MAGNEO: A Projected-Gradient Energy-Relaxation Framework with Adaptive Multi-Stream Penalties and Contrastive Integrality.

This paper introduces MAGNEO (Matrix-based Adaptive Gradient for Network Energy Optimisation), a general, problem-agnostic framework for combinatorial optimisation via differentiable continuous relaxation. Given a combinatorial problem over binary or q-ary decision variables, MAGNEO lifts it to a smooth energy-minimisation problem over a compact convex domain and solves the resulting non-convex programme via projected gradient descent with independently adaptive penalty coefficients. The framework addresses three fundamental difficulties of penalty-based continuous relaxation: 1. Constraint enforcement tension — penalty coefficients must be strong enough to enforce feasibility but not so large as to dominate the attraction landscape. 2. Fractional solutions — gradient descent on a continuous domain naturally converges to fractional optima; driving variables toward \0, 1\N without destroying accumulated structural information is non-trivial. 3. Global structural properties — constraints such as Hamiltonian connectivity cannot be enforced by separable per-variable penalties; classical local penalties are necessary but not sufficient. Mathematical Framework Target Energy Function Given a combinatorial problem encoded over the continuous relaxation domain x 0, 1N, the MAGNEO target energy is: E (x;\, (t) ) = E₀ₓₓ (x) + (t) \, P₂₎₍ₒₓₑ (x) where: P₂₎₍ₒₓₑ (x) = Pₒ₏₄₂ (x) + P₁₈₍ (x) - E₀ₓₓ (x) — the attraction term — encodes the combinatorial objective (or a surrogate guiding energy) and steers the search toward high-quality solutions. - Pₒ₏₄₂ (x) — the structural penalty — measures, in a smooth (at least C¹) way, the violation of problem-specific constraints (capacity, coverage, degree regularity, subtour elimination, etc. ). - P₁₈₍ (x) — the binary penalty — is problem-independent and pushes every relaxed variable toward \0, 1\: P₁₈₍ (x) = ₉=₁^N xⱼ\, (1 - xⱼ) Each summand is strictly positive for xⱼ (0, 1) and vanishes if and only if xⱼ \0, 1\. Three-Stream PGAC Architecture The Projected Gradient with Adaptive Coefficients (PGAC) energy extends the base framework to three independently regulated penalty streams: E₏₆₀₂ (x;\, t) = E₀ₓₓ (x) + ₒ₏₄₂ (t) \, Pₒ₏₄₂ (x) + ₁₈₍ (t) \, P₁₈₍^ctr (x) + ₂ₘ₂ (t) \, P₂ₘ₂ (x) The three coefficients evolve according to independent rules: - ₒ₏₄₂ (t) — updates reactively in proportion to the current measured constraint violation. - ₁₈₍ (t) — combines a reactive fractionality-driven term with a monotone floor that prevents stalling. - ₂ₘ₂ (t) — follows a smooth growth schedule (non-reactive), guiding the solution gradually toward globally connected structures. This independent regulation maintains a dynamic equilibrium among the three objectives throughout the optimisation trajectory, avoiding both premature constraint enforcement and stagnation in fractional solutions. Contrastive Integrality Penalty To produce sharp binarisation, MAGNEO introduces the contrastive binarisation penalty: P₁₈₍^ctr (x;\, ) = P₁₈₍ (x) + N₁₈₍ (x) ² = ⱼ xⱼ (1-xⱼ) + Nⱼ xⱼ (1-xⱼ) ² Setting B (x) = ⱼ xⱼ (1-xⱼ), the gradient is: P₁₈₍^ctr (x) = (1 + 2N\, B (x) ) \, P₁₈₍ (x), (P₁₈₍ (x) ) ⱼ = 1 - 2xⱼ The non-separable quadratic coupling NB (x) ² creates a self-reinforcing pressure: as variables become binary, the gradient pressure on the remaining fractional variables grows, producing an abrupt phase-transition-like convergence to binary solutions. Cyclic Spectral Penalty To enforce global connectivity and acyclicity on cycle-structured problems, MAGNEO develops a three-term cyclic spectral penalty: P₂ₘ₂ = PA + PB + PC Term A — Log-determinant (spanning-tree structure via effective resistances): PA = -A (L + I) where L is the weighted graph Laplacian constructed from the continuous edge variables and > 0 is a regularisation parameter. The closed-form gradient with respect to the edge variable x₈₉ is: PA x₈₉ = -A (L+ I) ^-1₈₈ - (L+ I) ^-1₈₉ - (L+ I) ^-1₉₈ + (L+ I) ^-1₉₉ This penalises disconnected structures via effective resistances. Term B — Trace powers (short cycle penalisation): PB = ₊=₃^K ₖ\, tr (Aᵏ) where A is the adjacency matrix of the current continuous solution. The matrix gradient satisfies: A\, tr (Aᵏ) = k\, A^k-1 giving closed-form per-entry gradients via the matrix power chain rule. Term C — Fiedler-value barrier (algebraic connectivity enforcement): PC = -C\, sp_\! (₂ (L) -), sp_ (z) = 1 (1 + e^ z) where ₂ (L) is the Fiedler value (second smallest eigenvalue of the Laplacian), > 0 is a target lower bound, and sp_ is the softplus approximation. Letting v₂ denote the Fiedler eigenvector, the gradient with respect to edge variable x₈₉ is: PC x₈₉ = -C\, sp_'\! (₂ -) (v₂, ₈ - v₂, ₉) ² All three gradients are derived in closed form through spectral matrix calculus. Problem-Specific Instantiations The framework is instantiated on five classical NP-hard problems by specifying the pair (E₀ₓₓ, \, Pₒ₏₄₂): | Problem | E₀ₓₓ | Pₒ₏₄₂ | |||| | Knapsack 0/1 | -₈=₁ⁿ vᵢ\, xᵢ | \! (0, \, ᵢ wᵢ xᵢ - W) ² | | Set Cover | -ⱼ xⱼ\, \|Sⱼ\|cⱼ | ₈ ₔ₉: \, ₈ ₒ䲛 (1-xⱼ) | | Vertex Cover | ₔ ₕ cᵤ\, xᵤ | (ₔ, ₕ) ₄ (1-xᵤ) (1-xᵥ) | | Max Cut | - (ₔ, ₕ) ₄ wₔₕ (xᵤ - xᵥ) ² | 0 | | TSP | ₈<₉ d₈₉\, x₈₉ | P₃₄₆ (X) + Pₒₔ₁ₓ₎ₔₑ^act (X) | Convergence Theory Theorem (PGAC Convergence — O (1/T) rate). Let \xᵗ\ be the sequence of projected gradient iterates of the PGAC energy with step size 1/L, where L is the Lipschitz constant of the gradient. Assume the penalty coefficients ₒ₏₄₂ (t), ₁₈₍ (t), ₂ₘ₂ (t) are non-decreasing and uniformly bounded above. Then for any T 1: 1Tₓ=₀^T-1\|xᵗ - ₃\! (xᵗ - \, E₏₆₀₂ (xᵗ;\, t) ) \|² \;\; C + T where C depends on the initial energy gap and is a finite constant that absorbs the cumulative perturbation from the time-varying coefficients via a telescoping perturbation argument. Crucially, is independent of T, so the adaptive-schedule setting achieves the same O (1/T) rate as classical fixed-coefficient projected gradient descent. The result extends to the full four-parameter PGAC energy (Corollary of the main theorem). The Lipschitz constants for all penalty terms — including PA, PB, and PC — are explicitly verified. MAGNEO-Certified: Hybrid Heuristic and SDP Certification MAGNEO-Certified is a two-phase certified solver: - Phase 1 — runs the PGAC heuristic to produce a high-quality upper bound UB (feasible tour). - Phase 2 — solves a moment–SOS semidefinite relaxation (MLM hierarchy) of the polynomial certification energy, producing a rigorous lower bound LBₖ at relaxation order k. The certified gap satisfies the full chain of inequalities: f₏₎₋ₘ^, TSP \;\; LBₖ \;\; f₏₎₋ₘ^, k \;\; UB where f₏₎₋ₘ^, TSP is the polynomial relaxation optimum and LBₖ converges monotonically to it as k. The certified gap UB - LBₖ is computable and rigorous. Fundamental incompatibility (Proposition). The non-polynomial spectral terms PA = -A (L+ I) and PC (softplus Fiedler barrier) cannot be exactly represented as polynomials and therefore cannot be directly certified via SOS methods. Trace-power approximations ₖ ₖ\, tr (Lᵏ) to fail near the boundary of the feasible set (quantified experimentally). Consequently, MAGNEO-Certified certifies proximity to the polynomial relaxation optimum, not to the full PGAC energy optimum. Extensions q-ary Variables via One-Hot Relaxation For problems with q-valued decision variables, each variable xᵢ \0, , q-1\ is encoded as a one-hot vector yᵢ ₐ-₁ (the probability simplex). The relaxation domain becomes the product simplex: D = (ₐ-₁) ⁿ, ₐ-₁ = \y 0, 1q \;: \; =₁q y_ = 1\ The one-hot integrality penalty is: P₈₍ₓ^q-ary (y) = ₈=₁ⁿ (1 - =₁q y₈, ²) which vanishes if and only if each yᵢ is a vertex of ₐ-₁. The projection ₃ becomes