Constrained Optimization

A Risk-Neutral Neural Operator for Arbitrage-Free SPX-VIX Term Structures ArXiv ID: 2511.06451 “View on arXiv” Authors: Jian’an Zhang Abstract We propose ARBITER, a risk-neutral neural operator for learning joint SPX-VIX term structures under no-arbitrage constraints. ARBITER maps market states to an operator that outputs implied volatility and variance curves while enforcing static arbitrage (calendar, vertical, butterfly), Lipschitz bounds, and monotonicity. The model couples operator learning with constrained decoders and is trained with extragradient-style updates plus projection. We introduce evaluation metrics for derivatives term structures (NAS, CNAS, NI, Dual-Gap, Stability Rate) and show gains over Fourier Neural Operator, DeepONet, and state-space sequence models on historical SPX and VIX data. Ablation studies indicate that tying the SPX and VIX legs reduces Dual-Gap and improves NI, Lipschitz projection stabilizes calibration, and selective state updates improve long-horizon generalization. We provide identifiability and approximation results and describe practical recipes for arbitrage-free interpolation and extrapolation across maturities and strikes. ...

Mean Field Game of Optimal Tracking Portfolio ArXiv ID: 2505.01858 “View on arXiv” Authors: Lijun Bo, Yijie Huang, Xiang Yu Abstract This paper studies the mean field game (MFG) problem arising from a large population competition in fund management, featuring a new type of relative performance via the benchmark tracking constraint. In the n-agent model, each agent can strategically inject capital to ensure that the total wealth outperforms the benchmark process, which is modeled as a linear combination of the population’s average wealth process and a market index process. That is, each agent is concerned about the performance of her competitors captured by the floor constraint. With a continuum of agents, we formulate the constrained MFG problem and transform it into an equivalent unconstrained MFG problem with a reflected state process. We establish the existence of the mean field equilibrium (MFE) using the partial differential equation (PDE) approach. Firstly, by applying the dual transform, the best response control of the representative agent can be characterized in analytical form in terms of a dual reflected diffusion process. As a novel contribution, we verify the consistency condition of the MFE in separated domains with the help of the duality relationship and properties of the dual process. ...

Bayesian Optimization for CVaR-based portfolio optimization ArXiv ID: 2503.17737 “View on arXiv” Authors: Unknown Abstract Optimal portfolio allocation is often formulated as a constrained risk problem, where one aims to minimize a risk measure subject to some performance constraints. This paper presents new Bayesian Optimization algorithms for such constrained minimization problems, seeking to minimize the conditional value-at-risk (a computationally intensive risk measure) under a minimum expected return constraint. The proposed algorithms utilize a new acquisition function, which drives sampling towards the optimal region. Additionally, a new two-stage procedure is developed, which significantly reduces the number of evaluations of the expensive-to-evaluate objective function. The proposed algorithm’s competitive performance is demonstrated through practical examples. ...

Breaking the Dimensional Barrier for Constrained Dynamic Portfolio Choice ArXiv ID: 2501.12600 “View on arXiv” Authors: Unknown Abstract We propose a scalable, policy-centric framework for continuous-time multi-asset portfolio-consumption optimization under inequality constraints. Our method integrates neural policies with Pontryagin’s Maximum Principle (PMP) and enforces feasibility by maximizing a log-barrier-regularized Hamiltonian at each time-state pair, thereby satisfying KKT conditions without value-function grids. Theoretically, we show that the barrier-regularized Hamiltonian yields O($ε$) policy error and a linear Hamiltonian gap (quadratic when the KKT solution is interior), and we extend the BPTT-PMP correspondence to constrained settings with stable costate convergence. Empirically, PG-DPO and its projected variant (P-PGDPO) recover KKT-optimal policies in canonical short-sale and consumption-cap problems while maintaining strict feasibility across dimensions; unlike PDE/BSDE solvers, runtime scales linearly with the number of assets and remains practical at n=100. These results provide a rigorous and scalable foundation for high-dimensional constrained continuous-time portfolio optimization. ...

Optimal reinsurance and investment via stochastic projected gradient method based on Malliavin calculus ArXiv ID: 2411.05417 “View on arXiv” Authors: Unknown Abstract This paper proposes a new approach using the stochastic projected gradient method and Malliavin calculus for optimal reinsurance and investment strategies. Unlike traditional methodologies, we aim to optimize static investment and reinsurance strategies by directly minimizing the ruin probability. Furthermore, we provide a convergence analysis of the stochastic projected gradient method for general constrained optimization problems whose objective function has Hölder continuous gradient. Numerical experiments show the effectiveness of our proposed method. ...

Time-limited Metaheuristics for Cardinality-constrained Portfolio Optimisation ArXiv ID: 2307.04045 “View on arXiv” Authors: Unknown Abstract A financial portfolio contains assets that offer a return with a certain level of risk. To maximise returns or minimise risk, the portfolio must be optimised - the ideal combination of optimal quantities of assets must be found. The number of possible combinations is vast. Furthermore, to make the problem realistic, constraints can be imposed on the number of assets held in the portfolio and the maximum proportion of the portfolio that can be allocated to an asset. This problem is unsolvable using quadratic programming, which means that the optimal solution cannot be calculated. A group of algorithms, called metaheuristics, can find near-optimal solutions in a practical computing time. These algorithms have been successfully used in constrained portfolio optimisation. However, in past studies the computation time of metaheuristics is not limited, which means that the results differ in both quality and computation time, and cannot be easily compared. This study proposes a different way of testing metaheuristics, limiting their computation time to a certain duration, yielding results that differ only in quality. Given that in some use cases the priority is the quality of the solution and in others the speed, time limits of 1, 5 and 25 seconds were tested. Three metaheuristics - simulated annealing, tabu search, and genetic algorithm - were evaluated on five sets of historical market data with different numbers of assets. Although the metaheuristics could not find a competitive solution in 1 second, simulated annealing found a near-optimal solution in 5 seconds in all but one dataset. The lowest quality solutions were obtained by genetic algorithm. ...