Dual Optimiser Concepts

• Dual optimiser is a concept from duality theory that deciphers the structure of a primal optimisation problem by leveraging convex duality to ensure convergence and robustness.
• It employs methodologies such as semi-infinite programming, martingale transport, and error sensitivity analysis to achieve optimal pricing, hedging, and recovery in various domains.
• Applications span finance, signal processing, and machine learning, where dual optimisers facilitate stable calibration, efficient set selection, and adaptive algorithm design.

A dual optimiser is a concept originating from duality theory in optimisation, where solutions to a dual optimisation problem provide critical insights into the structure and properties of the corresponding primal (original) problem. In the context of stochastic processes, mathematical finance, machine learning, and numerical methods, dual optimisers can be defined as solutions to dual problems—frequently convex, semi-infinite, or involving measure-valued constraints—that either encode optimal pricing, hedging, calibration, super-resolution recovery, or multi-objective efficiency. Dual optimisers often carry strong theoretical guarantees and can be constructed to ensure convergence, stability, and robustness features in both finite and infinite-dimensional settings.

1. Mathematical Definition and Foundational Role

Dual optimisers arise from the dualisation of optimisation problems, where one seeks to characterise the primal feasible/reward set using linear (or more generally, convex) constraints and form an associated dual problem, typically constrained by supermartingale deflators, pricing measures, or Bregman divergences. In mathematical finance, as in (Czichowsky et al., 2014), for proportional transaction cost portfolio optimisation, the dual optimiser is a process or family of price systems $Z \in Z_\mathrm{loc}$ such that the set of terminal liquidation values (payoffs) $C \subset L^0$ satisfies:

$$g \in C \iff \mathbb{E}[g Z_T] \leq 1 \quad \forall Z \in Z_\mathrm{loc},$$

where $Z$ is a dual variable—an $X$-consistent price system or supermartingale deflator. This bipolar characterisation lies at the heart of modern duality theory.

In the super-resolution regime (Chretien et al., 2019, Chrétien et al., 2020), dual optimisers are formed by the solution $\lambda^\star$ of a semi-infinite program,

$$\max_{\lambda \in \mathbb{R}^M} y^T \lambda \quad \text{subject to} \quad \sum_{j=1}^M \lambda_j \phi(t-s_j) \leq 1 \quad \forall t \in I,$$

where the dual certificate $q(s)$ drives spike recovery in the primal.

Dual optimisers are uniquely defined up to an equivalence (e.g., additive affine functions in martingale transport (Schachermayer et al., 27 Aug 2025)), and their existence underpins equivalence between primal and dual solutions. In multi-objective optimisation (Chaiblaine et al., 2020, Bencheikh et al., 2 Feb 2024), dual optimisers are efficient allocations or sequences satisfying cut and efficiency tests.

2. Dual Optimisers in Convex and Martingale Optimal Transport

The theory of dual optimisers is pivotal in martingale optimal transport and generalised calibration (Schachermayer et al., 27 Aug 2025). In stretched Brownian motion (SBM), the dual optimiser $\psi_{lim}$ is a convex function minimising the dual functional,

$$D(\psi) = \int\left[\int \psi(y)\, \pi_x(dy) - \phi^\psi(x)\right]\, \mu(dx),$$

with $$\phi^\psi(x) = \inf_{p} \left(\int \psi\, dp - \text{MCov}(p,\, \gamma) \right)$$, where probability measures $p$ with barycentre $x$ and $\text{MCov}$ denotes maximal covariance. The existence of $\psi_{lim}$ (unique modulo affine functions), and convergence of any dual optimising sequence $(\psi_n)_n$ (after suitable affine adjustments), guarantee robust calibration in financial models and martingale transport. Key technical ingredients include irreducibility, convex order $\mu \leq_c \nu$, and relative compactness of measure support, which enable pointwise and $L^1(\nu)$ convergence except possibly on the relative boundary of the convex hull of $\nu$.

3. Stability, Perturbation, and Sensitivity Analysis

A critical outcome of dual optimisation analyses is the quantification of stability against perturbations, either in dual variables (approximate solutions) or input measurements/noise (Chretien et al., 2019, Chrétien et al., 2020). For super-resolution, a precise relationship links errors in dual optimisers to errors in the recovered primal solutions:

$$|t - t^*| \leq C_{t^*} \|\lambda - \lambda^*\|_2, \quad \|a^\text{est} - a^*\|_2 \leq c \|t^\text{est} - t^*\|_2 + o(\|t^\text{est} - t^*\|_2^2),$$

where $C_{t^*}$ depends on local curvature of the dual certificate $q''(t^*)$ and measurement count $M$. In the presence of additive measurement noise, error propagation is linearly controlled:

$$\|\lambda - \lambda^*\|_2 \leq C_{\lambda^*} \|w\|_2.$$

The analytical backbone is typically a quantitative implicit function theorem, providing explicit constants to control parameter changes.

4. Applications in Finance, Signal Processing, AutoML, and Machine Learning

Dual optimisers have broad relevance:

• Finance: Shadow price construction via dual optimisers and supermartingale deflators (Czichowsky et al., 2014), robust hedging and pricing in discrete and continuous time (Cox et al., 2017), and calibration in local volatility models via martingale optimal transport (Schachermayer et al., 27 Aug 2025).
• Signal Processing: Dual certificates for non-negative super-resolution enable controlled recovery of sparse signals under physical convolution and noise (Chretien et al., 2019, Chrétien et al., 2020).
• Multi-objective and Bi-objective Optimisation: Dual optimiser algorithms for efficient set selection in combinatorial quadratic and linear fractional programming (Chaiblaine et al., 2020, Bencheikh et al., 2 Feb 2024); Pareto-front computation and resource allocation in heterogeneous data-parallel platforms (Khaleghzadeh et al., 2019).
• Learning and AutoML: In neural optimiser search (Morgan et al., 10 Apr 2024), dual optimisers are joint solutions over update equation and decay schedule spaces, evolved via particle-based genetic algorithms, delivering architectures outperforming Adam and SGD families.
• Meta-optimisation: Mirror descent approaches (Gao et al., 2022) learn Bregman divergences to meta-tune dual update rules. Optimizer amalgamation (Huang et al., 2022) aggregates behaviour across multiple optimisers into a single, adaptively optimal policy.

5. Algorithmic Formulation and Implementation Strategies

In discrete optimisation and numerical contexts, dual optimisers are obtained via linear programming or branch-and-cut strategies, often involving:

• Efficient cut construction (using reduced gradients or monotonicity checks) to eliminate dominated candidate points (Chaiblaine et al., 2020, Bencheikh et al., 2 Feb 2024).
• Sequential branching on fractional solutions to ensure exploration of the integer solution space.
• Primal attainment proofs via compactness and Kolmogorov–Riesz theorems in infinite-dimensional settings (Cox et al., 2017).
• Use of advanced data structures (e.g., Pareto memory arrays (Khaleghzadeh et al., 2019)) and search-space pruning to manage complexity.

In continuous optimisation and machine learning:

• Semi-infinite programming techniques or bundle methods are applied to obtain and refine dual variables, especially when the constraint family is infinite (Chretien et al., 2019, Chrétien et al., 2020).
• Meta-learning frameworks tune update rules in mirror descent, parameterising geometric aspects of the dual space (Gao et al., 2022).
• Particle-based genetic algorithms simultaneously evolve update and schedule components for joint dual optimisation (Morgan et al., 10 Apr 2024).

6. Comparative Analysis, Robustness, and Practical Impact

Empirical studies consistently demonstrate the advantages of dual optimiser frameworks in stability, convergence, and practicality:

• DualOptim strategies substantially reduce worst-case variance in model parameters for machine unlearning, leading to more robust and consistent forgetting and retention (Zhong et al., 22 Apr 2025).
• Dual approaches in heterogeneous workload allocation provide continuous Pareto-fronts, illuminating non-balanced solutions that offer substantial energy/performance gains over classical load-balancing (Khaleghzadeh et al., 2019).
• Duality in embedding and hedging (e.g., Skorokhod embedding) is crucial for achieving model-independent pricing in finance, and the transfer of discrete-to-continuous optimisers is theoretically justified (Cox et al., 2017).
• The explicit error propagation formulas and stability bounds allow direct control of approximation and measurement noise effects in practical inverse problems (Chretien et al., 2019, Chrétien et al., 2020).
• In auto-generated optimiser design, joint dual evolution of update and schedule components enables the discovery of strategies that outperform standard hand-crafted algorithms across broad domains (Morgan et al., 10 Apr 2024).

7. Future Directions and Research Opportunities

The continued development and generalisation of dual optimisers is anticipated to directly impact the robustness, scalability, and transferability of optimisation algorithms:

• The modularity of dual optimiser frameworks suggests compatibility with more general constraint structures and higher-order (nonlinear) dual formulations.
• Application in infinite-dimensional and measure-valued settings will increasingly require advances in functional analysis, measure theory, and stochastic processes.
• The technical approaches for pointwise and $L^1$ convergence on domain boundaries may inform the handling of irregularities in high-dimensional transport and calibration.
• Dual optimiser hybridisation and amalgamation offer promising directions for ensemble-based learning and adaptive control in machine learning, especially as problem landscapes become more heterogeneous and nonstationary.
• Extensions of stability results—quantifying sensitivity to perturbations—will play a fundamental role in real-world deployment in privacy, decontamination, and robust learning tasks.

In sum, the theory and practice of dual optimisers provide an analytic and algorithmic backbone for a diverse range of contemporary optimisation problems, ensuring principled behaviour, tractable error analysis, and strong generalisation properties.