We consider the problem of minimizing a smooth and convex function over the $n$-dimensional spectrahedron -- the set of real symmetric $n\times n$ positive semidefinite matrices with unit trace, which underlies numerous applications in statistics, machine learning and additional domains. Standard first-order methods often require high-rank matrix computations which are prohibitive when the dimension $n$ is large. The well-known Frank-Wolfe method on the other hand, only requires efficient rank-one matrix computations, however suffers from worst-case slow convergence, even under conditions that enable linear convergence rates for standard methods. In this work we present the first Frank-Wolfe-based algorithm that only applies efficient rank-one matrix computations and, assuming quadratic growth and strict complementarity conditions, is guaranteed, after a finite number of iterations, to converges linearly, in expectation, and independently of the ambient dimension.

Factor graph optimization serves as a fundamental framework for robotic perception, enabling applications such as pose estimation, simultaneous localization and mapping (SLAM), structure-from-motion (SfM), and situational awareness. Traditionally, these methods solve unconstrained least squares problems using algorithms such as Gauss-Newton and Levenberg-Marquardt. However, extending factor graphs with native support for equality constraints can improve solution accuracy and broaden their applicability, particularly in optimal control. In this paper, we propose a novel extension of factor graphs that seamlessly incorporates equality constraints without requiring additional optimization algorithms. Our approach maintains the efficiency and flexibility of existing second-order optimization techniques while ensuring constraint feasibility. To validate our method, we apply it to an optimal control problem for velocity tracking in autonomous vehicles and benchmark our results against state-of-the-art constraint handling techniques. Additionally, we introduce ecg2o, a header-only C++ library that extends the widely used g2o factor graph library by adding full support for equality-constrained optimization. This library, along with demonstrative examples and the optimal control problem, is available as open source at https://github.com/snt-arg/ecg2o

The alignment of large language models with human values presents a critical challenge, particularly when balancing conflicting objectives like helpfulness and harmlessness. Existing approaches, such as Reinforcement Learning from Human Feedback (RLHF) and Direct Preference Optimization (DPO), face notable limitations: RLHF suffers from instability and inefficiency in multi-objective optimization, while DPO lacks mechanisms for dynamic trade-offs. To address these challenges, we propose Post-Training Extrapolation Optimization (PEO), a novel and efficient framework for bi-factorial alignment. PEO generates a family of Pareto-optimal policies in a single training pass by leveraging a three-phase pipeline: (1) aspect-specific learning, (2) generalist initialization via interpolation, and (3) post-training optimization via extrapolation. PEO enables dynamic adaptation to diverse user preferences at inference time without retraining. Our comprehensive experiments across multiple LLMs demonstrate that PEO achieves superior Pareto fronts compared to baselines, offering improved flexibility and computational efficiency. Theoretical analyses further highlight PEO's capacity to overcome optimization bottlenecks, paving the way for scalable, personalized alignment.

We study the problem of estimating Dynamic Discrete Choice (DDC) models, also known as offline Maximum Entropy-Regularized Inverse Reinforcement Learning (offline MaxEnt-IRL) in machine learning. The objective is to recover reward or $Q^*$ functions that govern agent behavior from offline behavior data. In this paper, we propose a globally convergent gradient-based method for solving these problems without the restrictive assumption of linearly parameterized rewards. The novelty of our approach lies in introducing the Empirical Risk Minimization (ERM) based IRL/DDC framework, which circumvents the need for explicit state transition probability estimation in the Bellman equation. Furthermore, our method is compatible with non-parametric estimation techniques such as neural networks. Therefore, the proposed method has the potential to be scaled to high-dimensional, infinite state spaces. A key theoretical insight underlying our approach is that the Bellman residual satisfies the Polyak-Lojasiewicz (PL) condition -- a property that, while weaker than strong convexity, is sufficient to ensure fast global convergence guarantees. Through a series of synthetic experiments, we demonstrate that our approach consistently outperforms benchmark methods and state-of-the-art alternatives.

To date, the multi-objective optimization literature has mainly focused on conflicting objectives, studying the Pareto front, or requiring users to balance tradeoffs. Yet, in machine learning practice, there are many scenarios where such conflict does not take place. Recent findings from multi-task learning, reinforcement learning, and LLMs training show that diverse related tasks can enhance performance across objectives simultaneously. Despite this evidence, such phenomenon has not been examined from an optimization perspective. This leads to a lack of generic gradient-based methods that can scale to scenarios with a large number of related objectives. To address this gap, we introduce the Aligned Multi-Objective Optimization framework, propose new algorithms for this setting, and provide theoretical guarantees of their superior performance compared to naive approaches.

Time series foundation models provide a universal solution for generating forecasts to support optimization problems in energy systems. Those foundation models are typically trained in a prediction-focused manner to maximize forecast quality. In contrast, decision-focused learning directly improves the resulting value of the forecast in downstream optimization rather than merely maximizing forecasting quality. The practical integration of forecast values into forecasting models is challenging, particularly when addressing complex applications with diverse instances, such as buildings. This becomes even more complicated when instances possess specific characteristics that require instance-specific, tailored predictions to increase the forecast value. To tackle this challenge, we use decision-focused fine-tuning within time series foundation models to offer a scalable and efficient solution for decision-focused learning applied to the dispatchable feeder optimization problem. To obtain more robust predictions for scarce building data, we use Moirai as a state-of-the-art foundation model, which offers robust and generalized results with few-shot parameter-efficient fine-tuning. Comparing the decision-focused fine-tuned Moirai with a state-of-the-art classical prediction-focused fine-tuning Morai, we observe an improvement of 9.45% in average total daily costs.

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Recent research into solving vehicle routing problems (VRPs) has gained significant traction, particularly through the application of deep (reinforcement) learning for end-to-end solution construction. However, many current construction-based neural solvers predominantly utilize Transformer architectures, which can face scalability challenges and struggle to produce diverse solutions. To address these limitations, we introduce a novel framework beyond Transformer-based approaches, i.e., Adversarial Generative Flow Networks (AGFN). These models are trained alternately in an adversarial manner to improve the overall solution quality, followed by a proposed hybrid decoding method to construct the solution. We apply the AGFN framework to solve the capacitated vehicle routing problem (CVRP) and the travelling salesman problem (TSP), and our experimental results demonstrate that AGFN surpasses the popular construction-based neural solvers, showcasing strong generalization capabilities on synthetic and real-world benchmark instances. Our code is available at https://github.com/ZHANG-NI/AGFN . The vehicle routing problem (VRP) represents a fundamental and intricate combinatorial optimization challenge with extensive real-world implications (Toth & Vigo, 2014), including supply chain management (Lee et al., 2006), last-mile delivery services (Koc et al., 2020), and public transportation (Hassold & Ceder, 2014). Given its widespread occurrence across numerous domains, the VRPs have been the subject of extensive research for decades within the Operations Research (OR) community. Particularly, practitioners employ both exact and heuristic methods to tackle complex optimization problems including VRPs. Exact methods, such as branch-and-bound (Lawler & Wood, 1966), branch-and-cut (Tawarmalani & Sahinidis, 2005), and column generation (Barnhart et al., 1998), guarantee optimal solutions but often face computational limitations for large-scale instances.

Leveraging machine learning (ML) to predict an initial solution for mixed-integer linear programming (MILP) has gained considerable popularity in recent years. These methods predict a solution and fix a subset of variables to reduce the problem dimension. Then, they solve the reduced problem to obtain the final solutions. However, directly fixing variable values can lead to low-quality solutions or even infeasible reduced problems if the predicted solution is not accurate enough. To address this challenge, we propose an A lternating p redictio n-correction neural sol ving framewo rk (Apollo-MILP) that can identify and select accurate and reliable predicted values to fix. In each iteration, Apollo-MILP conducts a prediction step for the unfixed variables, followed by a correction step to obtain an improved solution (called reference solution) through a trust-region search. By incorporating the predicted and reference solutions, we introduce a novel U ncertainty-based E rror upper BO und (UEBO) to evaluate the uncertainty of the predicted values and fix those with high confidence. A notable feature of Apollo-MILP is the superior ability for problem reduction while preserving optimality, leading to high-quality final solutions. Experiments on commonly used benchmarks demonstrate that our proposed Apollo-MILP significantly outperforms other ML-based approaches in terms of solution quality, achieving over a 50% reduction in the solution gap. Mixed-integer linear programming (MILP) is one of the most fundamental models for combinatorial optimization with broad applications in operations research (Bixby et al., 2004), engineering (Ma et al., 2019), and daily scheduling or planning (Li et al., 2024b). However, solving large-size MILPs remains time-consuming and computationally expensive, as many are NP-hard and have exponential expansion of search spaces as instance sizes grow. To mitigate this challenge, researchers have explored a wide suite of machine learning (ML) methods (Gasse et al., 2022). In practice, MILP instances from the same scenario often share similar patterns and structures, which ML models can capture to achieve improved performance (Bengio et al., 2021). Recently, extensive research has focused on using ML models to predict solutions for MILPs. Notable approaches include Neural Diving (ND) (Nair et al., 2020; Y oon, 2021; Paulus & Krause, 2023) and Predict-and-Search (PS) (Han et al., 2023; Huang et al., 2024), as illustrated in Figure 1. Given a MILP instance, ND and PS begin by employing an ML model to predict an initial solution. ND with SelectiveNet (Nair et al., 2020) assigns fixed values to a subset of variables based on the prediction, thereby constructing a reduced MILP problem with a reduced dimensionality of decision variables. Then, ND solves the reduced problem to obtain the final solutions.

This paper investigates the optimization of the Vehicle Routing Problem for Security Dispatch (VRPSD). VRPSD focuses on security and patrolling applications which involve challenging constraints including precise timing and strict time windows. We propose three algorithms based on different metaheuristics, which are Adaptive Large Neighborhood Search (ALNS), Tabu Search (TS), and Threshold Accepting (TA). The first algorithm combines single-phase ALNS with TA, the second employs a multiphase ALNS with TA, and the third integrates multiphase ALNS, TS, and TA. Experiments are conducted on an instance comprising 251 customer requests. The results demonstrate that the third algorithm, the hybrid multiphase ALNS-TS-TA algorithm, delivers the best performance. This approach simultaneously leverages the large-area search capabilities of ALNS for exploration and effectively escapes local optima when the multiphase ALNS is coupled with TS and TA. Furthermore, in our experiments, the hybrid multiphase ALNS-TS-TA algorithm is the only one that shows potential for improving results with increased computation time across all attempts.