Affordable mixed-integer Lagrangian methods: optimality conditions and convergence analysis

TL;DR

This paper extends optimality conditions and convergence analysis for mixed-integer nonlinear optimization, providing a foundation for affordable algorithms with guarantees to critical points.

Contribution

It introduces new optimality conditions and a convergence framework for mixed-integer nonlinear problems without convexity assumptions.

Findings

Characterization of local minimizers and critical points.

Development of sequential minimization algorithms with convergence guarantees.

Preliminary numerical results for augmented Lagrangian and interior point methods.

Abstract

Necessary optimality conditions in Lagrangian form and the sequential minimization framework are extended to mixed-integer nonlinear optimization, without any convexity assumptions. Building upon a recently developed notion of local optimality for problems with polyhedral and integrality constraints, a characterization of local minimizers and critical points is given for problems including also nonlinear constraints. This approach lays the foundations for developing affordable sequential minimization algorithms with convergence guarantees to critical points from arbitrary initializations. A primal-dual perspective, a local saddle point property, and the dual relationships with the proximal point algorithm are also advanced in the presence of integer variables. Preliminary numerical results are presented for an augmented Lagrangian and an interior point method.