Optimization-based lane-changing trajectory planning in mixed traffic with embedded input convex neural networks

Yuanxiang Yang; Yu Liu; Fabricio Oliveira; Yu Han; Claudio Roncoli

math.OCOct 28, 2026

Optimization-based lane-changing trajectory planning in mixed traffic with embedded input convex neural networks

Authors: Yuanxiang Yang, Yu Liu, Fabricio Oliveira, Yu Han, Claudio Roncoli

Organizations: Mobility and Industrial Management (MIM), Faculty of Engineering Science, KU Leuven, Leuven, 3001, Belgium · Department of Built Environment, Aalto University, Espoo, 02150, Finland · Department of Mathematics and Systems Analysis, Aalto University, Espoo, 02150, Finland · DTU Management, Technical University of Denmark, Lyngby, 2800, Denmark ·

Abstract

Lane-changing (LC) maneuvers are highly challenging vehicle interactions, with substantial implications for traffic safety and efficiency. Developing effective LC strategies for connected and automated vehicles in mixed traffic is complicated by the uncertainty associated with surrounding-vehicle behavior and the computational burden of embedding realistic prediction models within optimization schemes. To capture these complex interactions while retaining tractability, this study proposes a data-driven optimization framework that embeds neural network (NN) surrogates for safety and lateral comfort within a Bézier-curve-based trajectory planner. Specifically, the safety cost during the LC process is quantified via a surrogate composed of an input convex recurrent neural network (ICRNN) encoding the planned trajectory, a fully input convex neural network (FICNN) embedding the LC duration, and a partially input convex neural network (PICNN) decoder fusing them with historical traffic states; separately, another FICNN surrogate approximates the optimal lateral comfort cost as a function of the LC duration. Both surrogates are embedded via linear programming (LP) formulations, avoiding the binary variables required by conventional mixed-integer programming (MIP) approaches for NN-embedded optimization. We further theoretically establish their exactness at optimality and extend the framework to problems with additional analytical objective terms and multiple surrogates. Experiments on a hybrid dataset combining real-world and simulated trajectories demonstrate that the proposed method achieves surrogate accuracy competitive with unconstrained baselines while substantially improving optimization efficiency over conventional MIPs. Closed-loop simulations further show that the proposed planners markedly reduce the weighted planning cost and consistently outperform baseline methods.

Figures & tables

Figure 1: Schematic diagram of LC trajectory planning in mixed traffic environment, reproduced from Yang et al., (2025) .

Figure 2: Architecture of a generic

L

-layer fully input convex neural network (FICNN).

Figure 3: Architecture of a

T

-step input convex recurrent neural network (ICRNN).

Figure 4: Architecture of an

L

-layer partially input convex neural network (PICNN Type I).

Figure 5: Architecture of an

L

-layer partially input convex neural network (PICNN Type II).

Figure 6: Architecture of the safety surrogate model, comprising a non-convex historical encoder, a convex trajectory encoder, and a skip-connection-free PICNN decoder.

Figure 7: The geographical map of the site for SWIFTraj data collection, reproduced from Han et al., 2026b () .

Figure 8: Distribution of mean speeds of ego vehicles during the LC process, comparing real trajectory data and simulation data.

Figure 9: Distribution of TET-based safety metrics comparing real trajectory data and simulation data

Figure 10: Distribution of TIT-based safety metrics comparing real trajectory data and simulation data

Model	LC Trajectories Encoder	LC Duration Encoder	Decoder
Input Reduced Baseline	–	–	MLP
Baseline	LSTM	MLP	MLP
Proposed Model I	ICRNN	FICNN	PICNN Type I
Proposed Model II	ICRNN	FICNN	PICNN Type II

Table 1: Architectural comparison of the evaluated surrogate models.

Model	MAE	RMSE	$R^{2}$
Input Reduced Baseline	$0.6055\pm 0.0067$	$1.3785\pm 0.0061$	$0.8072\pm 0.0017$
Baseline	$0.1525\pm 0.0236$	$0.4580\pm 0.0342$	$0.9786\pm 0.0031$
Proposed Model I	$0.2865\pm 0.0072$	$0.7026\pm 0.0070$	$0.9499\pm 0.0010$
Proposed Model II	$0.3225\pm 0.0055$	$0.7626\pm 0.0073$	$0.9410\pm 0.0011$

Table 2: Prediction performance of surrogate models for TET-based safety metric evaluated on the test dataset. Values are presented as mean

\pm

standard deviation.

Model	MAE	RMSE	$R^{2}$
Input Reduced Baseline	$5.7773\pm 0.1306$	$14.3723\pm 0.0649$	$0.7650\pm 0.0021$
Baseline	$2.1371\pm 0.3413$	$7.2389\pm 1.1846$	$0.9388\pm 0.0196$
Proposed Model I	$2.9469\pm 0.0896$	$7.5187\pm 0.4438$	$0.9355\pm 0.0081$
Proposed Model II	$3.5098\pm 0.0882$	$9.6971\pm 0.2507$	$0.8930\pm 0.0055$

Table 3: Prediction performance of surrogate models for TIT-based safety metric evaluated on the test dataset. Values are presented as mean

\pm

standard deviation.

Model	MAE	RMSE	$R^{2}$
MLP	$0.0306\pm 0.0088$	$0.0772\pm 0.0188$	$0.9969\pm 0.0016$
FICNN	$0.0150\pm 0.0011$	$0.0408\pm 0.0012$	$0.9992\pm 0.0001$

Table 4: Prediction performance of FICNN and MLP models for the optimal lateral comfort cost as a function of LC duration.

Figure 11: Predicted optimal lateral comfort cost of the FICNN surrogate against the sampled ground truth as a function of the LC duration.

Model	Reformulation Type	Binary Variables	Continuous Variables	Constraints	Solution Time (s)
Proposed Model I	LP	–	255	255	$0.0560\pm 0.0156$
Proposed Model I	MIP	200	255	655	$2.2146\pm 1.0703$
Proposed Model II	LP	–	255	255	$0.0512\pm 0.0107$
Proposed Model II	MIP	200	255	655	$1.5420\pm 0.8623$

Table 5: Computational performance and structural statistics of the two safety-surrogate architectures, Proposed Models I and II, under LP and MIP reformulations. Values are presented as mean

\pm

standard deviation.

Planner	Weighted Cost (TET)	Weighted Cost (TIT)	Infeasible Scenarios	Collision Scenarios
Baseline 1	103.475	86.514	23	13
Baseline 2	61.957	44.525	2	7
I-TET	51.141	44.046	-	-
I-TIT	60.461	51.078	-	-
II-TET	42.950	35.800	-	-
II-TIT	44.593	35.246	-	-

Table 6: Comparison of six planners across 100 test scenarios. Mean weighted objective values are computed over the 61 scenarios in which all planners completed LC maneuvers without collisions. The numbers of infeasible and collision scenarios are reported over 100 test scenarios. Both weighted-cost columns are evaluated for every planner, independent of the safety metric used during optimization.

Planner	$C_{\mathrm{safety}}$ (TET)	$C_{\mathrm{safety}}$ (TIT)	$C_{\mathrm{lon}}$	$C_{\mathrm{lat}}$	$C_{\mathrm{eff}}$
Baseline 1	1.020	6.813	4.569	0.650	0.264
Baseline 2	0.961	6.120	1.551	0.369	$-$ 5.276
I-TET	0.493	3.513	1.648	0.599	4.019
I-TIT	0.560	3.725	1.645	0.917	6.833
II-TET	0.513	3.697	1.206	0.608	$-$ 0.829
II-TIT	0.602	4.147	1.220	0.864	$-$ 6.330

Table 7: Mean values of performance metrics for baseline methods and all proposed planners on common successful scenarios.

Figure 12: Longitudinal speed profiles generated by the proposed planners on representative LC scenarios.

Planner	Weighted Cost (TET)	Weighted Cost (TIT)
I-TET	20	20
I-TIT	10	14
II-TET	44	50
II-TIT	26	16

Table 8: Number of test scenarios in which each planner achieves the lowest weighted objective among all four planners.

Model	Label	MAE	RMSE
Proposed Model I	TET	0.6570	1.6297
Proposed Model II	TET	0.6383	1.1963
Proposed Model I	TIT	7.2254	17.1961
Proposed Model II	TIT	6.5026	14.5227

Table 9: Safety-cost prediction errors on each planner’s own planned trajectories over all test scenarios.

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