The OPF L-IPM is evaluated on 3-bus, 24-bus, 118-bus, and 2869-bus systems, with the 2869-bus European high-voltage transmission system serving as the principal large-scale case. Load scenarios are generated by Latin hypercube sampling. For each load bus $$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$2,

$$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$3

with $$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$4 sampled from scenario-dependent uniform ranges: $$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$5 for nominal, $$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$6 for high demand, $$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$7 for low demand, and $$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$8 for mixed demand. The paper motivates this by the combinatorial growth

$$P_i^{\min} \le p_i \le P_i^{\max}, \qquad \forall i \in \mathbb G,$$9

if exhaustive enumeration were attempted, and supplements it with sample-count heuristics such as

$$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$0

together with the variance diagnostic

$$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$1

The implementation uses MATPOWER 8.0 in MATLAB for IPM OPF simulations and Python for the GI-LSTM (Amani et al., 26 Aug 2025).

Prediction quality is reported through MSE, MAE, RMSE, and $$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$2. The $$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$3 values are $$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$4 and $$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$5 on the 3-bus ACOPF and DCOPF cases, $$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$6 and $$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$7 on 24-bus, $$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$8 and $$V_i^{\min} \le v_i \le V_i^{\max}, \qquad \forall i \in \mathbb B,$$9 on 118-bus, and $$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$0 and $$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$1 on 2869-bus. The lower ACOPF $$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$2 on the 2869-bus system is attributed partly to the model attempting to remain in the feasible region even when some demand scenarios are themselves infeasible (Amani et al., 26 Aug 2025).

The reported iteration reductions are substantial. On the 2869-bus system, DCOPF iterations decrease from $$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$3–$$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$4 to $$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$5, while ACOPF iterations decrease from $$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$6–$$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$7 to $$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$8–$$|f_{ij}| \le F_{ij}^{\max}, \qquad \forall (i,j) \in \mathbb L.$$9, yielding improvements from $P_{G,i}$0 to $P_{G,i}$1. On 118-bus, DCOPF drops from $P_{G,i}$2–$P_{G,i}$3 to $P_{G,i}$4, and ACOPF from $P_{G,i}$5–$P_{G,i}$6 to $P_{G,i}$7. Smaller systems show the same pattern, though with smaller absolute savings because the late-stage ill-conditioning is less dominant (Amani et al., 26 Aug 2025).

Runtime reductions mirror the iteration counts. The paper reports ACOPF/DCOPF runtime reductions of $P_{G,i}$8/$P_{G,i}$9 on 3-bus, $\theta_i$0/$\theta_i$1 on 24-bus, $\theta_i$2/$\theta_i$3 on 118-bus, and $\theta_i$4/$\theta_i$5 on 2869-bus. The 2869-bus ACOPF case is especially significant because the paper states that it has $\theta_i$6 variables, $\theta_i$7 dual variables, and a Newton linear system dimension of $\theta_i$8. In that regime, skipping late Newton steps has a disproportionate time impact (Amani et al., 26 Aug 2025).

The paper also emphasizes that L-IPM recovers the same objective value as IPM after refinement, rapidly decreases feasibility and complementarity errors, and can reduce the error below $\theta_i$9 after just one post-prediction IPM iteration in the 3-bus case. In extreme stressed 2869-bus scenarios where no feasible OPF solution exists, both IPM and L-IPM fail to converge, but L-IPM reaches the numerical failure in about 5 refinement iterations after the learned jump, whereas classical IPM may run for about 53 iterations. This supports a narrower claim: the hybridization accelerates failure detection as well as successful convergence (Amani et al., 26 Aug 2025).

6. Conceptual boundaries, misconceptions, and open problems

A common misconception is that any learning-augmented IPM is automatically feasibility-guaranteed. The literature does not support that conclusion. A learned approximation inside an IPM may reduce linear-solve cost or produce a useful warm start, but unless feasibility is enforced by exact correction, strict interiority logic, or a final certified solver stage, the network itself does not inherit the guarantees of the classical method. The OPF L-IPM is feasibility-guaranteed because its final answer is obtained by IPM refinement under the true constraints, not because the LSTM output alone is provably feasible (Amani et al., 26 Aug 2025, Gao et al., 2024).

A second misconception is that “feasibility” has a single technical meaning across the IPM literature. In fact, at least four distinct notions appear. There is strict interiority of primal or slack variables, as in barrier-domain methods. There is exact affine primal-dual feasibility at every iterate, as in corrected feasible-start LP IPMs. There is controlled infeasibility with residuals converging to zero, as in infeasible-start long-step IPMs. And there is final feasibility after a learned proposal is refined by a classical solver, as in the OPF L-IPM. Conflating these notions obscures what the guarantee actually covers.

The strongest open problem suggested by the literature is the construction of a learned IPM whose learned component is itself certified at each iteration. The LP results indicate one path: require a residual certificate and an exact correction vector $P_{ij}$0. The nonlinear scheduling results indicate another: force primal feasibility, complementarity, and barrier reduction to remain coupled. The strict-feasibility and geodesic papers suggest a third: parameterize the iterate so that interiority is automatic. A plausible synthesis would therefore combine learned prediction with residual-based acceptance, exact algebraic repair, and deterministic fallback (Dexter et al., 2022, Chowdhury et al., 2022, Haeser et al., 2017, Marchi et al., 2022, Permenter, 2020).

Another unresolved issue is scope. The OPF L-IPM is trained and tested on fixed network models and diverse operating points generated by load sampling. The paper does not claim transfer across different topologies, contingency structures, or broader classes of nonlinear programs. More generally, the current literature is much stronger on feasibility-preserving optimization scaffolds than on statistical generalization theory for learned IPM components. This suggests that, at present, the most defensible interpretation of a feasibility-guaranteed L-IPM is a conservative one: learning may accelerate the trajectory, but correctness remains anchored in explicit barrier geometry, certified residual control, or a final classical refinement step.