Geometry-Weighted Optimization

• Geometry-weighted optimization is a framework that integrates geometric structure into optimization models using metrics like curvature, distance, and manifold constraints.
• It applies to multi-objective geometric programming, image segmentation, sparse coding, and quantum error correction, demonstrating versatility across engineering and data sciences.
• Advanced algorithms leverage dual transformations, manifold projections, and conic representations to improve computational efficiency and solution tractability.

Geometry-weighted optimization refers to frameworks, methodologies, and mathematical techniques that systematically incorporate geometric structure—either of the problem domain, the parameter space, or the objective functions—into the formulation and solution of optimization problems. The term encompasses both direct “weighting” of geometric quantities in the objective (e.g., curvature or distance penalties), and deeper structures where the geometry of the variables, constraints, or coefficient functions fundamentally shapes the feasible set and solution algorithms. Geometry-weighted approaches appear across fields such as engineering design, computer vision, combinatorics, machine learning, and quantum information, supporting applications ranging from multi-objective geometric programming to the optimal embedding of combinatorial structures.

1. Multi-Objective Geometric Programming with Weighted Means

Geometry-weighted optimization emerges naturally in multi-objective geometric programming (GP). In this context, objective and constraint functions are expressed as posynomials, i.e., sums of monomial terms with decision variables raised to arbitrary real exponents and positive coefficients. For multi-objective problems, each objective is a posynomial, frequently representing physically meaningful quantities in engineering or management tasks (Ojha et al., 2010, Ojha et al., 2010).

To resolve the inherent trade-offs, the weighted mean (or weighted sum) method is employed: the vector of objectives $\{f_k(x)\}$ is combined into a single scalar objective

$Z(x) = \sum_{k=1}^p w_k f_k(x)$

where $w_k > 0$ and $\sum_{k=1}^p w_k = 1$. Decision makers modulate the $w_k$ to steer the optimization towards solutions with different Pareto compromises. A distinctive advance in (Ojha et al., 2010) is the incorporation of continuous cost coefficients: instead of constant $C_{k0t}$, the coefficients themselves are continuous functions, e.g., $g(t) = 2t + 2$. This captures real-world variability in system parameters directly within the GP structure.

The resulting weighted GP retains its posynomial form and is solved via duality: converting to the dual, which is linear in the new variables, typically yields a convex problem (of zero degree of difficulty) that is more tractable computationally. However, if the objective space is nonconvex, certain noninferior (Pareto-optimal) points may not be reachable through the weighted sum, highlighting a geometric limitation of the weighted approach.

2. Geometry-Weighted Regularization and Objective Construction

Objective functions in geometry-weighted optimization often include explicit geometric regularization or penalty terms, structured to favor preferred geometric properties:

• Weighted curvature minimization: In image segmentation, energy functionals incorporating terms like $\int_\mathcal{C} (a + b\kappa^2) ds$ are classical (El-Zehiry et al., 2010). The weighted variant replaces the uniform penalty with a contrast-weighted curvature:

$$w_{ijk} = w_{ijk} \cdot \exp(-\beta (g(i) - g(j))^2) \exp(-\beta (g(i) - g(k))^2)$$

allowing sharp boundaries when contrast supports high curvature, otherwise penalizing deviation from smoothness.

• Piecewise-linear regression on weighted lattices: In tropical geometry and mathematical morphology, optimization operators such as dilations, erosions, and residuation are governed by the lattice structure and associated “weights” (Maragos et al., 2019). This underlies efficient algorithms for fitting convex piecewise-linear models to data under max-plus algebra.
• Weighted $\ell_1$ penalties with geometric locality: For dictionary-based manifold learning, the objective

$$\min_{x \in \Delta^m} \left( \frac12 \|y - Ax\|^2 + \lambda \sum_{j=1}^m x_j \|y - a_j\|^2 \right)$$

penalizes reconstructions that draw on distant atoms, thereby enforcing geometric locality in representation (Tasissa et al., 2021).

These constructions ensure that geometric information, either in the data or domain, influences the optimal solution directly via carefully designed weights or regularization terms.

3. Optimization over Geometric and Manifold Structures

Modern geometry-weighted optimization often involves search spaces that are not flat Euclidean domains but manifolds or fiber bundles determined by geometric constraints:

• Manifold optimization in communications: In joint beamforming and phase optimization for intelligent reflecting surfaces (IRS), both the precoder and phase shift variables are constrained to lie on Riemannian manifolds: the complex sphere or oblique manifold for beamformers ($\mathcal{M}_1$) and the complex circle manifold for phase shifts ($\mathcal{M}_\theta$) (Zhang et al., 2021, Devapriya et al., 17 Sep 2024). Optimization proceeds via geometric conjugate gradient or manifold-adapted meta-learning strategies, employing tangent space projections and retractions to maintain feasibility and improve convergence.
• Optimization geometry for cost function families: Rather than treating each instance in isolation, (Manton, 2012) organizes a whole family of cost functions $f(x;\theta)$ into a fiber bundle $M = X \times \Theta$ with projection $\pi$. The minimization occurs along fibers, and the geometry of the family (as opposed to individual convexity) dictates algorithmic complexity. Real-time optimization tracks how solutions evolve with parameters, employing homotopy and Newton updates, revealing that “convex versus nonconvex” is less relevant than the topology of solution sets over $\Theta$.
• Shape optimization via multi-mesh FEM: When constraints are defined by partial differential equations (PDEs) on domains with complex or changing geometry, the use of multiple Nitsche-coupled finite element meshes enables flexible adaptation without global remeshing (Dokken et al., 2018). Optimization is performed over the space of admissible domains, with shape sensitivities computed via Hadamard formulas and adjoint approaches.

4. Algorithmic Acceleration, Regularization, and Quantum Codes

Geometry-weighted approaches also underpin advanced numerical optimization and code construction:

• Barrier-aware and manifold-aware approaches: The BCQN algorithm (Zhu et al., 2017) for mesh deformation optimizes highly nonconvex energies by blending Sobolev- and L-BFGS-derived directions, employing barrier-aware filtering in line-search to prevent element inversion, and using a characteristic gradient norm that scales with problem geometry. This yields order-of-magnitude performance gains in practice.
• Anderson acceleration in geometry optimization: Anderson acceleration, effective for fixed-point iterations in mesh optimization and physics simulation, leverages prior iterates to build multi-secant updates, thereby reducing the iterations needed for convergence even in highly nonlinear or globally coupled geometric models (Peng et al., 2018).
• Geometry-weighted bounds in quantum coding: The minimum distance $d$ of a quantum code constructed from a weighted projective variety is bounded not only by code length ($n$) and dimension ($k$) as in the quantum Singleton bound, but further reduced by a correction $\epsilon$ reflecting the sum of entropy contributions from each orbifold (singular) point (Shaska, 11 Aug 2025):

$d \leq \frac{n - k + 2}{2} - \frac{\epsilon}{2},$

with $$\epsilon = \frac12 \sum_p \left(1 - \frac1{|G_p|}\right)$$. This quantifies the role of geometric defects in limiting code performance and provides a design lever for optimized trade-offs.

5. Combinatorial Optimization and Weighted Metric Spaces

A geometric perspective on optimization reveals deep connections between metric invariants and the solvability of quadratic forms, particularly in weighted finite metric spaces:

• Weighted Hamming cubes: For subsets $X \subset H_W$ (weighted Hamming cubes), explicit formulas for the determinant of the distance matrix, M-constant, and cofactor sum encode the geometric structure, e.g.,

$$\operatorname{det}(D_X) = (-1)^{2n} n^{n-1} \left(\prod_{k=1}^n w_k\right) \det(B)$$

and $M(X) = 2/(\sum_k w_k)$ (Doust et al., 10 Apr 2024). Invertibility (strict 1-negative type) and the absence of polygonal equalities are equivalent to affine independence, directly linking geometry to combinatorial optimization, such as the sparsest cut problem.

• Minimum mediated sets and conic representations: For constraints involving products of variables (weighted geometric means), conversion to second-order cone representations relies on constructing minimum mediated sets—combinatorial objects tracking the minimal number of quadratic constraints needed (Wang, 2022). The resulting efficiency improvements in polynomial, matrix, or quantum information optimization scale with the complexity of the geometric configuration.

6. Implications, Limitations, and Future Directions

Geometry-weighted optimization provides powerful mechanisms for embedding application-driven geometric structure directly into problem formulations and algorithms. The flexibility of such methods enables:

• Adaptation to physical or domain-specific cost variations by modeling coefficients as continuous functions (as in engineering design GP).
• Construction of regularized objectives capturing domain geometry (image segmentation, dictionary learning, or neural network training with path-based regularization).
• Use of geometric and topological invariants to bound or accelerate optimization (negative type for embedding, orbifold contributions for code distance).

However, several caveats arise:

• Weighted sum approaches in multi-objective settings may not capture Pareto points in nonconvex regions of the objective space (Ojha et al., 2010, Ojha et al., 2010).
• Assumptions about the geometric structure (e.g., existence of a unique Delaunay triangulation in dictionary learning (Tasissa et al., 2021)) are critical to correctness and performance.
• Computational cost, particularly in projection-based geometric algorithms for high-dimensional or highly-constrained systems, can be substantial—demanding efficient numerical linear algebra and incremental updating.

Ongoing research explores extensions to non-smooth and combinatorial settings, development of global transformations (problem “pre-conditioning” via geometric insight), incorporation of information-geometric metrics and connections, and applications to high-dimensional or non-Euclidean machine learning problems. In quantum information, exploiting the full spectrum of geometric corrections promises richer trade-offs in the design of error-correcting codes and post-quantum cryptographic systems.

Table: Key Approaches and Corresponding Domains

Geometry-Weighted Strategy	Domain / Application	Core Mathematical Tool
Weighted mean in GP (with continuous cost coefficients)	Multi-objective engineering design	Posynomial modeling, GP duality
Weighted curvature minimization	Image segmentation	Discrete graph-based curvature models
Manifold optimization (sphere/oblique/complex circle)	Communications (IRS/RIS systems)	Riemannian geometry, CG, meta-learning
Weighted $\ell_1$ simplex regularization	Sparse manifold learning	Locality-enforcing sparse codes
Weighted graph Laplacians and spectral alignment	Network robustness, clustering	Heat kernel, $L^1$-distortion criteria
Mediated set methods in conic and semidefinite programming	Polynomial and matrix optimization	Recursive decomposition, conic lifting
Orbifold-corrected quantum Singleton bound	Quantum error correction, cryptography	Algebraic geometry, homological algebra

Geometry-weighted optimization thus constitutes a unifying thread that harnesses geometric structures—quantitative, combinatorial, and topological—to enable new forms of tractable, expressive, and efficient optimization across diverse domains.