Generalized Scaling for the Constrained Maximum-Entropy Sampling Problem

TL;DR

This paper introduces a generalized scaling technique for the constrained maximum-entropy sampling problem, enhancing bounds in branch-and-bound algorithms through a flexible positive vector of parameters, supported by theoretical and computational results.

Contribution

It extends standard scaling to generalized scaling with multiple parameters, improving bound tightness in optimization for experimental design.

Findings

Generalized scaling reduces bounds gaps more effectively than standard scaling.

Mathematical results support algorithms for computing optimal generalized scalings.

Computational experiments show improved performance on benchmark instances.

Abstract

The best practical techniques for exact solution of instances of the constrained maximum-entropy sampling problem, a discrete-optimization problem arising in the design of experiments, are via a branch-and-bound framework, working with a variety of concave continuous relaxations of the objective function. A standard and computationally-important bound-enhancement technique in this context is (ordinary) scaling, via a single positive parameter. Scaling adjusts the shape of continuous relaxations to reduce the gaps between the upper bounds and the optimal value. We extend this technique to generalized scaling, employing a positive vector of parameters, which allows much more flexibility and thus potentially reduces the gaps further. We give mathematical results aimed at supporting algorithmic methods for computing optimal generalized scalings, and we give computational results demonstrating the performance of generalized scaling on benchmark problem instances.