Robust Data-Driven Quasiconcave Optimization

TL;DR

This paper develops an efficient binary search-based method for robust data-driven quasiconcave optimization, leveraging the structure of the problem to achieve finite convergence and practical applicability.

Contribution

It introduces a novel binary search approach that exploits quasiconcavity to solve robust optimization problems more efficiently than traditional methods.

Findings

Binary search method converges finitely to the global optimum.

The approach is computationally efficient, solving a logarithmic number of convex problems.

Numerical experiments demonstrate practical effectiveness on real-world problems.

Abstract

We investigate a data-driven quasiconcave maximization problem where information about the objective function is limited to a finite sample of data points. We begin by defining an ambiguity set for admissible objective functions based on available partial information about the objective. This ambiguity set consists of those quasiconcave functions that majorize a given data sample, and that satisfy additional functional properties (monotonicity, Lipschitz continuity, and permutation invariance). We then formulate a robust optimization (RO) problem which maximizes the worst-case objective function over this ambiguity set. Based on the quasiconcave structure in this problem, we explicitly construct the upper level sets of the worst-case objective at all levels. We can then solve the resulting RO problem efficiently by doing binary search over the upper level sets and solving a logarithmic number of convex feasibility problems. This numerical approach differs from traditional subgradient descent and support function based methods for this problem class. While these methods can be applied in our setting, the binary search method displays superb finite convergence to the global optimum, whereas the others do not. This is primarily because binary search fully exploits the specific structure of the worst-case quasiconcave objective, which leads to an explicit and general convergence rate in terms of the number of convex optimization problems to be solved. Our numerical experiments on a Cobb-Douglas production efficiency problem and a fair resource allocation problem demonstrate the tractability of our approach.