Cross-entropy optimization with prioritized constraints

Francisco Roldan Sanchez; Pau de las Heras Molins; David Fridovich-Keil; Georgios Bakirtzis

cs.ROOct 1, 2026

Cross-entropy optimization with prioritized constraints

Authors: Francisco Roldan Sanchez, Pau de las Heras Molins, David Fridovich-Keil, Georgios Bakirtzis

Organizations: LTCI, Télécom Paris, Institut Polytechnique de Paris · The University of Texas at Austin

Abstract

When constraints conflict, an optimizer must determine which requirements to preserve and which to relax. On the one hand, a priority ordering specifies which requirements take precedence. On the other hand, penalty-based formulations encode their relative importance through numerical weights. Depending on these weights, a solution can improve its weighted score while violating intended priorities. We introduce TierCEM, a variant of the cross-entropy method that incorporates strict constraint priorities directly into elite selection without requiring per-constraint importance weights. TierCEM works by sequentially filtering sampled candidates, from highest- to lowest-priority constraint. If and when a constraint eliminates all remaining candidates, TierCEM returns to the last nonempty set and selects elites with the smallest violations of that blocking constraint, recursively preserving satisfaction of all higher-priority constraints. We evaluate TierCEM on 2D navigation and contact-rich pushing tasks in proprioceptive and learned world-model settings. Experiments show that reversing the constraint ordering changes which constraints are violated under conflict. Prioritizing progress toward the task objective also enables TierCEM to relax lower-priority constraints when they would otherwise prevent further progress.

Figures & tables

Figure 1: At each iteration, TierCEM samples candidate trajectories from a Gaussian distribution and filters them from highest- to lowest-priority constraint, producing nested candidate sets

\mathcal{K}_{n}

. If a tier yields an empty set (

\mathcal{K}_{n^{*}}=\emptyset

), TierCEM returns to the preceding nonempty set

\mathcal{K}_{n^{*}-1}

and selects elites with the smallest violations of the constraint at tier

n^{*}

, preserving all higher-priority constraints already enforced. Otherwise, elites are selected from the final set according to the smallest violations of the constraint at tier

N

. The elites update the Gaussian sampling distribution for the next iteration; at the final iteration, a plan is selected for execution.

ordered margins

m_{1}\succ\dots\succ m_{N}

candidate samples

K

, elites

\ell

, iterations

I

Algorithm 1 TierCEM

	task_obstacle_lane				task_lane_obstacle
Method	Reach	V ↑	V ↓	Steps	Reach	V ↑	V ↓	Steps
TierCEM	20/20	0	20	34	20/20	0	20	32
CCE ( Wen and Topcu, 2021 )	0/20	0	0	160	0/20	0	0	160
Soft-constraint CEM, $\lambda=0.5$	20/20	20	12	30	20/20	11	20	21
Hierarchical scalarized CEM, (0.5, 0.02)/(0.02, 0.5)	20/20	0	20	38	20/20	0	20	16

Table 1: TierCEM versus the Soft-Constraint CEM, Constrained Cross-Entropy ( Wen and Topcu, 2021 ) (Section 4.2 ), and the hierarchical scalarized CEM (Section 4.2 ) on the task described in Section 4.3 across 20 random seeds, using 10 CEM iterations, initial Gaussian variance per dimension 1.00, planning horizon

H=8

, 4 committed actions before re-planning, 1200 samples and 24 elites per iteration. “Reach" shows the number of seeds where the method reaches the goal out of the 20 different seeds, V ↑ / V ↓ count seeds where the higher-/lower-priority constraint are violated and “Steps” reflects the average number of steps required to complete the task. Hierarchical scalarized CEM shows the weights as tuples (

\lambda_{O}

\lambda_{L}

) for each constraint priority order.

Figure 2: Effect of the prescribed priority ordering in the two-dimensional navigation problem. When task progress has lower priority than the spatial constraints, TierCEM can preserve both by stalling. Prioritizing task progress forces a spatial constraint to relax; reversing the spatial ordering reverses whether the trajectory leaves the lane or crosses the forbidden region.

Figure 3: Visualization of the competing constraints used in the Push-T environment. The goal is to move the block toward the goal position, ignoring orientation. First image shows the initial position, the second image shows the expected behavior when staying within the lane boundary is prioritized over the obstacle constraint, and the third image shows the expected behavior when the obstacle constraint is prioritized over staying within the lane boundary.

Method/configuration	Metric	Goal $x=256$	Goal $x=320$	Goal $x=360$
TierCEM, task $\succ$ obstacle $\succ$ lane	Position error (px)	$20.63\pm 16.17$	$16.06\pm 19.06$	$30.96\pm 34.31$
	Obstacle violation (%)	$0.50\pm 1.50$	$3.75\pm 6.78$	$3.08\pm 6.71$
	Lane violation (%)	$64.75\pm 34.26$	$71.58\pm 32.98$	$57.25\pm 33.70$
CCE ( Wen and Topcu, 2021 ) , joint constraints	Position error (px)	$21.29\pm 18.13$	$56.35\pm 30.77$	$87.80\pm 36.79$
	Obstacle violation (%)	$0.38\pm 1.19$	$1.51\pm 2.32$	$3.45\pm 4.56$
	Lane violation (%)	$0.25\pm 1.09$	$1.57\pm 2.59$	$3.34\pm 3.44$

Table 3: Goal-distance sensitivity in exact-state Push-T, with all planner hyperparameters fixed. Values are mean

\pm

standard deviation across 20 random seeds. Position error is measured in pixels; obstacle and lane violations are percentages of executed controls. Increasing goal distance turns jointly safe satisfaction into lower task accuracy for CCE, while TierCEM can continue by relaxing its designated lower-priority lane constraint.

Panel of Figure 4 — Figure 4: TierCEM and baselines execution in the Push-T scenario using DINO-WM latents. Frames progress from left to right. The red disk blocks the blue lane, the green T region is the goal, and the blue T is the block. TierCEM relaxes the lower-priority constraint to complete the task without requiring a per-constraint weight tuning as hierarchical CEM, CCE preserves safety without progressing toward the goal, and the Soft-Constraint CEM does not have any mechanism to prioritize one constraint over the other.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value	Description
$K$ (samples)	1200	CEM samples per iteration
$\ell$ (elites)	24	elites kept per iteration
$H$ (horizon)	8	planning horizon
iterations	10	CEM iterations
num_act_stepped	4	actions executed per replan
var_scale	1.0	proposal variance scale

Appendix

Table 5: Base configuration. The planner and cascade parameters are those of Algorithm 1 .

Geometry	Config	R	V ↑	V ↓	Steps
base (reference)	task_obstacle_lane	10	0	10	46.4
	task_lane_obstacle	10	0	10	40.0
	task_last	0	0	—	160
	plain	9	—	—	74.8
off-diagonal disk	task_obstacle_lane	10	0	0	42.0
	task_lane_obstacle	10	0	4	34.0

Appendix

Table 6: Geometry robustness of the tier-order at the recommended configuration (Table 5 ) across 10 random seeds. The reference disk-on-lane geometry is re-run alongside three perturbations (an off-diagonal disk (center

(1.2,2.2)

r=1.3

), two overlapping disks, and a lane narrowed to half-width

0.5

) by generalizing the exact signed-distance model to an arbitrary list of disks. R = seeds reaching the goal; V ↑ = seeds where the higher -priority geometry constraint went negative; V ↓ = seeds where the lower-priority constraint went negative; Steps = mean episode steps. For task_last the sacrificed tier is the task (both geometry constraints are protected, so V ↓ is blank), and for plain the cascade is off and neither constraint is protected.

Figure 5: Visualization of trajectories for different constraint configurations: each row is a geometry, each column a configuration. The shaded disks are the obstacles, the dashed lines bound the lane are the lane boundary, the green point is the start and the red star is the goal. Across every geometry plain (first column) smashes through at least one constraint, task_last (second column) stalls preserving both, task_obstacle_lane (third column) arcs around the obstacle sacrificing the lane while the obstacle stays clean, and task_lane_obstacle (fourth column) drives straight sacrificing the obstacle while the lane stays clean.

Setting	Hyperparameters $(K,\ell,H,\mathrm{it},\mathrm{cm},\sigma)$	task_obstacle_lane			task_lane_obstacle
		R	V ↓	V ↑	R	V ↓	V ↑
base	$(1200,\,24,\,8,\,10,\,4,\,0.75)$	2	2	0	0	0	0
samples 600	$(\mathbf{600},\,24,\,8,\,10,\,4,\,0.75)$	2	2	0	1	1	0
samples 2400	$(\mathbf{2400},\,24,\,8,\,10,\,4,\,0.75)$	0	0	0	0	0	0
samples 4800	$(\mathbf{4800},\,24,\,8,\,10,\,4,\,0.75)$	0	0	0	0	0	0
elites 12	$(1200,\,\mathbf{12},\,8,\,10,\,4,\,0.75)$	0	0	0	1	1	0

Appendix

Table 7: Grid Search of the mechanism analysis over 5 random seeds as one hyperparameter varies around the base configuration of Table 5 . Each row lists the full hyperparameter tuple

(K,\ell,H,\mathrm{iters},\mathrm{commit},\sigma)

; bold marks the single value varied away from the base. R = seeds reaching the goal; V ↓ = seeds where the low-priority geometry constraint went negative (lane for task_obstacle_lane , obstacle for task_lane_obstacle ); V ↑ = seeds where the higher-priority constraint went negative.

$\lambda$	Reach	V obs	V lane
0.05	20/20	20	5
0.1	20/20	20	3
0.2	20/20	20	7
0.5	20/20	20	12
1.0	20/20	20	14
2.0	20/20	20	16

Appendix

Table 8: The Soft-Constraint CEM baseline, sweeping the scalar weight

\lambda

(same hyperparameters as Table 1 across 20 random seeds). “Reach" is the number of seeds reaching the goal, and V obs and V lane are the number of seeds where the forbidden disk and the lane are violated respectively. At every

\lambda\in[0.05,2]

the disk is clipped in all 20 seeds, while the lane is violated throughout; the only

\lambda

that stops the disk from being clipped makes the planner stall (reach drops to 2/20 at

\lambda=5

	task_obstacle_lane				task_lane_obstacle
$(\lambda_{\mathrm{O}},\lambda_{\mathrm{L}})$ (ratio)	Reach	V ↑	V ↓	Steps	Reach	V ↑	V ↓	Steps
(0.5, 0.02), 25:1	20/20	0	20	38	20/20	0	20	16
(1.0, 0.05), 20:1	20/20	1	20	41	20/20	1	20	23
(0.5, 0.05), 10:1	20/20	1	20	35	20/20	0	20	25
(1.0, 0.1), 10:1	20/20	1	20	22	20/20	1	20	34
(2.0, 0.2), 10:1	20/20	0	20	32	20/20	2	20	35

Appendix

Table 9: The results of the hierarchical scalarized CEM (same hyperparameters as Table 1 across 20 random seeds). Each row is a pair

(\lambda_{\mathrm{O}},\lambda_{\mathrm{L}})

, and task_obstacle_lane is run with that pair and task_lane_obstacle with the reversed pair, meaning the first weight is always associated with the highest priority constraint of each of the configurations. “Reach" is the number of seeds reaching the goal, V ↑ and V ↓ are the number of seeds where the highest and lower priorities are violated respectively, and “Steps" is the average number of steps executed to complete the episode.

Figure 6: The soft-constraint baseline versus the lexicographic cascade on the mechanism analysis environment (rows: Soft-constraint CEM at $\lambda=0.5$ , the hierarchical scalarized CEM at the tuned 25:1 ratio of Table 9 , the Constrained Cross Entropy (CCE), and TierCEM; columns: the two tier orders). The Soft-Constraint CEM carries a single scalar weight and cuts the forbidden disk (negative min_obs); TierCEM protects the higher-priority tier and sacrifices only the lower one. The hierarchical scalarized CEM, given per-constraint weights in the correct 25:1 ratio, also protects the higher-priority tier at this seed, but the ratio that achieves this must be tuned per environment whereas TierCEM encodes the ordering directly. CCE stalls to a safe state but never completes the task.

Body	Speed $s$	$E(s)$	Reach	min obs	min lane
Kinematic
kinematic	0.4	—	0/20	$+0.59$	$+0.31$
kinematic	1.4	—	0/20	$+0.54$	$+0.31$
kinematic	2.0	—	0/20	$+0.64$	$+0.25$
Inertial, gap = 1.60
inertial	0.4	0.07	16/20	$+0.14$	$-9.23$

Appendix

Table 10: Zero-motion escape routes are only available to kinematic bodies. For inertial bodies we report stopping distance

E(s)=s^{2}/(2A_{\mathrm{max}})

with

A_{\mathrm{max}}=1.2

. “Reach" is successes / 20 seeds. “min obs" / “min lane" are the minimum signed distances to the forbidden disk and lane boundary. Negative = constraint violation. Results are across 20 random seeds.

Figure 7: The kinematic control never reaches the goal (left), meaning no tier ever empties and, therefore, the planner stalls and both geometry constraints stay positive. The inertial body sacrifices the lane constraint in order to complete the goal (center), unless the gap between the starting point and the obstacle is too narrow for the initial speed $s$ (right).

Scenario	min dist	min obs	min ws	Verdict
plain	0.21	$-0.85$	$+0.64$	reaches goal, crashes through
oracle	0.23	$+0.67$	$+0.85$	reaches goal, always safe
mlp_head	0.23	$+0.68$	$+0.80$	reaches goal, always safe
mlp_shifted	3.31	$+0.61$	$+0.71$	never arrives, over-cautious
noisy_head	3.94	$+1.00$	$+1.25$	never arrives, over-cautious
model_mismatch	0.23	$-0.27$	$-0.45$	arrives, violates (3/5 seeds)

Appendix

Table 11: Second controlled environment (with workspace-boundary tier). min dist is the minimum distance to the goal along the executed path, min obs / min ws the minimum signed distance to the obstacle / workspace boundary (negative = violated). Results are across 5 random seeds.

Figure 8: Imperfect constraint signals. plain cuts through the disk; oracle and mlp_head reach the goal safely; mlp_shifted and noisy_head stall short, hallucinating phantom unsafe regions; model_mismatch plans a safe arc but the executed path clips the disk and goes outside the workspace boundary occasionally.

Hyperparameter	Value
Input dimension	384
Hidden dimensions	$(256,64)$
Output dimension	2
Hidden activation	GELU
Normalization	LayerNorm after each hidden linear layer
Dropout	0.2

Appendix

Table 12: Hyperparameters of the shared DINO-WM classifier used for the forbidden-zone and lane-boundary heads.

Panel of Figure 9 — Figure 9: First two rows: resulting KDE of the MLP prediction for the forbidden disk and the lane boundary; third row: the predicted block to goal distance versus the true block to goal distance.

Hyperparameter	Value
Input dimension	384
Hidden dimensions	$(32,32)$
Output dimension	1
Hidden activation	GELU
Normalization	LayerNorm after each hidden linear layer
Dropout	0.5

Appendix

Table 13: Hyperparameters of the DINO-WM task-progress MLP. Values are the defaults used by the fixed-conflict training pipeline.

Panel of Figure 10 — Figure 10: Image representation samples of the Push-T environment. 10a shows the image observation used as an input for the world models. For the rest of images, red zones are forbidden regions, while the blue zone represents the workspace boundary. When a violation happens, it is marked around the end-effector with a circle surrounding it (red for an obstacle violation, and blue for a workspace boundary violation). 10b shows an example where all constraints are satisfied, 10c shows a workspace boundary violation, and 10d shows an obstacle violation.

$\lambda$	O (%)	WB (%)	Pos Dist (px)	Angle Error (°)
0.05	$10.9\pm 1.7$	$35.0\pm 3.4$	$15.2\pm 3.9$	$12.6\pm 3.9$
0.1	$10.4\pm 1.4$	$31.8\pm 3.5$	$15.9\pm 2.3$	$12.8\pm 2.4$
0.3	$10.2\pm 1.8$	$27.7\pm 2.6$	$16.7\pm 2.8$	$14.3\pm 2.9$
0.5	$10.3\pm 2.3$	$26.6\pm 3.2$	$17.3\pm 3.1$	$13.6\pm 3.1$
1.0	$11.0\pm 2.6$	$23.1\pm 2.4$	$19.2\pm 3.2$	$15.6\pm 4.1$

Appendix

Table 14: Soft-constraint CEM with scalarized penalty, evaluated for 100 episodes across 5 random seeds.

$\lambda_{\text{O}}$ / $\lambda_{\text{WB}}$	O (%)	WB (%)	Pos Dist (px)	Angle Error (°)
0.25 / 0.01	$11.2\pm 1.7$	$37.5\pm 2.2$	$15.6\pm 2.4$	$13.1\pm 4.0$
0.5 / 0.02	$9.9\pm 1.6$	$34.8\pm 2.2$	$16.3\pm 3.1$	$13.7\pm 4.2$
1 / 0.04	$10.1\pm 1.6$	$32.0\pm 2.5$	$16.7\pm 3.3$	$14.0\pm 3.7$
0.5 / 0.1	$9.8\pm 1.8$	$29.2\pm 2.9$	$16.9\pm 2.8$	$13.7\pm 3.3$

Appendix

Table 15: Hierarchical scalarized CEM, evaluated for 100 episodes across 5 random seeds.

Panel of Figure 11 — Figure 11: Pareto presentation of the Push-T sweeps of Tables 14 , 15 , and the obstacle as highest priority column of 16 (means over 100 episodes across 5 random seeds). These results do not use OOD detection. (a) Total violation rate against position error, the dashed step line is the Pareto front. (b) The same sweeps in the ordering plane.

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Jun 28, 2026, cs.LGJ/K move · Enter open · S save

Not All Objectives Are Born Equal: Priority-Constrained Descent for Hierarchical Multi-Objective Optimization

Dara Varam, Mohamed I. Alhajri

Department of Computer Science & Engineering, American University of Sharjah · Senseable City Laboratory, Massachusetts Institute of Technology

Multi-Objective Optimization Objectives Hierarchical

Open paper AskPDF arXiv

Deep learning problems rarely involve objectives that are equal in importance. A primary objective defines the goal, whilst secondary objectives, such as sparsity, compression, or robustness constrain the solution. While existing multi-objective methods have proven effective in practice, they have a clear symmetry problem and neglect the inherent objective hierarchy built into these objective spaces. We introduce Priority-Constrained Descent (PCD), a gradient-based optimization framework designed to explicitly exploit hierarchical objective structures. PCD preserves the direction of primary descent whilst allowing for the minimal distortion necessary to guarantee progress on secondary objectives, controlled by a single

τ\in [0, 1]

that dictates the strength of the distortion. The resulting formulation is invariant to objective scaling and admits exact closed-form solutions for problems with two and three objectives. We evaluate PCD within structured network compression settings, unstructured sparsity and low-rankness, and across a variety of synthetic experiments, showing Pareto dominance and better per-objective performance with secondary progress guarantees over existing methods, further exhibiting the interpretable trade-off that

τ

provides.

Abstract

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