Harmonizing Optimization Methods

• Harmonizing Optimization is a design pattern that couples interdependent subproblems, integrating data-driven and moment-based methods through adaptive weighting.
• It extends traditional optimization by combining stochastic programming, reinforcement learning, and discrete-continuous hybrid control for balanced outcomes.
• Empirical studies in traffic management, image compression, and robotic coordination show significant performance gains over decoupled approaches.

Harmonizing Optimization (HO) denotes a class of optimization formulations in which components that are usually treated separately are deliberately coupled within a single decision procedure. In the cited literature, the term is used for at least three distinct but structurally related settings: blending Sample Average Approximation (SAA) with moment-based Distributionally Robust Optimization (DRO) through a sample-size–dependent weight $\lambda_N$ (Jin et al., 26 Aug 2025); adaptively combining MSE and MS-SSIM during learned image compression training by reinforcement learning (Sun et al., 2021); and, in a closely related usage, tightly coupling discrete coalition assignment with continuous hybrid control in multi-agent systems through combinatorial-hybrid optimization (Tang et al., 2023). A related control formulation for automated vehicles exemplifies the same harmonization principle by coordinating each vehicle’s speed/time trajectory with its predecessor under a hard rear-end safety constraint (Malikopoulos et al., 2016).

1. Conceptual scope

The literature uses HO to address a recurring structural difficulty: important optimization subproblems are interdependent, yet conventional methods often optimize them separately. In stochastic programming, SAA is purely data-driven and can suffer when data are insufficient, whereas moment-based DRO uses partial distributional information but can become conservative even with ample data; HO is introduced specifically to bridge these approaches by “adaptively adjusting the weights of data and information based on sample size $N$” (Jin et al., 26 Aug 2025). In learned image compression, PSNR and MS-SSIM reflect different aspects of human perception, and separate training for each metric is described as common but suboptimal; HLIC instead trains a single model while dynamically adjusting the relative weight between MSE and MS-SSIM (Sun et al., 2021).

In multi-agent systems, the analogous separation occurs between coalition selection and collaborative control design. The CHO framework states that coordination often involves “selecting appropriate sub-teams for different tasks” and “designing collaborative control strategies to execute these tasks,” with existing work frequently assuming one aspect is given while optimizing the other. CHO addresses this by optimizing “both the discrete modes of collaboration and the continuous control parameters simultaneously and iteratively” (Tang et al., 2023). In freeway control, the harmonization idea appears as speed coordination: each automated vehicle computes an analytically optimal acceleration/deceleration trajectory while respecting predecessor-dependent safety timing (Malikopoulos et al., 2016).

Taken together, these works indicate that HO is not a single universally standardized algorithm. Rather, it is a domain-specific design pattern in which optimization layers, metrics, or information sources are brought into explicit interaction instead of being fixed in advance.

2. Canonical mathematical formulations

A central stochastic-programming formulation defines HO over a decision vector $x \in X \subset \mathbb{R}^n$, random parameter $\xi \in \Xi \subset \mathbb{R}^m$ with unknown distribution $P^*$, loss $\ell(x,\xi)$, iid samples $\{\xi_i\}_{i=1}^N$, and a moment-based ambiguity set $P$. Its simplest form is

$$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$

where $\lambda_N \in [0,1]$ depends on sample size (Jin et al., 26 Aug 2025). The same source notes that one may equivalently define $N$0 as the weight on information and $N$1 on data; the notation can therefore be reversed without changing the underlying philosophy.

In learned image compression, HLIC preserves the standard rate–distortion structure but makes the distortion term adaptive. The basic loss is

$N$2

and HLIC replaces the fixed distortion term by

$N$3

with $N$4 and $N$5 to guarantee positivity (Sun et al., 2021). At epoch $N$6, the end-to-end loss is written as $N$7 with epoch-dependent $N$8.

In multi-agent CHO, the coupled structure is explicitly hybrid. With agents $N$9, tasks $x \in X \subset \mathbb{R}^n$0, and parameterized modes $x \in X \subset \mathbb{R}^n$1, a hybrid plan for task $x \in X \subset \mathbb{R}^n$2 is

$x \in X \subset \mathbb{R}^n$3

where $x \in X \subset \mathbb{R}^n$4 is the coalition and $x \in X \subset \mathbb{R}^n$5 is the continuous parameter for one mode. The task cost is

$x \in X \subset \mathbb{R}^n$6

subject to discrete-time dynamics, terminal condition $x \in X \subset \mathbb{R}^n$7, and the no-overlap constraint $x \in X \subset \mathbb{R}^n$8 for all $x \in X \subset \mathbb{R}^n$9. The overall CHO objective is

$\xi \in \Xi \subset \mathbb{R}^m$0

(Tang et al., 2023).

In automated-vehicle speed harmonization, the state and control for vehicle $\xi \in \Xi \subset \mathbb{R}^m$1 are position $\xi \in \Xi \subset \mathbb{R}^m$2, speed $\xi \in \Xi \subset \mathbb{R}^m$3, and acceleration $\xi \in \Xi \subset \mathbb{R}^m$4, with dynamics $\xi \in \Xi \subset \mathbb{R}^m$5 and $\xi \in \Xi \subset \mathbb{R}^m$6. The cost is

$\xi \in \Xi \subset \mathbb{R}^m$7

subject to boundary conditions at control-zone entry and speed-reduction-zone entry, box constraints on acceleration and speed, and the rear-end safety constraint

$\xi \in \Xi \subset \mathbb{R}^m$8

where $\xi \in \Xi \subset \mathbb{R}^m$9 (Malikopoulos et al., 2016).

3. Algorithmic mechanisms for harmonization

The stochastic-programming HO framework makes sample size the primary control signal for balancing data and prior information. Its heuristic derivation states that when $P^*$0 is small, the empirical average is high variance and the method should rely more on the moment-based DRO term; when $P^*$1, the empirical term converges to $P^*$2 at rate $P^*$3, so the DRO contribution should be down-weighted proportionally to $P^*$4 (Jin et al., 26 Aug 2025). The formal prescription is $P^*$5 for some constant $P^*$6, with practical methods for choosing $P^*$7 including K-fold cross-validation, confidence-interval gap tightening, and a rule-of-thumb $P^*$8.

The same paper introduces a finite-sample coverage construction through the convex hull

$P^*$9

where $\ell(x,\xi)$0 is the empirical distribution. A bisection-search procedure computes the minimal $\ell(x,\xi)$1 such that the Gelbrich-distance boundary of $\ell(x,\xi)$2 exceeds $\ell(x,\xi)$3, and the optimization itself is implemented by building a CVX (or conic) model and calling a conic solver such as MOSEK (Jin et al., 26 Aug 2025).

HLIC uses online loss adaptation by reinforcement learning. Its observation vector $\ell(x,\xi)$4 is computed on a held-out validation set each epoch and contains current bpp, current PSNR, current MS-SSIM, gradient-norm of distortion loss, total variation of reconstructions, and related features. The action is $\ell(x,\xi)$5, produced by a small MLP policy $\ell(x,\xi)$6 that outputs Gaussian parameters $\ell(x,\xi)$7; the sampled internal variables are exponentiated to obtain positive $\ell(x,\xi)$8 (Sun et al., 2021). The reward is a composite function of current bpp relative to a target bit-rate and relative to two fixed baseline rate–distortion curves, one PSNR-only and one SSIM-only. Policy updates use PPO with GAE and the clipped objective

$\ell(x,\xi)$9

with $\{\xi_i\}_{i=1}^N$0.

CHO implements harmonization through two interleaved layers. The discrete coalition-formation layer optimizes an assignment

$\{\xi_i\}_{i=1}^N$1

using an estimated coalition cost $\{\xi_i\}_{i=1}^N$2 and global cost $\{\xi_i\}_{i=1}^N$3. Agents perform switch operations $\{\xi_i\}_{i=1}^N$4 that move agent $\{\xi_i\}_{i=1}^N$5 to task $\{\xi_i\}_{i=1}^N$6. The continuous hybrid layer, given a tentative coalition $\{\xi_i\}_{i=1}^N$7, solves for $\{\xi_i\}_{i=1}^N$8 under fixed mode intervals and system dynamics (Tang et al., 2023). Coordination proceeds iteratively from an initial assignment, querying the hybrid layer for the task whose estimate is currently the $\{\xi_i\}_{i=1}^N$9-th largest, updating estimates or attempting improving switches, restarting from $P$0 when earlier target coalitions change, and terminating at a Nash-stable assignment.

Within the CHO hybrid search, feasibility and cost are approximated at multiple granularities. The framework uses an initial cheap estimate such as a sum of agent-to-object distances, together with a two-level heuristic for “A*-style” discrete mode search: a global admissible heuristic $P$1 true-cost-to-go and a local differentiable heuristic $P$2 satisfying $P$3 within radius $P$4. These are blended into a balanced heuristic

$P$5

and node expansion alternates sampled primitive parameters with $P$6 iterations of continuous optimization for parameter refinement (Tang et al., 2023).

In automated-vehicle speed harmonization, the algorithmic structure is decentralized and analytic rather than iterative. Each vehicle entering the control zone reads its initial state and the predecessor’s exit time, computes a safe exit time $P$7 through the stated recursion, forms a $P$8 linear system from boundary conditions, solves analytically for the coefficients $P$9, and applies the affine control $$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$0 with periodic re-solving or replanning under perturbations (Malikopoulos et al., 2016).

4. Guarantees, stability, and approximation properties

The stochastic-programming HO paper gives both finite-sample and asymptotic guarantees. Under a moment-based ambiguity set

$$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$1

Theorem 3.1 states a non-asymptotic inclusion result: under mild light-tail assumptions on $$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$2, there exist constants $$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$3 such that, for a minimal $$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$4 determined by the Gelbrich-distance boundary condition, one has

$$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$5

with

$$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$6

Theorem 3.2 then states asymptotic optimality for $$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$7: if $$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$8 is the true optimum and $$\min_{x\in X}\left\{\lambda_N\cdot \frac{1}{N}\sum_{i=1}^N \ell(x,\xi_i) + (1-\lambda_N)\cdot \sup_{Q\in P}E_Q[\ell(x,\xi)]\right\},$$9 the HO optimum, then $\lambda_N \in [0,1]$0 with probability $\lambda_N \in [0,1]$1 and $\lambda_N \in [0,1]$2 (Jin et al., 26 Aug 2025).

CHO provides two distinct guarantees. First, the final assignment $\lambda_N \in [0,1]$3 produced by its switching process is Nash-stable under the true cost $\lambda_N \in [0,1]$4: no single switch $\lambda_N \in [0,1]$5 yields a lower global cost than $\lambda_N \in [0,1]$6. The proof sketch given is constructive, relying on the fact that each accepted switch strictly reduces the current $\lambda_N \in [0,1]$7, so once no further switch exists the Nash condition holds. Second, the hybrid-search layer admits an approximation bound: if, along a search-tree path, each step $\lambda_N \in [0,1]$8 uses

$\lambda_N \in [0,1]$9

then the balanced heuristic satisfies

$N$00

so the cost is within $N$01 of the true optimum (Tang et al., 2023).

In speed harmonization, safety is enforced as a hard constraint through entry-time scheduling rather than by directly embedding the continuous-time gap constraint in the closed-form optimizer. The authors prove that if the gap is safe at entry $N$02 and at the scheduled time $N$03, then under the unconstrained optimal acceleration arc the gap constraint never becomes active inside $N$04 (Malikopoulos et al., 2016). This result is central to real-time tractability, because it allows analytic trajectory generation without solving a continuously constrained online optimal-control problem.

HLIC does not state an analogous optimality theorem in the supplied summary, but it does provide methodological convergence claims tied to the optimization machinery. Exponentiating $N$05 ensures that the loss weights remain positive and vary smoothly on a log-scale, while PPO with GAE is reported to control gradient variance and enforce small policy changes via clipping. An ablation is also reported in which PPO with a trajectory length of $N$06 outperforms both REINFORCE and PPO-1 (Sun et al., 2021). This suggests that, in this setting, harmonization quality depends not only on the loss parameterization but also on the variance-control properties of the policy-gradient estimator.

5. Empirical behavior across application domains

The reported experiments cover transportation control, stochastic programming, learned image compression, and cooperative robotics. Although the metrics are domain-specific, each study compares harmonized optimization against a decoupled, fixed, or single-source alternative.

Domain	Harmonized formulation	Reported comparison
Stochastic programming	SAA + moment-based DRO	Better solution quality than Wasserstein-based DRO; effective scenario reduction (Jin et al., 26 Aug 2025)
Learned image compression	Adaptive MSE + MS-SSIM via PPO	Better visual quality and higher VMAF than separate-metric baselines (Sun et al., 2021)
Multi-agent systems	Discrete coalition formation + continuous hybrid control	Better completion time and mean cost than Greedy Assignment and Fixed-Mode (Tang et al., 2023)

In collaborative transportation and dynamic capture, CHO is evaluated against Greedy Assignment and Fixed-Mode baselines. For collaborative transportation with 16 agents and 6 boxes in clutter, the total planning time is approximately $N$07, including 16 hybrid solves; completion time is $N$08 versus $N$09 for Greedy Assignment and $N$10 for Fixed-Mode, while mean cost is $N$11 versus $N$12 and $N$13 (Tang et al., 2023). For dynamic capture with 10 pursuers and 3 evaders, CHO captures in 74 timesteps versus 186 for Greedy and 108 for Fixed-Mode, with mean cost $N$14 versus $N$15 and $N$16. The paper summarizes these outcomes as a 20–60% improvement over decoupled baselines.

In automated-vehicle speed harmonization, the simulation environment is a single-lane 2,000 m corridor in VISSIM with a 300 m control zone and a 300 m speed-reduction zone, connected to MATLAB through a DLL/COM interface. Three traffic volumes are tested: 1,620 veh/h, 1,800 veh/h, and 1,980 veh/h. Relative to human-driven baseline traffic, fuel consumption per vehicle is reduced by 19–22% and travel time by 26–30%; relative to the variable speed limit algorithm, fuel consumption is reduced by 12–17% and travel time by 3–19%; relative to vehicular SPD-HARM, fuel consumption is reduced by 18–34% and travel time by 31–39% (Malikopoulos et al., 2016). Throughput at the bottleneck is also reported to increase relative to all benchmarks.

In learned image compression, HLIC is evaluated on Kodak and Tecnick, and on both Ballé18 and Cheng20 architectures. Starting from an MS-SSIM-only baseline, an HLIC configuration tuned to maximize SSIM with $N$17, $N$18, $N$19 still yields a PSNR gain of 0.3–0.5 dB at typical bit-rates while further improving MS-SSIM by approximately 0.005 (Sun et al., 2021). Visual comparisons at low bit-rate, around 0.1 bpp, report that PSNR-only models oversmooth textures and SSIM-only models exhibit color shifts or text blurring, whereas HLIC retains sharp edges and fine texture without color artifacts. VMAF improves by 2–4 points over the MS-SSIM-only baseline at the same bpp.

In the stochastic-programming HO study, the mean-risk portfolio experiment uses $N$20 assets and evaluates out-of-sample loss on $N$21 test draws. HO-T, using the moment set and $N$22 selected by the stated procedures, is reported to uniformly outperform pure Wasserstein and W+M DRO across $N$23, with gains up to 5% when $N$24; after initial $N$25 selection, HO methods require less than 1 s, while W+M cross-validation takes more than 100 s (Jin et al., 26 Aug 2025). In network lot-sizing, with $N$26 stores and original scenario counts $N$27 reduced to $N$28, HO-SR achieves at most 5% approximation error even for $N$29, versus more than 200% for Random and more than 100% for Local-Search when $N$30 is small. For $N$31, total CPU time is approximately 380 s for HO-SR, versus more than 2 h for local-search and more than 20 h for full SAA.

6. Relation to neighboring methods and recurrent misconceptions

The surveyed literature consistently positions HO against methods that commit in advance to a single information source, a single objective surrogate, or a fixed decomposition. In stochastic programming, HO is contrasted with pure SAA, Wasserstein DRO, and moment-based DRO variants (Jin et al., 26 Aug 2025). In learned image compression, it is contrasted with training separate PSNR-only and MS-SSIM-only models, as well as with hand-tuned hybrid losses and the DSSIM heuristic (Sun et al., 2021). In multi-agent systems, it is contrasted with decoupled baselines such as Greedy Assignment and Fixed-Mode (Tang et al., 2023). In freeway control, it is contrasted with human-driven traffic, a variable speed limit strategy labeled “SPECIALIST,” and vehicular SPD-HARM (Malikopoulos et al., 2016).

A common misconception suggested by these comparisons is that harmonization is merely a static weighted average. The literature points to a more structured interpretation. In HLIC, the weights are not fixed: they are policy outputs updated every epoch from validation-state observations (Sun et al., 2021). In stochastic HO, the balance is explicitly tied to sample size through $N$32 or through finite-sample coverage calibration (Jin et al., 26 Aug 2025). In CHO, harmonization is not a convex combination at all, but an interleaving of discrete switches, true-cost feedback, and continuous hybrid replanning (Tang et al., 2023). In speed harmonization, the key mechanism is timing recursion plus closed-form trajectory generation under hard safety constraints rather than objective blending (Malikopoulos et al., 2016).

Another recurrent misconception is that the coupled parts can be optimized independently with negligible loss. The reported results do not support that view. The multi-agent study explicitly attributes its gains to “continuously querying the continuous (hybrid) optimizer only for those coalitions that look promising in the discrete layer” and then feeding back true costs, so that “cheap estimates guide coalition search” while “accurate hybrid solutions correct the assignment” (Tang et al., 2023). The image-compression study similarly argues that separate PSNR- and MS-SSIM-optimized models make it difficult to select the model with the best visual quality or overall performance, motivating online loss adaptation (Sun et al., 2021).

The available extensions are likewise domain-specific. For automated vehicles, the cited extensions include mixed traffic, lane changes, uncertainty in sensor/communication, and multi-lane corridors (Malikopoulos et al., 2016). For learned image compression, the stated generalization is to other low-level vision tasks by adapting weights among losses such as $N$33, adversarial terms, VGG terms, and MSE, again through a stable policy-gradient method (Sun et al., 2021). For stochastic optimization, HO is extended directly to scenario reduction through the reduced-HO formulation with $N$34, which preserves partial moment information of the full scenario set while discarding scenarios (Jin et al., 26 Aug 2025).

Across these literatures, the unifying implication is methodological rather than taxonomic: HO names a family of procedures that preserve interaction among optimization components that would otherwise be separated. The specific machinery—analytic optimal control, PPO-based online weight adaptation, sample-size–dependent interpolation between SAA and DRO, or interleaved discrete–continuous search—depends entirely on the structure of the application.