This paper reviews recent advances in the field of optimization under uncertainty via a modern data lens, highlights key research challenges and promise of data-driven optimization that organically integrates machine learning and mathematical programming for decision-making under uncertainty, and identifies potential r…
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48 results for “Mathematical Optimization”
Paper proposes a math-inspired L2O model for better generalization.
problem Overfitting and poor generalization of generic L2O models.
method Derived mathematical conditions for successful update rules and proposed a novel L2O model.
result Proposed model outperforms generic L2O models in out-of-distribution settings.
The optimal binning is the optimal discretization of a variable into bins given a discrete or continuous numeric target. We present a rigorous and extensible mathematical programming formulation for solving the optimal binning problem for a binary, continuous and multi-class target type, incorporating constraints not p…
Mathematical framework for language models processes text and predicts next tokens.
problem Understanding and optimizing the performance of large language models.
method Describes encoding, prediction models, learning from data, and deployment of LLMs.
result Demonstrates remarkable empirical successes and provides a platform for further research.
Abstract reviews mathematical fairness in machine learning.
problem Ensuring fairness in machine learning models.
method Independence-based approach to fairness definitions and methodologies.
result Optimal fair classifiers and predictors under equality of odds are derived.
Examines challenges and proposes new approaches in machine learning theory.
problem Challenges in machine learning as a function approximation and optimization.
method Mathematical analysis of gradient descent, fixed network limitations, and RNNs.
result New insights and mathematical approaches to improve machine learning.
Proves a tree of shapes for n-D images in optimal time.
problem Proving the mathematical structure of tree of shapes in n-D images.
method Optimal quasi-linear time algorithm for computing the tree of shapes.
result The tree of shapes is the self-dual morphological hierarchical structure of n-D gray-level images.
This study optimizes crypto-market trading conditions without assuming convexity.
problem Optimizing crypto-market trading conditions without convexity.
method Rigorous mathematical analysis of constant function market makers under quasilinear trade functions.
result Quasilinear trade functions can replicate convex functions' robustness against arbitrage.
Lecture Notes on Linear Neural Networks: A Tale of Optimization and Generalization in Deep Learningcs.LG
Lecture notes on linear neural networks for deep learning optimization and generalization.
problem Understanding optimization and generalization in deep learning models.
method Mathematical tools and dynamical systems theory.
result Potential of mathematical tools to enhance understanding of deep learning.
We propose a simple mathematical model for unemployment. Despite its simpleness, we claim that the model is more realistic and useful than recent models available in the literature. A case study with real data from Portugal supports our claim. An optimal control problem is formulated and solved, which provides some non…
Deep learning models are growing, posing new mathematical challenges.
problem Mathematical challenges in training, inference, generalization, and optimization of deep models.
method Formal mathematical analysis and communication with mathematicians, statisticians, and computer scientists.
result A set of new mathematical challenges in deep learning.
RL methods applied to option pricing using modified QLBS and RLOP models.
problem Applying reinforcement learning to price options accurately.
method Developed modified QLBS and RLOP models, implemented RL learning algorithm with neural networks.
result Optimal hedging strategies learned by RL outperform baseline models.
Calculation of an optimal tariff is a principal challenge for pricing actuaries. In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff. Our motivation comes from two important actuarial tasks, namely a) construction …
Exact Matrix Seriation through Mathematical Optimization: Stress and Effectiveness-Based Modelsmath.OC
Unified optimization framework for matrix seriation.
problem Discovering latent structure in relational data.
method Mathematical optimization models for seriation.
result Optimization models enhance solution quality and interpretability.
Survey of mathematical foundations for reinforcement learning.
problem Design and analysis of modern reinforcement learning algorithms.
method Organizes mathematical structures from probability, optimization, and operator theory.
result Unified mathematical entry point for researchers in various fields.
Dynamic Pricing for Real Estateq-fin.MF
Optimizes real estate prices with dynamic strategies.
problem Optimizing prices for limited real estate goods over time.
method Develops a mathematical model considering variable demand, time value, and growth of real estate value.
result Enhanced model for better revenue management in real estate.
Survey revisits Bachelier and Dupire, highlighting optimal transport's role.
problem Finding arbitrage-free models calibrated to volatility surfaces.
method Revisits mathematical finance principles, uses optimal transport results.
result Optimal transport provides rigorous foundations for Dupire's model.
Book introduces deep learning methods with math, theory, and applications.
problem Understanding deep learning algorithms and their mathematical foundations.
method Reviews various ANN architectures and optimization methods, covers theoretical aspects.
result Provides a solid mathematical foundation for deep learning.
Optimizes leveraged staking strategies in decentralized finance.
problem Maximizing returns on staked assets in decentralized lending platforms.
method Developed a mathematical framework to optimize leveraged staking strategies, reducing the multi-market problem to convex allocation over market exposures.
result Rebalanced leveraged positions can achieve up to 6.2% APY, significantly higher than unleveraged staking.
New method tackles inexact bilevel optimization for faster parameter learning.
problem Nested optimization problems in bilevel learning with computationally difficult exact solutions.
method Inexact derivative-free optimization algorithms for approximate lower-level solutions.
result Global convergence and worst-case complexity for the proposed approach.
Mathematical analysis shows Delisle-Euler map methods are optimal.
problem Comparing ancient and modern map drawing methods.
method Analyzing similarities and differences between ancient and modern map drawing methods.
result Delisle-Euler map methods are optimal among conical maps.
This paper studies dynamic stochastic optimization problems parametrized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of solutions and the absence of a duality gap. Our proof uses extended dynamic programm…
With the recent rise of Machine Learning as a candidate to partially replace classic Financial Mathematics methodologies, we investigate the performances of both in solving the problem of dynamic portfolio optimization in continuous-time, finite-horizon setting for a portfolio of two assets that are intertwined. In Fin…
Optimizes group testing for COVID-19 to reduce test numbers.
problem Minimizing tests for accurate infection detection.
method Bayesian approach with genetic algorithms and sub-modularity.
result Greedy-adaptive method provides theoretical guarantees.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.
The main purpose of this paper is to formalize the modelling process, analysis and mathematical definition of corruption when entering into a contract between principal agent and producers. The formulation of the problem and the definition of concepts for the general case are considered. For definiteness, all calculati…
Language models help text classification tasks by predicting next words.
problem Lack of theoretical understanding of why language models perform well on downstream tasks.
method Mathematical study of the connection between next word prediction and text classification, formalizing it and quantifying the benefit.
result Language models that are ε-optimal in cross-entropy learn features that can solve classification tasks with linear approximation.
Analog method solves portfolio optimization problems faster and more efficiently.
problem Accurate covariance matrix estimation and fast optimal portfolio selection for financial applications.
method Two-step process using equilibrium propagation and analog Hopfield networks.
result Fully analog pipeline calculates optimal portfolios in energy-efficient manner.
We solve the problem of optimal stopping of a Brownian motion subject to the constraint that the stopping time's distribution is a given measure consisting of finitely-many atoms. In particular, we show that this problem can be converted to a finite sequence of state-constrained optimal control problems with additional…
Mathematical framework for transfer learning feasibility and transfer risk.
problem Theoretical analysis of transfer learning.
method Reformulated transfer learning as an optimization problem, introduced transfer risk concept.
result Demonstrated the potential and benefits of incorporating transfer risk in transfer learning evaluation.
A new optimizer, Grad-Avg, converges faster than SGD and improves classification task performance.
problem Optimizing error functions in regression and classification tasks.
method Grad-Avg optimizer based on gradient averaging, with mathematical convergence proof and parameter scaling for classification.
result Grad-Avg converges faster than other optimizers for classification tasks on benchmark datasets.
Mathematical framework for differential machine learning in finance.
problem Theoretical assumptions in financial models and their impact on machine learning algorithms.
method Rigorous mathematical framework for differential machine learning in finance.
result Theoretical grounding enhances the predictive capabilities of neural networks in financial applications.
We consider an infinite dimensional optimization problem motivated by mathematical economics. Within the celebrated "Arbitrage Pricing Model", we use probabilistic and functional analytic techniques to show the existence of optimal strategies for investors who maximize their expected utility.
Mathematical models help keep vaccine prices low.
problem Pricing COVID-19 vaccines to ensure affordability and profitability.
method Optimization and game theory approaches modeling a duopoly market.
result Government can negotiate low prices while manufacturers earn profits.
Develops a mathematical model for CLMM dynamics in DeFi.
problem Analyzing CLMMs in continuous time trading.
method Modeling CLMM dynamics as measure-valued processes, examining three arbitrage models.
result Trading fees limit admissible price processes, impacting CLMM design.
Investor skill levels affect optimal portfolio size, study shows.
problem Optimal portfolio size for different skill levels of investors.
method Mathematical methods to study annual and continuous portfolio diversification, regression analysis.
result Strong investors should hold concentrated portfolios, poor investors should hold diversified portfolios.
Notes on generative modeling: flow matching, diffusion, optimal transport and Schr{ö}dinger bridgestat.ML
Abstract notes on generative modeling techniques.
problem Improving generative modeling techniques.
method Connections between optimal transport and Schrödinger bridge, flow matching.
result Showed connections between mathematical principles and generative modeling techniques.
MFGs explain and enhance generative models, revealing new model types.
problem Understanding and improving generative models.
method Mean-field games (MFGs) as a framework to explain and enhance generative models.
result Established connections between MFGs and generative flows, diffusions, and gradient flows.
A discrete time probabilistic model, for optimal equity allocation and portfolio selection, is formulated so as to apply to (at least) reinsurance. In the context of a company with several portfolios (or subsidiaries), representing both liabilities and assets, it is proved that the model has solutions respecting constr…
A new method solves complex financial problems using deep learning.
problem Optimal stopping and option pricing in finance.
method Compound BSDE method, based on reformulating BSDEs.
result The method offers accurate and efficient solutions for high-dimensional problems.
New risk measures improve portfolio diversification and stability.
problem Concentration risk in traditional portfolio optimization methods.
method Equal-correlation portfolio strategy with mathematical optimization.
result Improved risk diversification and stable returns.
Paper develops a new method for solving complex problems in generative modeling and mean-field games.
problem Solving complex problems in generative modeling and mean-field games.
method Reinterpreting Generalized Schrödinger Bridges (GSBs) as probabilistic models and using the nonlinear Feynman-Kac lemma.
result Demonstrates the efficacy of the new method in generative modeling and mean-field games.
Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus, approximation theory, optimization and linear algebra. This article provides a very…
Semi-static trading strategies make frequent appearances in mathematical finance, where dynamic trading in a liquid asset is combined with static buy-and-hold positions in options on that asset. We show that the space of outcomes of such strategies can have very poor closure properties when all European options for a f…
Mathematical analysis of SNE and t-SNE for dimension reduction.
problem Optimal mapping of high-dimensional data to low dimensions.
method Gradient flow of relative entropy to minimize the distance between points.
result The diameter of the evolving sets remains bounded for SNE but may blow up for t-SNE.
Georg de Buquoy, Lord de Vaux, lived in Nove Hrady, Prague and Cerveny Hradek for most of his productive life. From his extensive scientific contributions, both theoretical and experimental, we expand here the discussion of his contributions to mathematical economy. He is mainly celebrated as the first persons to defin…
Mathematical optimization is widely used in various research fields. With a carefully-designed objective function, mathematical optimization can be quite helpful in solving many problems. However, objective functions are usually hand-crafted and designing a good one can be quite challenging. In this paper, we propose a…
Paper relaxes optimal transport using convex functions for data science.
problem Optimal transport problem on finite spaces.
method Relaxation via strictly convex functions (Kullback-Leibler divergence, Bregman divergences). Gradient descent iterative process.
result Mathematical foundations and iterative process for the relaxed optimal transport problem.