Hyper-Adaptive Momentum Dynamics for Native Cubic Portfolio Optimization: Avoiding Quadratization Distortion in Higher-Order Cardinality-Constrained Search

Paper: arXiv 2603.15947

Authors: Greg Serbarinov

Abstract

We study cubic cardinality-constrained portfolio optimization, a higher-order extension of the standard Markowitz formulation where three-way sector co-movement terms augment the quadratic risk-return objective. Classical heuristics like simulated annealing (SA) and tabu search require Rosenberg quadratization of these cubic interactions. This inflates the variable count from n to 5n and introduces penalty terms that substantially distort the augmented search landscape. In contrast, Hyper-Adaptive Momentum Dynamics (HAMD) operates directly on the native higher-order objective using a hybrid pipeline combining continuous Hamiltonian search, exact cardinality-preserving projection, and iterated local search (ILS). On a cubic portfolio benchmark under matched 60-second CPU budgets, HAMD achieves substantially lower decoded native cubic objective values than SA and tabu search, yielding single-seed relative improvements of 87.9%, 71.2%, 59.5%, and 46.9% at n = 200, 300, 500, and 1000. In a detailed three-seed study at n = 200, HAMD attains a median native objective of 195.65 (zero variance), while SA and tabu yield 1208.07. Decoded-feasibility analysis shows SA satisfies all exact cardinality and Rosenberg auxiliary constraints, yet decodes to a native objective 80-88% worse than HAMD, demonstrating a surrogate-distortion effect rather than simple infeasibility. Exact calibration on small instances (n = 20, 25, 30) confirms HAMD finds the provably global optimum in 9/9 trials. These results demonstrate that native higher-order search offers a substantial advantage over quadratized surrogate optimization for constrained cubic portfolio problems.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly novel approach (HAMD) to higher-order portfolio optimization, directly tackling native cubic objectives without resorting to variable-inflating quadratization. It combines rigorous mathematical modeling with strong computational benchmarking, demonstrating clear performance gains and provable global optimality on smaller instances.

Research Flowchart

  flowchart TD
    A["Research Goal: Avoid quadratization distortion in cubic portfolio optimization"] --> B["Data & Setup: Cubic co-movement tensor, risk-return objective, & cardinality constraints"]
    B --> C["Approach Comparison: HAMD Native Search vs. Classical Quadratization SA/Tabu"]
    C --> D["HAMD Pipeline: Continuous Hamiltonian search, cardinality projection, & ILS"]
    C --> E["Baseline Pipeline: Rosenberg quadratization inflates variables 5x + penalty terms"]
    D --> F["Exact Calibration (n=20-30): HAMD finds global optimum in 9/9 trials"]
    D --> G["Key Outcome: HAMD outperforms SA/Tabu by 46.9%-87.9% without surrogate distortion"]

Cited References (29)

1. HAMD Community Edition : Reference Implementation of Hyper-Adaptive Momentum Dynamics for Native Higher-Order Binary Optimization (2026)
2. Portfolio Selection (1952) — link
3. Heuristics for cardinality constrained portfolio optimisation (2000) — link
4. Numerical Methods and Optimization in Finance (2011)
5. Pseudo- B oolean optimization (2002) — link
6. Tabu Search— P art I (1989) — link
7. Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems (2019)
8. High-performance combinatorial optimization based on classical mechanics (2021)

0 of 29 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.