A Graphical Global Optimization Framework for Parameter Estimation of Statistical Models with Nonconvex Regularization Functions

Optimization problems with norm-bounding constraints arise in a variety of applications, including portfolio optimization, machine learning, and feature selection. A common approach to these problems involves relaxing the norm constraint via Lagrangian relaxation, transforming it into a regularization term in the objective function. A particularly challenging class includes the zero-norm function, which promotes sparsity in statistical parameter estimation. Most existing exact methods for solving these problems introduce binary variables and artificial bounds to reformulate them as higher-dimensional mixed-integer programs, solvable by standard solvers. Other exact approaches exploit specific structural properties of the objective, making them difficult to generalize across different problem types. Alternative methods employ nonconvex penalties with favorable statistical characteristics, but these are typically addressed using heuristic or local optimization techniques due to their structural complexity. In this paper, we propose a novel graph-based method to globally solve optimization problems involving generalized norm-bounding constraints. Our approach encompasses standard $\ell_p$-norms for $p \in [0, \infty)$ and nonconvex penalties such as SCAD and MCP. We leverage decision diagrams to construct strong convex relaxations directly in the original variable space, eliminating the need for auxiliary variables or artificial bounds. Integrated into a spatial branch-and-cut framework, our method guarantees convergence to the global optimum. We demonstrate its effectiveness through preliminary computational experiments on benchmark sparse linear regression problems involving complex nonconvex penalties, which are not tractable using existing global optimization techniques.

Latent Variable Estimation in Bayesian Black-Litterman Models

We revisit the Bayesian Black-Litterman (BL) portfolio model and remove its reliance on subjective investor views. Classical BL requires an investor "view": a forecast vector $q$ and its uncertainty matrix $\Omega$ that describe how much a chosen portfolio should outperform the market. Our key idea is to treat $(q,\Omega)$ as latent variables and learn them from market data within a single Bayesian network. Consequently, the resulting posterior estimation admits closed-form expression, enabling fast inference and stable portfolio weights. Building on these, we propose two mechanisms to capture how features interact with returns: shared-latent parametrization and feature-influenced views; both recover classical BL and Markowitz portfolios as special cases. Empirically, on 30-year Dow-Jones and 20-year sector-ETF data, we improve Sharpe ratios by 50% and cut turnover by 55% relative to Markowitz and the index baselines. This work turns BL into a fully data-driven, view-free, and coherent Bayesian framework for portfolio optimization.

Integrating LLM-Generated Views into Mean-Variance Optimization Using the Black-Litterman Model

Portfolio optimization faces challenges due to the sensitivity in traditional mean-variance models. The Black-Litterman model mitigates this by integrating investor views, but defining these views remains difficult. This study explores the integration of large language models (LLMs) generated views into portfolio optimization using the Black-Litterman framework. Our method leverages LLMs to estimate expected stock returns from historical prices and company metadata, incorporating uncertainty through the variance in predictions. We conduct a backtest of the LLM-optimized portfolios from June 2024 to February 2025, rebalancing biweekly using the previous two weeks of price data. As baselines, we compare against the S&P 500, an equal-weighted portfolio, and a traditional mean-variance optimized portfolio constructed using the same set of stocks. Empirical results suggest that different LLMs exhibit varying levels of predictive optimism and confidence stability, which impact portfolio performance. The source code and data are available at https://github.com/youngandbin/LLM-MVO-BLM.

Deep Learning Models Meet Financial Data Modalities

Algorithmic trading relies on extracting meaningful signals from diverse financial data sources, including candlestick charts, order statistics on put and canceled orders, traded volume data, limit order books, and news flow. While deep learning has demonstrated remarkable success in processing unstructured data and has significantly advanced natural language processing, its application to structured financial data remains an ongoing challenge. This study investigates the integration of deep learning models with financial data modalities, aiming to enhance predictive performance in trading strategies and portfolio optimization. We present a novel approach to incorporating limit order book analysis into algorithmic trading by developing embedding techniques and treating sequential limit order book snapshots as distinct input channels in an image-based representation. Our methodology for processing limit order book data achieves state-of-the-art performance in high-frequency trading algorithms, underscoring the effectiveness of deep learning in financial applications.

Risk-aware black-box portfolio construction using Bayesian optimization with adaptive weighted Lagrangian estimator

Existing portfolio management approaches are often black-box models due to safety and commercial issues in the industry. However, their performance can vary considerably whenever market conditions or internal trading strategies change. Furthermore, evaluating these non-transparent systems is expensive, where certain budgets limit observations of the systems. Therefore, optimizing performance while controlling the potential risk of these financial systems has become a critical challenge. This work presents a novel Bayesian optimization framework to optimize black-box portfolio management models under limited observations. In conventional Bayesian optimization settings, the objective function is to maximize the expectation of performance metrics. However, simply maximizing performance expectations leads to erratic optimization trajectories, which exacerbate risk accumulation in portfolio management. Meanwhile, this can lead to misalignment between the target distribution and the actual distribution of the black-box model. To mitigate this problem, we propose an adaptive weight Lagrangian estimator considering dual objective, which incorporates maximizing model performance and minimizing variance of model observations. Extensive experiments demonstrate the superiority of our approach over five backtest settings with three black-box stock portfolio management models. Ablation studies further verify the effectiveness of the proposed estimator.

Factor-MCLS: Multi-agent learning system with reward factor matrix and multi-critic framework for dynamic portfolio optimization

Typical deep reinforcement learning (DRL) agents for dynamic portfolio optimization learn the factors influencing portfolio return and risk by analyzing the output values of the reward function while adjusting portfolio weights within the training environment. However, it faces a major limitation where it is difficult for investors to intervene in the training based on different levels of risk aversion towards each portfolio asset. This difficulty arises from another limitation: existing DRL agents may not develop a thorough understanding of the factors responsible for the portfolio return and risk by only learning from the output of the reward function. As a result, the strategy for determining the target portfolio weights is entirely dependent on the DRL agents themselves. To address these limitations, we propose a reward factor matrix for elucidating the return and risk of each asset in the portfolio. Additionally, we propose a novel learning system named Factor-MCLS using a multi-critic framework that facilitates learning of the reward factor matrix. In this way, our DRL-based learning system can effectively learn the factors influencing portfolio return and risk. Moreover, based on the critic networks within the multi-critic framework, we develop a risk constraint term in the training objective function of the policy function. This risk constraint term allows investors to intervene in the training of the DRL agent according to their individual levels of risk aversion towards the portfolio assets.

Diffusion Factor Models: Generating High-Dimensional Returns with Factor Structure

Financial scenario simulation is essential for risk management and portfolio optimization, yet it remains challenging especially in high-dimensional and small data settings common in finance. We propose a diffusion factor model that integrates latent factor structure into generative diffusion processes, bridging econometrics with modern generative AI to address the challenges of the curse of dimensionality and data scarcity in financial simulation. By exploiting the low-dimensional factor structure inherent in asset returns, we decompose the score function--a key component in diffusion models--using time-varying orthogonal projections, and this decomposition is incorporated into the design of neural network architectures. We derive rigorous statistical guarantees, establishing nonasymptotic error bounds for both score estimation at O(d^{5/2} n^{-2/(k+5)}) and generated distribution at O(d^{5/4} n^{-1/2(k+5)}), primarily driven by the intrinsic factor dimension k rather than the number of assets d, surpassing the dimension-dependent limits in the classical nonparametric statistics literature and making the framework viable for markets with thousands of assets. Numerical studies confirm superior performance in latent subspace recovery under small data regimes. Empirical analysis demonstrates the economic significance of our framework in constructing mean-variance optimal portfolios and factor portfolios. This work presents the first theoretical integration of factor structure with diffusion models, offering a principled approach for high-dimensional financial simulation with limited data.

Greedy Restart Schedules: A Baseline for Dynamic Algorithm Selection on Numerical Black-box Optimization Problems

In many optimization domains, there are multiple different solvers that contribute to the overall state-of-the-art, each performing better on some, and worse on other types of problem instances. Meta-algorithmic approaches, such as instance-based algorithm selection, configuration and scheduling, aim to close this gap by extracting the most performance possible from a set of (configurable) optimizers. In this context, the best performing individual algorithms are often hand-crafted hybrid heuristics which perform many restarts of fast local optimization approaches. However, data-driven techniques to create optimized restart schedules have not yet been extensively studied. Here, we present a simple scheduling approach that iteratively selects the algorithm performing best on the distribution of unsolved training problems at time of selection, resulting in a problem-independent solver schedule. We demonstrate our approach using well-known optimizers from numerical black-box optimization on the BBOB testbed, bridging much of the gap between single and virtual best solver from the original portfolio across various evaluation protocols. Our greedy restart schedule presents a powerful baseline for more complex dynamic algorithm selection models.

Why risk matters for protein binder design

Bayesian optimization (BO) has recently become more prevalent in protein engineering applications and hence has become a fruitful target of benchmarks. However, current BO comparisons often overlook real-world considerations like risk and cost constraints. In this work, we compare 72 model combinations of encodings, surrogate models, and acquisition functions on 11 protein binder fitness landscapes, specifically from this perspective. Drawing from the portfolio optimization literature, we adopt metrics to quantify the cold-start performance relative to a random baseline, to assess the risk of an optimization campaign, and to calculate the overall budget required to reach a fitness threshold. Our results suggest the existence of Pareto-optimal models on the risk-performance axis, the shift of this preference depending on the landscape explored, and the robust correlation between landscape properties such as epistasis with the average and worst-case model performance. They also highlight that rigorous model selection requires substantial computational and statistical efforts.

Semi-Decision-Focused Learning with Deep Ensembles: A Practical Framework for Robust Portfolio Optimization

I propose Semi-Decision-Focused Learning, a practical adaptation of Decision-Focused Learning for portfolio optimization. Rather than directly optimizing complex financial metrics, I employ simple target portfolios (Max-Sortino or One-Hot) and train models with a convex, cross-entropy loss. I further incorporate Deep Ensemble methods to reduce variance and stabilize performance. Experiments on two universes (one upward-trending and another range-bound) show consistent outperformance over baseline portfolios, demonstrating the effectiveness and robustness of my approach. Code is available at https://github.com/sDFLwDE/sDFLwDE