Interval constraint programming for globally solving catalog-based categorical optimization

TL;DR

The paper tackles catalog-based categorical optimization by recasting the problem into a continuous interval-constraint framework. It introduces Clutch, a catalog-based contractor that ensures local catalog-consistency by computing the convex hull of catalog items inside a property box, and integrates this with an arborescent branch-and-contract solver to achieve global optimality with numerical robustness. The approach avoids extra modeling effort, requires only continuous relaxations of catalog properties, and can be implemented on top of off-the-shelf interval solvers, demonstrated on a toy problem and two scenarios with a Julia prototype. Compared to binary-encoded MINLP formulations, the method delivers competitive or superior performance while maintaining rigor against roundoff errors and providing transparent, provable bounds on the optimum.

Abstract

In this article, we propose an interval constraint programming method for globally solving catalog-based categorical optimization problems. It supports catalogs of arbitrary size and properties of arbitrary dimension, and does not require any modeling effort from the user. A novel catalog-based contractor (or filtering operator) guarantees consistency between the categorical properties and the existing catalog items. This results in an intuitive and generic approach that is exact, rigorous (robust to roundoff errors) and can be easily implemented in an off-the-shelf interval-based continuous solver that interleaves branching and constraint propagation. We demonstrate the validity of the approach on a numerical problem in which a categorical variable is described by a two-dimensional property space. A Julia prototype is available as open-source software under the MIT license at https://github.com/cvanaret/CateGOrical.jl

Figures (6)

• Figure 1: Chromatic aberration in a refractive optical system.
• Figure 2: Common optical glasses in the space of Cauchy coefficients.
• Figure 3: FBBT/HC4Revise on the constraint $2x = z - y^2$.
• Figure 4: Property space of the toy problem.
• Figure 5: Scenario 1: alternation of filtering and branching phases.