arXiv subjects

Jourdain Lamperski

Publications and source records attributed to Jourdain Lamperski.

4 recordsLinked to original sources

arXiv2026

Unit-Demand Pricing for Min-Buying Customers

We study the unit-demand pricing problem under \textit{min-buying} behavior, where customers purchase the lowest-priced item that yields non-negative utility. In contrast to the quasi-linear model, we show that uniform pricing becomes substantially more effective under min-buying. First, we study symmetric environments. We show that for min-buying customers with independent and identically distributed valuations, a uniform price across items is optimal. We then extend this optimality result to all valuations drawn from exchangeable and affiliated distributions. However, uniform pricing is not optimal in all symmetric environments: for min-buying customers we construct an exchangeable and non-affiliated valuation distribution for which the gap between uniform pricing and optimal item pricing is logarithmic in the number of items. This gap is tight and matches the worst-case gap in the quasi-linear setting. Next, we consider asymmetric environments in which valuations are independent but not identically distributed. When these valuations are regular and customers are min-buying, we show that a uniform price guarantees approximately 69.86\% of the optimal item-pricing revenue and this is tight. Overall, our results highlight that the value of non-uniform pricing substantially diminishes, and often disappears entirely, under more conservative demand models in which customers engage in satisficing behaviors such as min-buying.

cs.GT↗

arXiv2022

A New Combinatorial Property of Geometric Unique Sink Orientations

A unique sink orientation (USO) is an orientation of the hypercube graph with the property that every face has a unique sink. A number of well-studied problems reduce in strongly polynomial time to finding the global sink of a USO; most notably, linear programming (LP) and the P-matrix linear complementarity problem (P-LCP). The former is not known to have a strongly polynomial-time algorithm, while the latter is not known to even have a polynomial-time algorithm, motivating the problem to find the global sink of a USO. Although, every known class of geometric USOs, arising from a concrete problem such as LP, is exponentially small, relative to the class of all USOs. Accordingly, geometric USOs exhibit additional properties that set them apart from general USOs, and it may be advantageous, if not necessary, to leverage these properties to find the global sink of a USO faster. Only a few such properties are known. In this paper, we establish a new combinatorial property of the USOs that arise from symmetric P-LCP, which includes the USOs that arise from linear and simple convex quadratic programming.

math.CO↗

arXiv2020

An Oblivious Ellipsoid Algorithm for Solving a System of (In)Feasible Linear Inequalities

The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of $m$ linear inequalities in $n$ variables $(P): A^{\top}x \le u$ when its set of solutions has positive volume. However, when $(P)$ is infeasible, the ellipsoid algorithm has no mechanism for proving that $(P)$ is infeasible. This is in contrast to the other two fundamental algorithms for tackling $(P)$, namely the simplex method and interior-point methods, each of which can be easily implemented in a way that either produces a solution of $(P)$ or proves that $(P)$ is infeasible by producing a solution to the alternative system $\mathrm{({\it Alt})}: Aλ= 0$, $u^{\top}λ< 0$, $λ\ge 0$. This paper develops an Oblivious Ellipsoid Algorithm (OEA) that either produces a solution of $(P)$ or produces a solution of $\mathrm{({\it Alt})}$. Depending on the dimensions and on other natural condition measures, the computational complexity of the basic OEA may be worse than, the same as, or better than that of the standard ellipsoid algorithm. We also present two modified versions of OEA, whose computational complexity is superior to that of OEA when $n \ll m$. This is achieved in the first modified version by proving infeasibility without actually producing a solution of $\mathrm{({\it Alt})}$, and in the second modified version by using more memory.

math.OC↗

arXiv2019

Certifiably Optimal Sparse Inverse Covariance Estimation

We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity. We give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality. The approach uses techniques from mixed-integer optimization and convex optimization, and provides a high-quality solution with a guarantee on its suboptimality, even if the algorithm is terminated early. Using a variety of synthetic and real datasets, we demonstrate that our approach can solve problems where the dimension of the inverse covariance matrix is up to 1,000s. We also demonstrate that our approach produces significantly sparser solutions than Glasso and other popular learning procedures, makes less false discoveries, while still maintaining state-of-the-art accuracy.

stat.ML↗