Solving equilibrium problems in economies with financial markets, home production, and retention

We propose a new methodology to compute equilibria for general equilibrium problems on exchange economies with real financial markets, home-production, and retention. We demonstrate that equilibrium prices can be determined by solving a related maxinf-optimization problem. We incorporate the non-arbitrage condition for financial markets into the equilibrium formulation and establish the equivalence between solutions to both problems. This reduces the complexity of the original by eliminating the need to directly compute financial contract prices, allowing us to calculate equilibria even in cases of incomplete financial markets. We also introduce a Walrasian bifunction that captures the imbalances and show that maxinf-points of this function correspond to equilibrium points. Moreover, we demonstrate that every equilibrium point can be approximated by a limit of maxinf points for a family of perturbed problems, by relying on the notion of lopsided convergence. Finally, we propose an augmented Walrasian algorithm and present numerical examples to illustrate the effectiveness of this approach. Our methodology allows for efficient calculation of equilibria in a variety of exchange economies and has potential applications in finance and economics

An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure

We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common theory, leading to novel methods for working with several important models of interest in statistics, machine learning and computer vision. In Part 1, we review the concepts of network flows and submodular function optimization theory foundational to our results. We then examine the connections between network flows and the minimum-norm algorithm from submodular optimization, extending and improving several current results. This leads to a concise representation of the structure of a large class of pairwise regularized models important in machine learning, statistics and computer vision. In Part 2, we describe the full regularization path of a class of penalized regression problems with dependent variables that includes the graph-guided LASSO and total variation constrained models. This description also motivates a practical algorithm. This allows us to efficiently find the regularization path of the discretized version of TV penalized models. Ultimately, our new algorithms scale up to high-dimensional problems with millions of variables