Generalized Surrogate Duals

• Generalized surrogate duals are frameworks that extend classical surrogate methods by aggregating multiple constraints in MINLP and decoding duality in quantum modular forms.
• They provide computational benefits with tighter lower bounds, as demonstrated by improved gap closures with increasing aggregation levels.
• Advanced algorithms such as Benders-type methods and MILP relaxations optimize these frameworks while their modular decompositions reveal rich structures in number theory and quantum topology.

A generalized surrogate dual is a theoretical and algorithmic framework extending the classical surrogate dual construction, originally designed for relaxations of mixed-integer nonlinear programs (MINLPs), to handle multiple aggregations of constraints simultaneously. In a parallel but distinct context, the term also denotes the duals of generalized quantum modular forms, which admit intricate connections with mixed mock modular forms and quantum invariants. Both settings exploit duality and aggregation principles, but the methodologies, structural results, and application domains are sharply distinguished. This article presents both paradigms, systematically organizing structural results, algorithms, low-parameter behaviors, and their embedding in broader research frameworks (Mortenson et al., 2022, Müller et al., 2019).

1. Generalized Surrogate Duals in Mixed-Integer Nonlinear Programming

Mixed-integer nonlinear programming problems (MINLPs) take the generic form: $$\text{minimize}\quad c^\top x \quad\text{subject to}\quad g_i(x) \le 0,\,\,\,x \in X := \{ x \in \mathbb{R}^{n-p} \times \mathbb{Z}^p \,|\, Ax \le b \}$$ where $g_i$ are continuous factorable nonlinear functions, $X$ is the mixed-integer linear feasible region.

The classical surrogate relaxation introduces a single nonnegative multiplier vector $\lambda \in \mathbb{R}_+^m$, and replaces $m$ nonlinear constraints with one aggregated surrogate constraint: $\sum_{i=1}^m \lambda_i g_i(x) \le 0$. The minimum of $c^\top x$ over $X$ subject to this constraint defines $S(\lambda)$. The surrogate dual (SD) is the supremal such relaxation: $$\text{(SD)} \qquad \sup_{\lambda \in \mathbb{R}_+^m}\, S(\lambda)$$ This yields a valid lower bound, with $S(\lambda) \ge L(\lambda)$ for all $\lambda \ge 0$, where $L(\lambda)$ is the classical Lagrangian dual (Müller et al., 2019).

The generalized surrogate dual further extends this by including $K$ simultaneous aggregations, introducing $K$ nonnegative vectors $\lambda^1, \dots, \lambda^K \in \mathbb{R}_+^m$ to form $K$ surrogate constraints. The feasible set enforced is the intersection of individual surrogate-aggregated feasible regions, and the dual is taken over all such $K$-tuples (with normalization to avoid degenerate scaling).

$$\text{(GSD}_K)\qquad \sup_{\{\lambda^k\}_{k=1}^K \subset \mathbb{R}_+^m}\, \min_{x \in L_K(\lambda)} c^\top x$$

where $$L_K(\lambda) = \bigcap_{k=1}^K \{x \in X : \sum_i \lambda^k_i g_i(x) \le 0\}$$.

2. Hierarchical Structure and Theoretical Properties

Key structural properties govern the family of generalized surrogate dual bounds:

• Weak duality: Every surrogate relaxation, for any choice of $\lambda^k$, provides a valid lower bound for MINLP.
• Strengthening with increasing $K$: For $K=1$, the classical surrogate dual is recovered; as $K$ increases, the bounds can only tighten:

$$\sup_\lambda S(\lambda) \leq \sup_\lambda S_2(\lambda) \leq \cdots \leq \sup_\lambda S_m(\lambda) = \text{MINLP optimum}$$

• (Non)quasi-concavity: The single-aggregation surrogate dual is quasi-concave but generally discontinuous as a function of $\lambda$; for $K>1$ (multiple aggregations), quasi-concavity fails, though lower-semicontinuity is retained (Proposition 3.1 in (Müller et al., 2019)).
• Zero duality gap at envelope: For $K = m$, the generalized surrogate dual attains the exact MINLP optimum even without additional convexity or regularity assumptions.

3. Algorithmic Framework: Benders-Type Methods

Benders-type algorithms decompose the dual maximization into iterative subproblem/master problem steps. For $K=1$, the master is a linear program (LP); for $K>1$, it becomes a mixed-integer linear program (MILP) with indicator variables encoding the disjunctions from multiple surrogate constraints.

Basic algorithmic cycle:

1. Subproblem: Solve the surrogate relaxation for the current $\lambda$ (or $\{\lambda^k\}$), obtaining primal solution $x^*$ and objective value.
2. Master problem: Generate a new multiplier (or tuple) by maximizing the slack (violation margin) across all aggregated solutions, subject to normalizations to control scaling and symmetry.
3. Termination: Stop when violation margin drops below a prescribed $\epsilon$.

For $K=1$, classical results (Karwan 1976) guarantee finite or limit-point convergence for linear constraints. For general $K$, the MILP master incorporates disjunctions or big-$M$ logic, and convergence to the supremum of $S_K$ is similarly assured (Theorem 4.1, (Müller et al., 2019)).

Algorithmic enhancements with pronounced practical effect include:

• Refined MILP relaxations (cutting planes, RLT, SDP cuts),
• Early stopping criteria in subproblem and master,
• Warm-starts and stabilization (support and trust-region constraints),
• Symmetry-breaking among $\{\lambda^k\}$,
• Primal/dual bound-based early master termination.

4. Numerical Performance and Experimental Outcomes

Computational studies on 1,671 MINLPLib instances demonstrate the practical impact of generalized surrogate duals:

• With 1-hour time budgets and $K=1$, the average root-node gap closed is 18.4%, increasing to 21.4% ($K=2$) and 23.4% ($K=3$) over all instances.
• On the subset of 469 affected problems (those with at least 1% gap closure), surplus closure intensifies: 35.0% ($K=1$), 42.2% ($K=2$), 46.9% ($K=3$).
• Experimental ablations establish the critical role of master early stopping (NOEARLY disables, yielding 88.2% drop in gap closure) and solution stabilization (support/trust-region) for practical convergence.
• In problems with convex nonlinearities (e.g., power-flow), the nonconvex surrogate constraint admits tractable MIQCQP solution and produces tight bounds, outperforming standard spatial branch-and-bound.

The following table summarizes average gap closure (GC) values for select configurations (Müller et al., 2019):

K	All Instances GC (%)	Affected Instances GC (%)
1	18.4	35.0
2	21.4	42.2
3	23.4	46.9

5. Generalized Surrogate Duals in Quantum Modular Forms

In the context of quantum and mock modular forms, generalized surrogate duals refer specifically to the duals of certain generalized quantum modular forms introduced by Hikami and Lovejoy. These appear as convergent $q$-hypergeometric series and are explicated through explicit Hecke–Appell-type sums: $$F_t(q) = q^t \sum_{k_t > \cdots > k_1 \geq 0} \frac{q^{Q(k_1, \ldots, k_t)}}{(q)_{k_1} \cdots (q)_{k_t-k_{t-1}}}$$ where $F_t(q)$ is quantum modular, and its value at $N$-th roots of unity coincides with

$$U_t(x;q) = q^{-t} \sum_{k_t > \cdots > k_1 \geq 1} (-xq)_{k_{t-1}-1}(-q/x)_{k_{t-1}-1} q^{k_1+\cdots+k_t} x^\#\,.$$

These duals admit Hecke–Appell representations, packaging into alternating sums. A key outcome is that (for $1 \leq m \leq t$) the normalized dual

$(1-x) U_t^{(m)}(-x;q) = f_{t+1,m}(x)$

can be decomposed as a finite linear combination of Appell–Lerch sums and theta functions. Explicitly, for $t \geq 2,\, 1 < m < t$ ((Mortenson et al., 2022), Theorem 1.7): $$f_{t,m}(x) = \sum_{k=0}^{2t-1} (-1)^k q^{(k-t)^2} \theta_{k-t,m}(q) \, m(q^{2t-1} x^2, q^{k-t}x; q) + \text{theta products}$$ where each $\theta_{p,m}(q)$ is an indefinite theta function.

6. Mixed Mock Modular Structure and Explicit Examples

The (normalized) generalized surrogate duals $(1-x) U_t^{(m)}(-x;q)$ are mixed mock modular forms of weight 2 on congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$. Their modular transformations mix Appell–Lerch and theta components, with Eichler integrals supplying non-holomorphic completions ("shadows"). For $t=1$, the duals reduce to classical theta series and are modular without mock contributions. For $t=2$, $m=1$, explicit expansions display genuine mixed mock modular behavior, with connections to Ramanujan's universal mock theta function of order 3.

7. Connections, Generalizations, and Significance

Generalized surrogate duals, in both optimization and modular form contexts, provide structured relaxations or expansions that are strictly tighter or richer than traditional (single-aggregation or modular) analogs. In MINLP, generalized surrogate relaxations systematically strengthen dual bounds and offer computational leverage within branch-and-bound or cutting-plane algorithms. In quantum modularity, these duals underpin quantum invariants for torus knots and encode holomorphic anomaly structure essential for applications in topology, quantum topology, and the enumeration of black-hole microstates.

The theory of Hecke-type double sums, Appell–Lerch sum decompositions, and indefinite theta completions applies widely in the study of mock modularity, extending to umbral and higher rank settings in representation theory and mathematical physics. The duality concept also generalizes to other combinatorial and quantum modular settings, exposing connections to knot theory, wall-crossing, and the geometry of indefinite theta series (Mortenson et al., 2022, Müller et al., 2019).