A condensing approach for linear-quadratic optimization with geometric constraints

TL;DR

This paper introduces a condensing method for linear-quadratic optimization problems with complex, possibly nonconvex constraints, enhancing computational efficiency while maintaining convergence guarantees.

Contribution

It develops a novel condensing approach integrated with augmented Lagrangian methods to handle nonconvex constraints efficiently in quadratic optimization.

Findings

Significant computational performance improvements achieved.

Method effectively handles nonconvex and complex constraints.

Convergence guarantees are maintained with the new approach.

Abstract

Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and cardinality constraints, among others. In particular, we cover also situations where parts of the constraints are nonconvex and possibly complicated, but it is practical to compute projections onto this nonconvex set. Our approach combines the augmented Lagrangian framework with a solver-agnostic structure-exploiting subproblem reformulation. While convergence guarantees follow from the former, the proposed condensing technique leads to significant improvements in computational performance.