Coefficient Optimization Techniques

Updated 30 November 2025

Coefficient optimization is the process of inferring, selecting, or adapting real-valued parameters in mathematical models, leveraging techniques like gradient descent and convex reformulation.
Methodologies include analytical gradients, fixed-point iterations, convex programming, and metaheuristic search, enabling effective calibration in PDEs, control systems, and signal processing.
Challenges such as nonconvexity and ill-posedness are mitigated through regularization, smoothing, and hybrid algorithms, which improve computational efficiency and solution accuracy.

Coefficient optimization refers to a broad class of inverse and direct optimization problems where the primary task is to infer, select, or adapt coefficients—typically real-valued parameters—within mathematical models, physical systems, or computational architectures to best achieve a specified objective. This task appears across domains such as partial differential equation (PDE) inverse problems, engineered system calibration, statistical regression/combinatorial schemes, control of dynamical systems, signal processing, and modern data-driven learning methods. Rigorous approaches to coefficient optimization exploit analytical structure, gradient or fixed-point schemes, convex-programming, metaheuristic search, and problem-specific regularization or smoothing to overcome ill-posedness, nonconvexity, or computational bottlenecks.

1. General Mathematical Formulation

In a coefficient optimization problem, the decision variable is a coefficient vector or function (scalar in some models) parameterizing a forward model:

Inverse coefficient problems: Given data $y_{\text{obs}}$ and forward map $y=f(c)$ (where $c$ is the unknown coefficient/parameter vector), infer $c$ such that a loss $L(y_{\text{obs}}, f(c))$ is minimized, possibly subject to constraints $c \in \mathcal{C}$ .
Direct coefficient adaptation: Select coefficients $c$ to optimize system behavior or resource usage (e.g., filter bank coefficient reuse, network connectivity coefficients).
Combinatorial coefficient selection: Identify the subset of coefficients (e.g., features, outlier points) maximizing a statistical metric.

Key challenges often include nonconvexity (e.g., nonlinear PDEs), ill-posedness, data/model noise, or computational scalability. Solutions exploit convexification, fixed-point theory, gradient smoothing, submodular maximization, or metaheuristic search depending on regime and domain.

2. Gradient-based and Fixed-Point Techniques in Physical and Engineering Models

Coefficient optimization in engineered and physical systems often exploits analytical gradients or fixed-point iterates.

Online friction coefficient identification: In legged robot locomotion, $\mu$ (Coulomb friction coefficient) is identified by minimizing the sum of state-prediction residuals over a buffer of measurements, formulated as:

$\min_{\mu \in [\mu_{\min}, \mu_{\max}]} \frac{1}{2} \sum_{i=1}^{H-1} \|\hat{x}_{i+1} - x_{i+1}\|^2_\Sigma$

where $\hat{x}_{i+1}$ is predicted state from rigid body contact dynamics parameterized by $\mu$ . To overcome the non-informative gradient problem from complementarity constraints, a smoothed analytic gradient is constructed by replacing the KKT complementarity with a single smooth equality, enabling robust and efficient Gauss–Newton SQP optimization. Data rejection based on high post-impact normal velocity is critical to avoid spurious updates on non-slippery terrain. Experimental results show subsecond convergence and improved consistency versus nonsmooth or Monte Carlo methods (Kim et al., 24 Feb 2025).

IMRT optimization with truncated coefficient matrices: Fixed-point iteration compensates for dose-deposition matrix truncation in radiation therapy planning. By alternating minimization over a “major” matrix and correction via the truncated “minor” part, the method converges (under a spectral radius condition) to a solution much closer to the full-matrix optimum than a naive truncated solve. The fixed-point map

$x^{(k+1)} = G(x^{(k)}) = \arg\min_{x \ge 0} F(D_1 x + D_2 x^{(k)} - T)$

is shown to be a contraction under mild assumptions, with rapid convergence in practice, reduced memory and computation, and minimal loss in plan accuracy (Tian et al., 2013).

3. Convex and Semidefinite Programming Approaches

Inverse coefficient estimation in continuum models presents challenges of nonlinearity and nonconvexity. Recent advances exploit convex reformulations:

Inverse elliptic coefficient identification: For PDE models such as the Robin–transmission problem, the coefficient function (e.g., piecewise-constant interface parameter $\gamma$ ) can be recovered through a convex nonlinear semidefinite program:

$\min \mathbf{1}^{\top} x \ \text{subject to} \ a \le x \le b, \ F(\hat{x}) - F(x) \succeq 0$

where $F(x)$ is a matrix of Neumann-to-Dirichlet measurements. Theoretical guarantees ensure uniqueness, global solvability, and explicit stability error bounds, with avoidance of spurious local minima. The number of required measurements can be determined via eigenvalue tests on PDE variational derivatives (Harrach, 2021).

4. Metaheuristic and Population-based Algorithms

For problems with weak/no gradients or complex, multimodal landscapes, metaheuristic optimization methods are highly effective in coefficient calibration:

Calibration of drag and heat transfer coefficients: In coastal wave-vegetation modeling and metallurgical solidification, metaheuristics such as Grey Wolf Optimizer, Moth-Flame Optimizer, Particle Swarm Optimization, DE, and others are used to minimize RMSE-type objectives between observed and model-predicted quantities (e.g., wave height, temperature profiles) over parameterized coefficient laws:

$\min_{c \in [c_{\text{min}}, c_{\text{max}}]} \text{RMSE}(c)$

Metaheuristics accelerate and automate calibration compared to manual fitting (orders-of-magnitude faster convergence, reproducibility, easy parallelization), with results for MFO, GWO, and PSO achieving precision as good as or superior to Bayesian MCMC posteriors or empirical formulas (Amini et al., 18 Jan 2024, Stieven et al., 2020). Guidelines on population sizing, stopping criteria, and hybrid empirical–optimization workflows optimize convergence and robustness.

5. Combinatorial and Statistical Coefficient Optimization

Coefficient optimization also includes the discrete selection of coefficients maximizing statistical metrics:

Robust $R^2$ maximization in outlier-prone regression: The “quadratic sweep” algorithm finds the $k$ -subset with the highest coefficient of determination $R^2$ among $n$ planar points by exploiting the conjecture that such inliers are always separated by a conic in $\mathbb{R}^2$ , corresponding to a hyperplane in $\mathbb{R}^5$ . The method lifts points to 5D, enumerates all separating 5-tuples, and solves for the best $R^2$ in $\Theta(n^5\log n)$ time, with empirical evidence for tightness up to $n=30$ , providing a deterministic combinatorial optimum absent convex relaxations (Harary, 12 Oct 2024).
Network coefficient optimization via edge rewiring: Maximizing or minimizing the assortativity coefficient $r$ in a graph through $k$ degree-preserving edge rewirings is formulated as a 0–1 integer program and efficiently approximated via a greedy algorithm exploiting monotonicity and submodularity. For example, the greedy rewiring approach increases $r$ by $0.60$ (from near $0$ to $0.6$) via $10\%$ edge rewiring in Erdős–Rényi networks (Zou et al., 2023).

6. Analytical and Structural Coefficient Optimization

Special classes of coefficient optimization arise in convex analysis, variational mechanics, and signal processing:

Symmetry coefficient in Bregman optimization: The symmetry coefficient $\alpha(h)$ of a Legendre reference function $h$ delineates the maximal allowed step size in Bregman/NoLips-type schemes for non-Lipschitz objectives. Recent work provides calculus rules, dimension-independence, and an efficient root-finding algorithm for $\alpha(\|x\|_2^p)$ , showing that as $p \to \infty$ , $\alpha(\|x\|_2^p) \sim 1/(2p)$ . This enables practical step-size optimization and safe parameterization for a large class of proximal-like methods (Nilsson et al., 25 Apr 2025).
Optimized sharing of coefficients in parallel filter banks: In digital signal processing, structural rearrangement of filter coefficients across $K$ filters of length $M$ enables a two-stage grouping scheme. By exploiting binary coefficient patterns and subset intersections, the total number of MACs is minimized:

$\min_G G M + K 2^{K/G}$

For moderate $K$ , this halves computational resource demand on FPGAs, without increased clock rate or introducing multirate constraints (Arslan et al., 2019).

7. Hybrid and Squeezing Schemes in Nonlinear Inverse Problems

Hybrid coefficient optimization algorithms combine bracketing/fixed-point and nonlinear optimization stages to leverage the strengths of both global exploration and local refinement:

In fluorescence photoacoustic tomography (FPAT), the absorption coefficient is iteratively squeezed between monotone bounding sequences (SIM), rapidly bracketing the solution before switching to gradient-based nonlinear optimization for final accuracy. The hybrid method combines robustness to poor initial guesses and stability under noise or limited data. Multi-measurement extensions further enhance performance (Wang et al., 2018).

Coefficient optimization is thus a unifying paradigm that subsumes a wide spectrum of theoretical and applied problems, ranging from the global inversion of nonlinear coefficients in PDEs, through online and fixed-point estimation in dynamical systems, to discrete combinatorial selection for statistical robustness, and architectural resource minimization in engineered systems. Advances continue to leverage problem-specific structure—smoothness, monotonicity, convexity, submodularity, and analytical gradient flow—alongside sophisticated algorithmic frameworks, to address emerging challenges in scientific computing, engineering, and data science.