In these processes, an evaluator estimates an individual's value to an institution.

We develop a theoretically-grounded learning method for the Generalized Linear Programming Value Function (GVF), which models the optimal value of a linear programming (LP) problem as its objective and constraint bounds vary. This function plays a fundamental role in algorithmic techniques for large-scale optimization, particularly in decomposition for two-stage mixed-integer linear programs (MILPs). This paper establishes a structural characterization of the GVF that enables it to be modeled as a particular neural network architecture, which we then use to learn the GVF in a way that benefits from three notable properties. First, our method produces a true under-approximation of the value function with respect to the constraint bounds. Second, the model is input-convex in the constraint bounds, which not only matches the structure of the GVF but also enables the trained model to be efficiently optimized over using LP. Finally, our learning method is unsupervised, meaning that training data generation does not require computing LP optimal values, which can be prohibitively expensive at large scales. We numerically show that our method can approximate the GVF well, even when compared to supervised methods that collect training data by solving an LP for each data point. Furthermore, as an application of our framework, we develop a fast heuristic method for large-scale two-stage MILPs with continuous second-stage variables, via a compact reformulation that can be solved faster than the full model linear relaxation at large scales and orders of magnitude faster than the original model.

Hierarchical clustering has usually been addressed by discrete optimization using heuristics or continuous optimization of relaxed scores for hierarchies. In this work, we propose to optimize expected scores under a probabilistic model over hierarchies.

Ranking methods or models based on their performance is of prime importance but is tricky because performance is fundamentally multidimensional. In the case of classification, precision and recall are scores with probabilistic interpretations that are both important to consider and complementary. The rankings induced by these two scores are often in partial contradiction. In practice, therefore, it is extremely useful to establish a compromise between the two views to obtain a single, global ranking. Over the last fifty years or so,it has been proposed to take a weighted harmonic mean, known as the F-score, F-measure, or $F_β$. Generally speaking, by averaging basic scores, we obtain a score that is intermediate in terms of values. However, there is no guarantee that these scores lead to meaningful rankings and no guarantee that the rankings are good tradeoffs between these base scores. Given the ubiquity of $F_β$ scores in the literature, some clarification is in order. Concretely: (1) We establish that $F_β$-induced rankings are meaningful and define a shortest path between precision- and recall-induced rankings. (2) We frame the problem of finding a tradeoff between two scores as an optimization problem expressed with Kendall rank correlations. We show that $F_1$ and its skew-insensitive version are far from being optimal in that regard. (3) We provide theoretical tools and a closed-form expression to find the optimal value for $β$ for any distribution or set of performances, and we illustrate their use on six case studies.

Federated learning (FL) and split learning (SL) are two effective distributed learning paradigms in wireless networks, enabling collaborative model training across mobile devices without sharing raw data. While FL supports low-latency parallel training, it may converge to less accurate model. In contrast, SL achieves higher accuracy through sequential training but suffers from increased delay. To leverage the advantages of both, hybrid split and federated learning (HSFL) allows some devices to operate in FL mode and others in SL mode. This paper aims to accelerate HSFL by addressing three key questions: 1) How does learning mode selection affect overall learning performance? 2) How does it interact with batch size? 3) How can these hyperparameters be jointly optimized alongside communication and computational resources to reduce overall learning delay? We first analyze convergence, revealing the interplay between learning mode and batch size. Next, we formulate a delay minimization problem and propose a two-stage solution: a block coordinate descent method for a relaxed problem to obtain a locally optimal solution, followed by a rounding algorithm to recover integer batch sizes with near-optimal performance. Experimental results demonstrate that our approach significantly accelerates convergence to the target accuracy compared to existing methods.

A sampling-based optimization method for quadratic functions is proposed.

Empirical risk minimization (ERM) is ubiquitous in machine learning and underlies most supervised learning methods.

Given demand models of multiple products, the role of optimization is to find the best price strategy.

Precise expressions for the asymptotic performance of LASSO have been obtained for a number of different cases, in particular when the elements of the dictionary matrix are sampled independently from a Gaussian distribution.